Research ArticleThe Solution of Two-Phase Inverse Stefan Problem Based ona Hybrid Method with Optimization

Yang Yu Xiaochuan Luo and Haijuan Cui

State Key Laboratory of Synthetical Automation for Process Industries Northeastern University ShenyangLiaoning 110819 China

Correspondence should be addressed to Xiaochuan Luo luoxchmailneueducn

Received 20 March 2015 Revised 30 April 2015 Accepted 4 May 2015

Academic Editor Evangelos J Sapountzakis

Copyright copy 2015 Yang Yu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The two-phase Stefan problem is widely used in industrial fieldThis paper focuses on solving the two-phase inverse Stefan problemwhen the interface moving is unknown which is more realistic from the practical point of view With the help of optimizationmethod the paper presents a hybrid method which combines the homotopy perturbation method with the improved Adomiandecomposition method to solve this problem Simulation experiment demonstrates the validity of this method Optimizationmethod plays a very important role in this paper so we propose a modified spectral DY conjugate gradient method And theconvergence of this method is given Simulation experiment illustrates the effectiveness of this modified spectral DY conjugategradient method

1 Introduction

The process of solidificationmelting occurs in many indus-trial fields such as steel-making continuous casting In thecontinuous casting process [1 2] molten steel comes intoslab by solidification And this process is shown in Figure 1Because of the heat symmetry we only consider 12 of slabthickness and we divide this domain into two regions bythe freezing front which is shown in Figure 2 119863

1stands

for the solid phase and 1198632is taken by the liquid phase

The solidification problem of continuous casting is associatedwith Stefan problem which is a model of the process with aphase change

Many researches have focused on solving Stefan problems[3ndash7] Grzymkowski and Slota [8 9] applied the Adomiandecomposition method (ADM) combined with optimizationfor solving the direct and the inverse one-phase Stefan prob-lem and Hetmaniok et al [10] used the Adomian decom-position method (ADM) and variational iteration method tosolve the one-phase Stefan moving boundary problems Incomparison to the studies on the one-phase Stefan problemsresearches on two-phase Stefan problems [11] are much more

scarce Słota [12] applied homotopy perturbation method(HPM) to solve the two-phase inverse Stefan problem How-ever in their formulation these two-phase inverse Stefanproblems were solved when the position of the movinginterface is knownWhile the position of themoving interfaceis unknown and no temperature or heat flux boundaryconditions are specified onone part of the boundary this two-phase inverse Stefan problem may be difficult to be solvedThis problem is close to the problem which occurs in thecontinuous casting Therefore this paper focuses on solvingthis problem by a hybrid method with optimization whichcombines the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM)

This paper is organized as follows In Section 2 we givethe description of the two-phase Stefan problem And thehybrid method for solving this problem is introduced inSection 3 We give a modified spectral DY conjugate gradientmethod and the convergence of this method in Section 4 InSection 5 some numerical results are given to illustrate thevalidity of the hybrid method

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 319054 13 pageshttpdxdoiorg1011552015319054

2 Mathematical Problems in Engineering

x

z

Secondarycooling

zone

Mold

Figure 1 The solidification process of continuous casting slab (119909stands for the direction of heat transfer and 119911 stands for castingdirection)

2 The Description of Two-Phase InverseStefan Problem

In this section the two-phase inverse Stefan problem isconsidered The two domains are described (see Figure 2)

1198631 = (119909 119905) 119909 isin [0 120585 (119905)] 119905 isin [0 119905lowast]

1198632 = (119909 119905) 119909 isin [120585 (119905) 119889] 119905 isin [0 119905lowast]

(1)

and their boundaries with unknown function 119909 = 120585(119905) are

Γ1 = (119909 0) 119909 isin (0 119904) 119904 = 120585 (0) (2)

Γ2 = (119909 0) 119909 isin (119904 119889) 119904 = 120585 (0) (3)

Γ3 = (0 119905) 119905 isin [0 119905lowast] (4)

Γ4 = (119889 119905) 119905 isin [0 119905lowast] (5)

Γ5 = (119909 119905) 119905 isin [0 119905lowast] 119909 = 120585 (119905) (6)

Γ6 = (119909 119905lowast) 119909 isin [0 120585 (119905lowast)] (7)

In practice the measurement of temperature on theboundary Γ4 may not be easily obtained and the position ofthe moving interface described by means of function 119909 =

120585(119905) is unknown Therefore the designed two-phase inverseStefan problem consists in the calculation of the temperaturedistribution in the domains 1198631 and 1198632 respectively as wellas in the reconstruction of the functions describing movinginterface 119909 = 120585(119905) In order to solve this inverse problem theadditional information is given In such a situation the finaltime internal temperature at 119905 = 119905

lowast on the boundary Γ6 isknown which can be measured in practice The temperaturedistribution is determined by the functions 1199061(119909 119905) and1199062(119909 119905) which are defined in the domains 1198631 and 1198632

Γ1 s Γ2 d

Γ3

Γ5

0

D2

x

Γ4

D1

120585(t) =

tlowastΓ6

t

120585(tlowast)

Figure 2Thedomain formulation of the two-phase Stefan problem

respectively and they satisfy the following heat conductionequations

1205971199061 (119909 119905)

120597119905= 1205721

12059721199061 (119909 119905)

1205971199092in 1198631 (8)

1205971199062 (119909 119905)

120597119905= 1205722

12059721199062 (119909 119905)

1205971199092in 1198632 (9)

where 119905 and 119909 refer to time and spatial location and 1205721 and1205722 are the thermal diffusivity in the solid phase and the liquidphase respectively On the boundaries Γ1 and Γ2 the initialconditions are described in the following

1199061 (119909 0) = 1205931 (119909) on Γ1 (10)

1199062 (119909 0) = 1205932 (119909) on Γ2 (11)

120585 (0) = 119904 (12)

On the boundary Γ3 it satisfies the Dirichlet boundary orNeumann boundary condition

1199061 (0 119905) = 1205791 (119905) on Γ3 (13)

On the boundary Γ4 it satisfies the Dirichlet boundaryconditions

1199062 (119889 119905) = 120579 (119905) on Γ4 (14)

On the phase change moving interface (Γ5) it satisfies thecondition of temperature continuity and the Stefan condition

1199061 (120585 (119905) 119905) = 1199062 (120585 (119905) 119905) = 119906lowast (15)

where 119906lowast is the phase change temperature Hence

120581119889120585 (119905)

119905= 1198962

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

minus 11989611205971199061 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(16)

where 120581 is the latent heat of fusion per unit volume and 1198961and 1198962 are the thermal conductivity in the solid and liquidphase respectivelyThe final time internal temperature on theboundary Γ6 satisfies

1199061 (119909 119905lowast) = 119906119879

1 (119909) on Γ6 (17)

Mathematical Problems in Engineering 3

3 The Solution of Two-Phase InverseStefan Problem

31 The Homotopy Perturbation Method The homotopy per-turbation method was presented in [13 14] it arose as acombination of two other methods the perturbation methodand the homotopy technique from topology Homotopyperturbation method was applied on differential equationspartial differential equations and the direct and inverseproblems of heat transfer by many authors [15ndash17]

Homotopy perturbation method was used to find theexact or the approximate solutions of the linear and nonlinearintegral equations [16 18] the two-dimensional integralequations [19ndash21] and the integrodifferential equations [22ndash25] This method enables seeking a solution of the followingoperator equation

119861 (119908) minus 120577 (119911) = 0 119911 isin Ω (18)

where 119861 is an operator 119908 is a sought function and 120577 isa known function on the domain of Ω The operator 119861 ispresented in the form of sum

119861 (V) = 119877 (119908) +119876 (119908) (19)

where 119877 defines the linear operator and 119876 is the remainingpart of operator 119861 So (18) can be written as

119877 (119908) +119876 (119908) minus 120577 (119911) = 0 119911 isin Ω (20)

We define a new operator 119867 which is called the homotopyoperator in the following

119867(V 119901) = (1minus119901) (119877 (V) minus 119877 (1199080)) + 119901 (119861 (V) minus 120577 (119911))

= 0(21)

where 119901 isin [0 1] is an embedding parameter V(119911 119901) Ω times

[0 1] rarr R and 1199080 defines the initial approximation of asolution of (18) (21) is transformed into the following form

119867(V 119901) = 119877 (V) minus 119877 (1199080) + 119901119877 (1199080)

+ 119901 (119876 (V) minus 120577 (119911)) = 0(22)

Therefore

119867(V 0) = 119877 (V) minus 119877 (1199080)

119867 (V 1) = 119877 (1199080) +119876 (V) minus 120577 (119911) (23)

For 119901 = 0 the solution of operator equation 119867(V 0) = 0 isequivalent to solution of problem 119877(V) minus 119877(1199080) = 0 whereasfor 119901 = 1 the solution of operator equation 119867(V 1) = 0 isequivalent to solution of problem (20) Thus a monotonouschange of parameter119901 from zero to one is just that continuouschange of the trivial problem 119877(V) minus119877(1199080) = 0 to the originalproblem In topology this is called homotopy According tothe HPM we assume that the solution of (21) and (22) can bewritten as a power series in 119901

V =infin

sum

119894=0119901119894119908119894= 1199080 +1199011199081 +119901

21199082 + sdot sdot sdot (24)

setting 119901 = 1 the approximate solution of (20) is obtained

119908 = lim119901rarr 1

V =infin

sum

119894=0119908119894= 1199080 +1199081 +1199082 + sdot sdot sdot (25)

in most cases the series in (25) is convergent and in [15] anadditional remark on the convergence of this series can befound

32 The Improved Adomian Decomposition Method TheAdomian decompositionmethod was presented by Adomian[26] This method was used to solve a wide variety oflinear and nonlinear problems Several other researchers haddeveloped a modification to the ADM [27]Themodificationusually involves a slight change and it is aimed at improvingthe accuracy of the series solution

The Adomian decomposition method is able to solve awide class of nonlinear operator equations [28] in the form

119865 (119906) = 120578 (26)

where 119865 119867 rarr 119866 is a nonlinear operator 120578 is a knownelement from Hilbert space 119866 and 119906 is the sought elementfrom Hilbert space119867 The operator 119865(119906) can be written as

119865 (119906) = 119871 (119906) +119877 (119906) +119873 (119906) (27)

where 119871 is an invertible linear operator 119877 is a linear operatorand119873 is a nonlinear operatorThe solution of (26) admits thedecomposition into an infinite series of components

119906 =

infin

sum

119894=0119892119894 (28)

the nonlinear term119873 is equated to an infinite series

119873(119906) =

infin

sum

119894=0119860119894 (29)

119860119899is the Adomian polynomials which can be determined by

1198600 = 119873 (1198920) (30)

119860119899=

1119899[119889119899

119889120582119899119873(

119899

sum

119894=0120582119894119892119894)]

120582=0

119899 ge 1 (31)

Substituting (28) (29) (30) and (31) into operator equation(26) and using the inverse operator 119871

minus1 the followingequation can be obtained

1198920 = 119892lowast+119871minus1(120578)

119892119899= minus119871minus1119877 (119892119899minus1) minus 119871

minus1(119860119899minus1) 119899 ge 1

(32)

where 119892lowast is the function dependent on the initial and

boundary conditionsThe improved Adomian decomposition method [27] it

is important to note that the improved Adomian decompo-sition method is based on the assumption that the function

4 Mathematical Problems in Engineering

119910 = 119892lowast+ 119871minus1(120578) can be divided into two parts namely 1199101

and 1199102 Under this assumption the following equation canbe obtained

119910 = 1199101 +1199102 (33)

Accordingly a slight variation is proposed on the components1198920 and 1198921 The suggestion is that only the part 1199101 is assignedto the component 1198920 whereas the remaining part 1199102 iscombined with other terms given in (33) to define 1198921Consequently the recursive relation is written as follows

1198920 = 1199101

1198921 = 1199102 minus119871minus1119877 (1198920) minus 119871

minus1(1198600)

119892119899+2 = minus119871

minus1119877 (119892119899+1) minus 119871

minus1(119860119899+1) 119899 ge 0

(34)

33 The Solution of the Two-Phase Inverse Stefan ProblemIn this section we introduce the solution of the two-phaseinverse Stefan problemWe split the two-phase inverse Stefanproblem into two problems The first is the determination of1199061(119909 119905) in domain1198631 and the moving interface 120585(119905) which issatisfying (7) (8) (10) and (13) Once the boundary 119909 = 120585(119905)and the heat flux (1205971199062(119909 119905)120597119909)|119909=120585(119905) have been obtained thesecond problem for determining the temperature 1199062(119909 119905) indomain1198632 is solved

The solution of 1199061(119909 119905) and moving interface 120585(119905) basedon HPM because the heat flux on the moving position isunknown in this two-phase inverse Stefan problem theHPMis used to compute 1199061(119909 119905) In this section the homotopyperturbation method is applied to solve the inverse problemWe make the homotopy map for (22)

1198671 (V1 119901) =1205972V11205971199092

minus120597211990610

1205971199092+119901(

120597211990610

1205971199092+

11205721

120597V119894

120597119905) (35)

the solution to this equation

1198671 (V1 119901) = 0 (36)

is sought in the form of a power series in 119901 and it satisfies

V1 =infin

sum

119894=01199011198941199061119894 (37)

where 119901 isin [0 1] by substituting (37) into (36) the followingequation can be obtained

infin

sum

119894=011990111989412059721199061119894

1205971199092=120597211990610

1205971199092minus119901

120597211990610

1205971199092+

11205721

infin

sum

119894=1

1205971199061119894minus1

120597119905 (38)

Next by comparing the same powers of parameter 119901 in (38)the following equation can be obtained

120597211990611

1205971199092=

11205721

12059711990610

120597119905minus120597211990610

1205971199092 (39)

12059721199061119894

1205971199092=

11205721

1205971199061119894minus1

120597119905for 119894 ge 2 (40)

Equations (39) and (40) must be supplemented by theboundary conditionsTherefore for (39) we set the followingboundary conditions

11990610 (0 119905) + 11990611 (0 119905) = 1205791 (119905) (41)

11990610 (120585 (119905) 119905) + 11990611 (120585 (119905) 119905) = 119906lowast (42)

and for (40) we set the conditions in the following (119894 ge 2)

1199061119894 (0 119905) = 0 (43)

1199061119894 (120585 (119905) 119905) = 0 (44)

The above conditions are selected So we give the 119899-orderapproximate solution

1119899 =119899

sum

119894=11199061119894 (45)

because 1199061119894 are determined by the unknown function 120585(119905)This function is derived in the form

120585 (119905) =

119898

sum

119894=1119888119894120595119894(119905) (46)

Thus we define a cost function 1198691 according to (10) and (17)

1198691 (1198881 119888119898) = int120585(0)

01199061119899 (119909 0) minus 1205931 (119909)

2119889119909

+int

120585(119905lowast

)

01199061119899 (119909 119905

lowast) minus 119906119879

1 (119909)2119889119909

(47)

Therefore 119888119894are chosen according to the minimum of (47)

The moving interface 120585(119905) and the solution of 1199061 in domain1198631 can be obtained

The boundary condition (13) on the boundary Γ3 and thecondition of temperature continuity (15) should be fulfilledfor 119899 ge 1 In this way while looking for the solution of thisproblem in domain 1198631 the initial approximation 11990611 will befound by solving the system of (39) with conditions (41) and(42) Conversely functions1199061119894 119894 = 1 2 are determined bythe recurrent solution of (40) with conditions (42) and (44)Consequently themoving interface 120585(119905) and the solution of1199061in domain1198631 can be obtained by the homotopy perturbationmethodwith optimization According to the Stefan condition(16) on the phase changemoving interface Γ5 the temperature1199062(119909 119905) in domain 1198632 can be recovered by IADM withoptimization method

The solution of 1199062(119909 119905) based on IADM with optimiza-tion in this problemwe consider the operator equations (27)where

119871 (1199062) =12059721199062

1205971199092

119877 (1199062) = minus11205722

1205971199062120597119905

119873 (1199062) = 0

120578 = 0

(48)

Mathematical Problems in Engineering 5

The inverse operator 119871minus1 is given by the following equation

119871minus1(1199062) = int

119909

119889

int

119909

120585(119905)

1199062 (119909 119905) 119889119909 119889119909 (49)

The boundary condition and the Stefan condition from (14)are used to obtain the following equation

119871minus1119871(

12059721199062

1205971199092) = 1199062 (119909 119905) minus 120579 (119905)

minus1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

(50)

Hence

119892lowast= 120579 (119905) +

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889) (51)

then the following recursive relation can be obtained

1198920 = 1199062 (119889 119905) = 120579 (119905)

1198921 =1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

+11205722

int

119909

119889

int

119909

120585(119905)

1205971198920120597119905

119889119909 119889119909

119892119899+2 =

11205722

int

119909

119889

int

119909

120585(119905)

120597119892119899minus1120597119905

119889119909 119889119909 119899 ge 0

(52)

because 1199061(119909 119905) and 120585(119905) are determined by the aboveHPM with optimization method (1205971199062(119909 119905)120597119909)|119909=120585(119905) can beobtained by (16) An approximation solution is expressed inthe form

1199062119899 =119899

sum

119894=0119892119894

119899 isin 119873 (53)

and in this inverse Stefan problem 120579(119905) is unknown on theboundary Γ4 and it is derived in the following form

120579 (119905) =

119898

sum

119894=1119902119894120601119894(119905) (54)

where 119902119894isin 119877 and 120601

119894(119905) are the basis functionThe coefficients

119902119894are selected to show minimal deviation of function (55)

from conditions (11) and (15) Thus 119902119894are chosen according

to the minimum of the following functional

1198692 (1199021 119902119898) = int119905lowast

01199062119899 (120585 (119905) 119905) minus 119906

lowast2119889119905

+int

119889

120585(0)1199062119899 (119909 0) minus 1205932 (119909)

2119889119909

(55)

where 1198692(1199021 119902119898) is a nonlinear function The optimiza-tion method plays a very important role in solving this two-phase inverse Stefan problem Therefore a modified spectral

DY conjugate gradientmethod is presented In Section 4 thisoptimization method is introduced

RemarkThe IADMwith optimizationmethod can be used tosolve the direct and inverse one-phase Stefan problem whichis designed in the literature [8 9] And this method is betterthan ADM with optimization method In Section 5 we givetwo examples of one-phase Stefan problem to illustrate theeffectiveness of the IADM with optimization method

4 A Modified Spectral DY ConjugateGradient Method

Conjugate gradient methods are very efficient for solvingthe unconstrained optimization problemmin

119909isin119877119899

119891(119909) Theiriterative formula is given as follows

119909119896+1 = 119909119896 +120572119896119889119896

119889119896=

minus119892119896 if 119896 = 1

minus119892119896+ 120573119896119889119896minus1

if 119896 ge 1

(56)

where step-size 120572119896is positive 119892

119896= nabla119891(119909

119896) and 120573

119896is a scalar

In addition 120572119896is a step length which is calculated by the line

search 120573119896is a very important factor in conjugate gradient

methods the FR PRP DY and CD methods [29] are usedfor choosing this scalar Among them the DY method isregarded as one of the efficient conjugate gradient methodsThe scalar 120573

119896is given by

120573DY119896

=

10038171003817100381710038171198921198961003817100381710038171003817

119889119879

119896minus1119910119896minus1 (57)

where 119910119896minus1 = 119892

119896minus 119892119896minus1 and sdot stands for Euclidean

norm Recently Birgin andMartłnez [30] presented a spectralconjugate gradient method which is combining conjugategradient method and spectral gradient method And thedirection 119889

119896is obtained by

119889119896= minus 120579119896119892119896+120573119896119889119896minus1

120573119896=(120579119896119910119896minus1 minus 119904119896minus1)

119879

119892119896

119889119879

119896minus1119910119896minus1

(58)

where 120579119896= 119904119879

119896minus1119904119896minus1119904119879

119896minus1119910119896minus1 is a parameter with 119904119896minus1 =

119909119896minus 119909119896minus1 and 120579119896 is regarded as spectral gradient However

according to [31] the spectral conjugate gradient method cannot guarantee producing the descent directions

Based on the above analysis in the following we describea modified spectral DY conjugate gradient method and givethe convergence of this method with the Wolfe type linesearch rules

41 The Modified Spectral DY Conjugate Gradient MethodThe new method is introduced

119889119896=

minus119892119896 if 119896 = 1

minus120579119896119892119896+ 120573119896119889119896minus1 if 119896 ge 1

(59)

6 Mathematical Problems in Engineering

where 120579119896= 1 + 119892

119896minus121198921198962 and 120573

119896= 120573

DY119896

This method iscalled the MDY method

Algorithm 1 Consider the following

Step 1 The following are given 1199090 isin 119877119899 120576 gt 0 and the max

iteration step119873max

Step 2 Set 119896 = 1 compute 1198891 = minus1198921 = minusnabla119891(1199090) if 1198921 le 120576then stop

Step 3 Find 120572119896according to Wolfe type line search rule

Step 4 Let 119909119896+1 = 119909119896+120572119896119889119896 and 119892119896+1 = 119892(119909119896+1) if 119892119896+1 le 120576

then stop

Step 5 Compute 119889119896+1 by (59)

Step 6 Set 119896 = 119896 + 1 go to Step 3

42 Convergence Analysis From the above the MDY con-jugate gradient method provides a descent direction for theobjective function In the following we give the convergenceof this algorithmwith theWolfe type line search rules Firstlywe introduce some assumptions

(i) The level setΩ = 119909 isin 119877119899| 119891(119909) le 119891(1199091) is bounded

(ii) In some neighborhood 119873 of Ω 119891(119909) is continuouslydifferentiable and its gradient is Lipschitz continuousnamely there exists a constant 119871 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus119910

1003817100381710038171003817 forall119909 119910 isin 119873 (60)

It is implied from the above assumptions that there exist twopositive constants 120573 and 120574 such that

119909 le 119861

1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120574

forall119909 isin Ω

(61)

TheWolfe type line search [29] is

119891 (119909119896) minus119891 (119909

119896+120572119896119889119896) ge 120575120572

2119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(62)

119892 (119909119896+120572119896119889119896)119879

119889119896ge minus 2120575120572

119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(63)

and the strong Wolfe line search is

119891 (119909119896+120572119896119889119896) le 119891 (119909

119896) + 120575120572

119896119892119879

119896119889119896 (64)

100381610038161003816100381610038161003816119892 (119909119896+120572119896119889119896)119879

119889119896

100381610038161003816100381610038161003816le10038161003816100381610038161003816120590119892119879

119896119889119896

10038161003816100381610038161003816 (65)

where 0 lt 120575 lt 120590 lt 1

Lemma 2 Let the sequences 119892119896 and 119889

119896 be generated by the

proposed algorithm (Algorithm 1) with Wolfe type line searchrules then 119892119879

119896119889119896lt 0 and

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (66)

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= 120573DY119896

[minus120579119896119889119896minus1 (119892

119879

119896minus119892119879

119896minus1) + 119892119879

119896119889119896minus1]

= 120573DY119896

[119892119879

119896minus1119889119896minus1 minus

1003817100381710038171003817119892119896minus11003817100381710038171003817

2

119889119896minus1119910119879

119896minus11003817100381710038171003817119892119896

1003817100381710038171003817

2]

le 120573DY119896119892119879

119896minus1119889119896minus1

(67)

If we choose 119896 = 1 then

119892119879

1 1198891 = minus10038171003817100381710038171198921

1003817100381710038171003817

2

lt 0 (68)

according to (65) we have

119910119879

1 1198891 ge (120590 minus 1) 119892119879

1 1198891 ge 0 (69)

So 120573DY2 ge 0 For 119896 gt 1 according to (67) we have

119892119879

119896119889119896lt 0 (70)

by induction So we obtain

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (71)

the proof is completed

Theorem 3 Let the sequences 119892119896 and 119889

119896 be generated

by the proposed algorithm (Algorithm 1) with Wolfe type linesearch rules then

119892119879

119896119889119896le minus119862 119892

1198962 (72)

where 119862 gt 0 is a constant

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= minus1003817100381710038171003817119892119896minus1

1003817100381710038171003817

2

+(119892119879

119896119889119896minus1

119889119879

119896minus1119910119896minus1minus 1) 1003817100381710038171003817119892119896

1003817100381710038171003817

2

le (119892119879

119896minus1119889119896minus1

119889119879

119896minus1119910119896minus1)1003817100381710038171003817119892119896

1003817100381710038171003817

2

(73)

From the Wolfe type line search rules (65) we can obtain

minus (120590 + 1) 119892119879119896minus1119889119896minus1 ge 119889

119879

119896minus1 (119892119896 minus119892119896minus1) gt 0 (74)

according to (73) and (74) we have

119892119879

119896119889119896le minus

1120590 + 1

10038171003817100381710038171198921198961003817100381710038171003817

2

(75)

we define 119862 = 1(120590 + 1) Therefore we can obtain inequality(72)

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

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2 Mathematical Problems in Engineering

x

z

Secondarycooling

zone

Mold

Figure 1 The solidification process of continuous casting slab (119909stands for the direction of heat transfer and 119911 stands for castingdirection)

2 The Description of Two-Phase InverseStefan Problem

In this section the two-phase inverse Stefan problem isconsidered The two domains are described (see Figure 2)

1198631 = (119909 119905) 119909 isin [0 120585 (119905)] 119905 isin [0 119905lowast]

1198632 = (119909 119905) 119909 isin [120585 (119905) 119889] 119905 isin [0 119905lowast]

(1)

and their boundaries with unknown function 119909 = 120585(119905) are

Γ1 = (119909 0) 119909 isin (0 119904) 119904 = 120585 (0) (2)

Γ2 = (119909 0) 119909 isin (119904 119889) 119904 = 120585 (0) (3)

Γ3 = (0 119905) 119905 isin [0 119905lowast] (4)

Γ4 = (119889 119905) 119905 isin [0 119905lowast] (5)

Γ5 = (119909 119905) 119905 isin [0 119905lowast] 119909 = 120585 (119905) (6)

Γ6 = (119909 119905lowast) 119909 isin [0 120585 (119905lowast)] (7)

In practice the measurement of temperature on theboundary Γ4 may not be easily obtained and the position ofthe moving interface described by means of function 119909 =

120585(119905) is unknown Therefore the designed two-phase inverseStefan problem consists in the calculation of the temperaturedistribution in the domains 1198631 and 1198632 respectively as wellas in the reconstruction of the functions describing movinginterface 119909 = 120585(119905) In order to solve this inverse problem theadditional information is given In such a situation the finaltime internal temperature at 119905 = 119905

lowast on the boundary Γ6 isknown which can be measured in practice The temperaturedistribution is determined by the functions 1199061(119909 119905) and1199062(119909 119905) which are defined in the domains 1198631 and 1198632

Γ1 s Γ2 d

Γ3

Γ5

0

D2

x

Γ4

D1

120585(t) =

tlowastΓ6

t

120585(tlowast)

Figure 2Thedomain formulation of the two-phase Stefan problem

respectively and they satisfy the following heat conductionequations

1205971199061 (119909 119905)

120597119905= 1205721

12059721199061 (119909 119905)

1205971199092in 1198631 (8)

1205971199062 (119909 119905)

120597119905= 1205722

12059721199062 (119909 119905)

1205971199092in 1198632 (9)

where 119905 and 119909 refer to time and spatial location and 1205721 and1205722 are the thermal diffusivity in the solid phase and the liquidphase respectively On the boundaries Γ1 and Γ2 the initialconditions are described in the following

1199061 (119909 0) = 1205931 (119909) on Γ1 (10)

1199062 (119909 0) = 1205932 (119909) on Γ2 (11)

120585 (0) = 119904 (12)

On the boundary Γ3 it satisfies the Dirichlet boundary orNeumann boundary condition

1199061 (0 119905) = 1205791 (119905) on Γ3 (13)

On the boundary Γ4 it satisfies the Dirichlet boundaryconditions

1199062 (119889 119905) = 120579 (119905) on Γ4 (14)

On the phase change moving interface (Γ5) it satisfies thecondition of temperature continuity and the Stefan condition

1199061 (120585 (119905) 119905) = 1199062 (120585 (119905) 119905) = 119906lowast (15)

where 119906lowast is the phase change temperature Hence

120581119889120585 (119905)

119905= 1198962

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

minus 11989611205971199061 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(16)

where 120581 is the latent heat of fusion per unit volume and 1198961and 1198962 are the thermal conductivity in the solid and liquidphase respectivelyThe final time internal temperature on theboundary Γ6 satisfies

1199061 (119909 119905lowast) = 119906119879

1 (119909) on Γ6 (17)

Mathematical Problems in Engineering 3

3 The Solution of Two-Phase InverseStefan Problem

31 The Homotopy Perturbation Method The homotopy per-turbation method was presented in [13 14] it arose as acombination of two other methods the perturbation methodand the homotopy technique from topology Homotopyperturbation method was applied on differential equationspartial differential equations and the direct and inverseproblems of heat transfer by many authors [15ndash17]

Homotopy perturbation method was used to find theexact or the approximate solutions of the linear and nonlinearintegral equations [16 18] the two-dimensional integralequations [19ndash21] and the integrodifferential equations [22ndash25] This method enables seeking a solution of the followingoperator equation

119861 (119908) minus 120577 (119911) = 0 119911 isin Ω (18)

where 119861 is an operator 119908 is a sought function and 120577 isa known function on the domain of Ω The operator 119861 ispresented in the form of sum

119861 (V) = 119877 (119908) +119876 (119908) (19)

where 119877 defines the linear operator and 119876 is the remainingpart of operator 119861 So (18) can be written as

119877 (119908) +119876 (119908) minus 120577 (119911) = 0 119911 isin Ω (20)

We define a new operator 119867 which is called the homotopyoperator in the following

119867(V 119901) = (1minus119901) (119877 (V) minus 119877 (1199080)) + 119901 (119861 (V) minus 120577 (119911))

= 0(21)

where 119901 isin [0 1] is an embedding parameter V(119911 119901) Ω times

[0 1] rarr R and 1199080 defines the initial approximation of asolution of (18) (21) is transformed into the following form

119867(V 119901) = 119877 (V) minus 119877 (1199080) + 119901119877 (1199080)

+ 119901 (119876 (V) minus 120577 (119911)) = 0(22)

Therefore

119867(V 0) = 119877 (V) minus 119877 (1199080)

119867 (V 1) = 119877 (1199080) +119876 (V) minus 120577 (119911) (23)

For 119901 = 0 the solution of operator equation 119867(V 0) = 0 isequivalent to solution of problem 119877(V) minus 119877(1199080) = 0 whereasfor 119901 = 1 the solution of operator equation 119867(V 1) = 0 isequivalent to solution of problem (20) Thus a monotonouschange of parameter119901 from zero to one is just that continuouschange of the trivial problem 119877(V) minus119877(1199080) = 0 to the originalproblem In topology this is called homotopy According tothe HPM we assume that the solution of (21) and (22) can bewritten as a power series in 119901

V =infin

sum

119894=0119901119894119908119894= 1199080 +1199011199081 +119901

21199082 + sdot sdot sdot (24)

setting 119901 = 1 the approximate solution of (20) is obtained

119908 = lim119901rarr 1

V =infin

sum

119894=0119908119894= 1199080 +1199081 +1199082 + sdot sdot sdot (25)

in most cases the series in (25) is convergent and in [15] anadditional remark on the convergence of this series can befound

32 The Improved Adomian Decomposition Method TheAdomian decompositionmethod was presented by Adomian[26] This method was used to solve a wide variety oflinear and nonlinear problems Several other researchers haddeveloped a modification to the ADM [27]Themodificationusually involves a slight change and it is aimed at improvingthe accuracy of the series solution

The Adomian decomposition method is able to solve awide class of nonlinear operator equations [28] in the form

119865 (119906) = 120578 (26)

where 119865 119867 rarr 119866 is a nonlinear operator 120578 is a knownelement from Hilbert space 119866 and 119906 is the sought elementfrom Hilbert space119867 The operator 119865(119906) can be written as

119865 (119906) = 119871 (119906) +119877 (119906) +119873 (119906) (27)

where 119871 is an invertible linear operator 119877 is a linear operatorand119873 is a nonlinear operatorThe solution of (26) admits thedecomposition into an infinite series of components

119906 =

infin

sum

119894=0119892119894 (28)

the nonlinear term119873 is equated to an infinite series

119873(119906) =

infin

sum

119894=0119860119894 (29)

119860119899is the Adomian polynomials which can be determined by

1198600 = 119873 (1198920) (30)

119860119899=

1119899[119889119899

119889120582119899119873(

119899

sum

119894=0120582119894119892119894)]

120582=0

119899 ge 1 (31)

Substituting (28) (29) (30) and (31) into operator equation(26) and using the inverse operator 119871

minus1 the followingequation can be obtained

1198920 = 119892lowast+119871minus1(120578)

119892119899= minus119871minus1119877 (119892119899minus1) minus 119871

minus1(119860119899minus1) 119899 ge 1

(32)

where 119892lowast is the function dependent on the initial and

boundary conditionsThe improved Adomian decomposition method [27] it

is important to note that the improved Adomian decompo-sition method is based on the assumption that the function

4 Mathematical Problems in Engineering

119910 = 119892lowast+ 119871minus1(120578) can be divided into two parts namely 1199101

and 1199102 Under this assumption the following equation canbe obtained

119910 = 1199101 +1199102 (33)

Accordingly a slight variation is proposed on the components1198920 and 1198921 The suggestion is that only the part 1199101 is assignedto the component 1198920 whereas the remaining part 1199102 iscombined with other terms given in (33) to define 1198921Consequently the recursive relation is written as follows

1198920 = 1199101

1198921 = 1199102 minus119871minus1119877 (1198920) minus 119871

minus1(1198600)

119892119899+2 = minus119871

minus1119877 (119892119899+1) minus 119871

minus1(119860119899+1) 119899 ge 0

(34)

33 The Solution of the Two-Phase Inverse Stefan ProblemIn this section we introduce the solution of the two-phaseinverse Stefan problemWe split the two-phase inverse Stefanproblem into two problems The first is the determination of1199061(119909 119905) in domain1198631 and the moving interface 120585(119905) which issatisfying (7) (8) (10) and (13) Once the boundary 119909 = 120585(119905)and the heat flux (1205971199062(119909 119905)120597119909)|119909=120585(119905) have been obtained thesecond problem for determining the temperature 1199062(119909 119905) indomain1198632 is solved

The solution of 1199061(119909 119905) and moving interface 120585(119905) basedon HPM because the heat flux on the moving position isunknown in this two-phase inverse Stefan problem theHPMis used to compute 1199061(119909 119905) In this section the homotopyperturbation method is applied to solve the inverse problemWe make the homotopy map for (22)

1198671 (V1 119901) =1205972V11205971199092

minus120597211990610

1205971199092+119901(

120597211990610

1205971199092+

11205721

120597V119894

120597119905) (35)

the solution to this equation

1198671 (V1 119901) = 0 (36)

is sought in the form of a power series in 119901 and it satisfies

V1 =infin

sum

119894=01199011198941199061119894 (37)

where 119901 isin [0 1] by substituting (37) into (36) the followingequation can be obtained

infin

sum

119894=011990111989412059721199061119894

1205971199092=120597211990610

1205971199092minus119901

120597211990610

1205971199092+

11205721

infin

sum

119894=1

1205971199061119894minus1

120597119905 (38)

Next by comparing the same powers of parameter 119901 in (38)the following equation can be obtained

120597211990611

1205971199092=

11205721

12059711990610

120597119905minus120597211990610

1205971199092 (39)

12059721199061119894

1205971199092=

11205721

1205971199061119894minus1

120597119905for 119894 ge 2 (40)

Equations (39) and (40) must be supplemented by theboundary conditionsTherefore for (39) we set the followingboundary conditions

11990610 (0 119905) + 11990611 (0 119905) = 1205791 (119905) (41)

11990610 (120585 (119905) 119905) + 11990611 (120585 (119905) 119905) = 119906lowast (42)

and for (40) we set the conditions in the following (119894 ge 2)

1199061119894 (0 119905) = 0 (43)

1199061119894 (120585 (119905) 119905) = 0 (44)

The above conditions are selected So we give the 119899-orderapproximate solution

1119899 =119899

sum

119894=11199061119894 (45)

because 1199061119894 are determined by the unknown function 120585(119905)This function is derived in the form

120585 (119905) =

119898

sum

119894=1119888119894120595119894(119905) (46)

Thus we define a cost function 1198691 according to (10) and (17)

1198691 (1198881 119888119898) = int120585(0)

01199061119899 (119909 0) minus 1205931 (119909)

2119889119909

+int

120585(119905lowast

)

01199061119899 (119909 119905

lowast) minus 119906119879

1 (119909)2119889119909

(47)

Therefore 119888119894are chosen according to the minimum of (47)

The moving interface 120585(119905) and the solution of 1199061 in domain1198631 can be obtained

The boundary condition (13) on the boundary Γ3 and thecondition of temperature continuity (15) should be fulfilledfor 119899 ge 1 In this way while looking for the solution of thisproblem in domain 1198631 the initial approximation 11990611 will befound by solving the system of (39) with conditions (41) and(42) Conversely functions1199061119894 119894 = 1 2 are determined bythe recurrent solution of (40) with conditions (42) and (44)Consequently themoving interface 120585(119905) and the solution of1199061in domain1198631 can be obtained by the homotopy perturbationmethodwith optimization According to the Stefan condition(16) on the phase changemoving interface Γ5 the temperature1199062(119909 119905) in domain 1198632 can be recovered by IADM withoptimization method

The solution of 1199062(119909 119905) based on IADM with optimiza-tion in this problemwe consider the operator equations (27)where

119871 (1199062) =12059721199062

1205971199092

119877 (1199062) = minus11205722

1205971199062120597119905

119873 (1199062) = 0

120578 = 0

(48)

Mathematical Problems in Engineering 5

The inverse operator 119871minus1 is given by the following equation

119871minus1(1199062) = int

119909

119889

int

119909

120585(119905)

1199062 (119909 119905) 119889119909 119889119909 (49)

The boundary condition and the Stefan condition from (14)are used to obtain the following equation

119871minus1119871(

12059721199062

1205971199092) = 1199062 (119909 119905) minus 120579 (119905)

minus1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

(50)

Hence

119892lowast= 120579 (119905) +

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889) (51)

then the following recursive relation can be obtained

1198920 = 1199062 (119889 119905) = 120579 (119905)

1198921 =1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

+11205722

int

119909

119889

int

119909

120585(119905)

1205971198920120597119905

119889119909 119889119909

119892119899+2 =

11205722

int

119909

119889

int

119909

120585(119905)

120597119892119899minus1120597119905

119889119909 119889119909 119899 ge 0

(52)

because 1199061(119909 119905) and 120585(119905) are determined by the aboveHPM with optimization method (1205971199062(119909 119905)120597119909)|119909=120585(119905) can beobtained by (16) An approximation solution is expressed inthe form

1199062119899 =119899

sum

119894=0119892119894

119899 isin 119873 (53)

and in this inverse Stefan problem 120579(119905) is unknown on theboundary Γ4 and it is derived in the following form

120579 (119905) =

119898

sum

119894=1119902119894120601119894(119905) (54)

where 119902119894isin 119877 and 120601

119894(119905) are the basis functionThe coefficients

119902119894are selected to show minimal deviation of function (55)

from conditions (11) and (15) Thus 119902119894are chosen according

to the minimum of the following functional

1198692 (1199021 119902119898) = int119905lowast

01199062119899 (120585 (119905) 119905) minus 119906

lowast2119889119905

+int

119889

120585(0)1199062119899 (119909 0) minus 1205932 (119909)

2119889119909

(55)

where 1198692(1199021 119902119898) is a nonlinear function The optimiza-tion method plays a very important role in solving this two-phase inverse Stefan problem Therefore a modified spectral

DY conjugate gradientmethod is presented In Section 4 thisoptimization method is introduced

RemarkThe IADMwith optimizationmethod can be used tosolve the direct and inverse one-phase Stefan problem whichis designed in the literature [8 9] And this method is betterthan ADM with optimization method In Section 5 we givetwo examples of one-phase Stefan problem to illustrate theeffectiveness of the IADM with optimization method

4 A Modified Spectral DY ConjugateGradient Method

Conjugate gradient methods are very efficient for solvingthe unconstrained optimization problemmin

119909isin119877119899

119891(119909) Theiriterative formula is given as follows

119909119896+1 = 119909119896 +120572119896119889119896

119889119896=

minus119892119896 if 119896 = 1

minus119892119896+ 120573119896119889119896minus1

if 119896 ge 1

(56)

where step-size 120572119896is positive 119892

119896= nabla119891(119909

119896) and 120573

119896is a scalar

In addition 120572119896is a step length which is calculated by the line

search 120573119896is a very important factor in conjugate gradient

methods the FR PRP DY and CD methods [29] are usedfor choosing this scalar Among them the DY method isregarded as one of the efficient conjugate gradient methodsThe scalar 120573

119896is given by

120573DY119896

=

10038171003817100381710038171198921198961003817100381710038171003817

119889119879

119896minus1119910119896minus1 (57)

where 119910119896minus1 = 119892

119896minus 119892119896minus1 and sdot stands for Euclidean

norm Recently Birgin andMartłnez [30] presented a spectralconjugate gradient method which is combining conjugategradient method and spectral gradient method And thedirection 119889

119896is obtained by

119889119896= minus 120579119896119892119896+120573119896119889119896minus1

120573119896=(120579119896119910119896minus1 minus 119904119896minus1)

119879

119892119896

119889119879

119896minus1119910119896minus1

(58)

where 120579119896= 119904119879

119896minus1119904119896minus1119904119879

119896minus1119910119896minus1 is a parameter with 119904119896minus1 =

119909119896minus 119909119896minus1 and 120579119896 is regarded as spectral gradient However

according to [31] the spectral conjugate gradient method cannot guarantee producing the descent directions

Based on the above analysis in the following we describea modified spectral DY conjugate gradient method and givethe convergence of this method with the Wolfe type linesearch rules

41 The Modified Spectral DY Conjugate Gradient MethodThe new method is introduced

119889119896=

minus119892119896 if 119896 = 1

minus120579119896119892119896+ 120573119896119889119896minus1 if 119896 ge 1

(59)

6 Mathematical Problems in Engineering

where 120579119896= 1 + 119892

119896minus121198921198962 and 120573

119896= 120573

DY119896

This method iscalled the MDY method

Algorithm 1 Consider the following

Step 1 The following are given 1199090 isin 119877119899 120576 gt 0 and the max

iteration step119873max

Step 2 Set 119896 = 1 compute 1198891 = minus1198921 = minusnabla119891(1199090) if 1198921 le 120576then stop

Step 3 Find 120572119896according to Wolfe type line search rule

Step 4 Let 119909119896+1 = 119909119896+120572119896119889119896 and 119892119896+1 = 119892(119909119896+1) if 119892119896+1 le 120576

then stop

Step 5 Compute 119889119896+1 by (59)

Step 6 Set 119896 = 119896 + 1 go to Step 3

42 Convergence Analysis From the above the MDY con-jugate gradient method provides a descent direction for theobjective function In the following we give the convergenceof this algorithmwith theWolfe type line search rules Firstlywe introduce some assumptions

(i) The level setΩ = 119909 isin 119877119899| 119891(119909) le 119891(1199091) is bounded

(ii) In some neighborhood 119873 of Ω 119891(119909) is continuouslydifferentiable and its gradient is Lipschitz continuousnamely there exists a constant 119871 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus119910

1003817100381710038171003817 forall119909 119910 isin 119873 (60)

It is implied from the above assumptions that there exist twopositive constants 120573 and 120574 such that

119909 le 119861

1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120574

forall119909 isin Ω

(61)

TheWolfe type line search [29] is

119891 (119909119896) minus119891 (119909

119896+120572119896119889119896) ge 120575120572

2119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(62)

119892 (119909119896+120572119896119889119896)119879

119889119896ge minus 2120575120572

119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(63)

and the strong Wolfe line search is

119891 (119909119896+120572119896119889119896) le 119891 (119909

119896) + 120575120572

119896119892119879

119896119889119896 (64)

100381610038161003816100381610038161003816119892 (119909119896+120572119896119889119896)119879

119889119896

100381610038161003816100381610038161003816le10038161003816100381610038161003816120590119892119879

119896119889119896

10038161003816100381610038161003816 (65)

where 0 lt 120575 lt 120590 lt 1

Lemma 2 Let the sequences 119892119896 and 119889

119896 be generated by the

proposed algorithm (Algorithm 1) with Wolfe type line searchrules then 119892119879

119896119889119896lt 0 and

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (66)

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= 120573DY119896

[minus120579119896119889119896minus1 (119892

119879

119896minus119892119879

119896minus1) + 119892119879

119896119889119896minus1]

= 120573DY119896

[119892119879

119896minus1119889119896minus1 minus

1003817100381710038171003817119892119896minus11003817100381710038171003817

2

119889119896minus1119910119879

119896minus11003817100381710038171003817119892119896

1003817100381710038171003817

2]

le 120573DY119896119892119879

119896minus1119889119896minus1

(67)

If we choose 119896 = 1 then

119892119879

1 1198891 = minus10038171003817100381710038171198921

1003817100381710038171003817

2

lt 0 (68)

according to (65) we have

119910119879

1 1198891 ge (120590 minus 1) 119892119879

1 1198891 ge 0 (69)

So 120573DY2 ge 0 For 119896 gt 1 according to (67) we have

119892119879

119896119889119896lt 0 (70)

by induction So we obtain

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (71)

the proof is completed

Theorem 3 Let the sequences 119892119896 and 119889

119896 be generated

by the proposed algorithm (Algorithm 1) with Wolfe type linesearch rules then

119892119879

119896119889119896le minus119862 119892

1198962 (72)

where 119862 gt 0 is a constant

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= minus1003817100381710038171003817119892119896minus1

1003817100381710038171003817

2

+(119892119879

119896119889119896minus1

119889119879

119896minus1119910119896minus1minus 1) 1003817100381710038171003817119892119896

1003817100381710038171003817

2

le (119892119879

119896minus1119889119896minus1

119889119879

119896minus1119910119896minus1)1003817100381710038171003817119892119896

1003817100381710038171003817

2

(73)

From the Wolfe type line search rules (65) we can obtain

minus (120590 + 1) 119892119879119896minus1119889119896minus1 ge 119889

119879

119896minus1 (119892119896 minus119892119896minus1) gt 0 (74)

according to (73) and (74) we have

119892119879

119896119889119896le minus

1120590 + 1

10038171003817100381710038171198921198961003817100381710038171003817

2

(75)

we define 119862 = 1(120590 + 1) Therefore we can obtain inequality(72)

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

3 The Solution of Two-Phase InverseStefan Problem

31 The Homotopy Perturbation Method The homotopy per-turbation method was presented in [13 14] it arose as acombination of two other methods the perturbation methodand the homotopy technique from topology Homotopyperturbation method was applied on differential equationspartial differential equations and the direct and inverseproblems of heat transfer by many authors [15ndash17]

Homotopy perturbation method was used to find theexact or the approximate solutions of the linear and nonlinearintegral equations [16 18] the two-dimensional integralequations [19ndash21] and the integrodifferential equations [22ndash25] This method enables seeking a solution of the followingoperator equation

119861 (119908) minus 120577 (119911) = 0 119911 isin Ω (18)

where 119861 is an operator 119908 is a sought function and 120577 isa known function on the domain of Ω The operator 119861 ispresented in the form of sum

119861 (V) = 119877 (119908) +119876 (119908) (19)

where 119877 defines the linear operator and 119876 is the remainingpart of operator 119861 So (18) can be written as

119877 (119908) +119876 (119908) minus 120577 (119911) = 0 119911 isin Ω (20)

We define a new operator 119867 which is called the homotopyoperator in the following

119867(V 119901) = (1minus119901) (119877 (V) minus 119877 (1199080)) + 119901 (119861 (V) minus 120577 (119911))

= 0(21)

where 119901 isin [0 1] is an embedding parameter V(119911 119901) Ω times

[0 1] rarr R and 1199080 defines the initial approximation of asolution of (18) (21) is transformed into the following form

119867(V 119901) = 119877 (V) minus 119877 (1199080) + 119901119877 (1199080)

+ 119901 (119876 (V) minus 120577 (119911)) = 0(22)

Therefore

119867(V 0) = 119877 (V) minus 119877 (1199080)

119867 (V 1) = 119877 (1199080) +119876 (V) minus 120577 (119911) (23)

For 119901 = 0 the solution of operator equation 119867(V 0) = 0 isequivalent to solution of problem 119877(V) minus 119877(1199080) = 0 whereasfor 119901 = 1 the solution of operator equation 119867(V 1) = 0 isequivalent to solution of problem (20) Thus a monotonouschange of parameter119901 from zero to one is just that continuouschange of the trivial problem 119877(V) minus119877(1199080) = 0 to the originalproblem In topology this is called homotopy According tothe HPM we assume that the solution of (21) and (22) can bewritten as a power series in 119901

V =infin

sum

119894=0119901119894119908119894= 1199080 +1199011199081 +119901

21199082 + sdot sdot sdot (24)

setting 119901 = 1 the approximate solution of (20) is obtained

119908 = lim119901rarr 1

V =infin

sum

119894=0119908119894= 1199080 +1199081 +1199082 + sdot sdot sdot (25)

in most cases the series in (25) is convergent and in [15] anadditional remark on the convergence of this series can befound

32 The Improved Adomian Decomposition Method TheAdomian decompositionmethod was presented by Adomian[26] This method was used to solve a wide variety oflinear and nonlinear problems Several other researchers haddeveloped a modification to the ADM [27]Themodificationusually involves a slight change and it is aimed at improvingthe accuracy of the series solution

The Adomian decomposition method is able to solve awide class of nonlinear operator equations [28] in the form

119865 (119906) = 120578 (26)

where 119865 119867 rarr 119866 is a nonlinear operator 120578 is a knownelement from Hilbert space 119866 and 119906 is the sought elementfrom Hilbert space119867 The operator 119865(119906) can be written as

119865 (119906) = 119871 (119906) +119877 (119906) +119873 (119906) (27)

where 119871 is an invertible linear operator 119877 is a linear operatorand119873 is a nonlinear operatorThe solution of (26) admits thedecomposition into an infinite series of components

119906 =

infin

sum

119894=0119892119894 (28)

the nonlinear term119873 is equated to an infinite series

119873(119906) =

infin

sum

119894=0119860119894 (29)

119860119899is the Adomian polynomials which can be determined by

1198600 = 119873 (1198920) (30)

119860119899=

1119899[119889119899

119889120582119899119873(

119899

sum

119894=0120582119894119892119894)]

120582=0

119899 ge 1 (31)

Substituting (28) (29) (30) and (31) into operator equation(26) and using the inverse operator 119871

minus1 the followingequation can be obtained

1198920 = 119892lowast+119871minus1(120578)

119892119899= minus119871minus1119877 (119892119899minus1) minus 119871

minus1(119860119899minus1) 119899 ge 1

(32)

where 119892lowast is the function dependent on the initial and

boundary conditionsThe improved Adomian decomposition method [27] it

is important to note that the improved Adomian decompo-sition method is based on the assumption that the function

4 Mathematical Problems in Engineering

119910 = 119892lowast+ 119871minus1(120578) can be divided into two parts namely 1199101

and 1199102 Under this assumption the following equation canbe obtained

119910 = 1199101 +1199102 (33)

Accordingly a slight variation is proposed on the components1198920 and 1198921 The suggestion is that only the part 1199101 is assignedto the component 1198920 whereas the remaining part 1199102 iscombined with other terms given in (33) to define 1198921Consequently the recursive relation is written as follows

1198920 = 1199101

1198921 = 1199102 minus119871minus1119877 (1198920) minus 119871

minus1(1198600)

119892119899+2 = minus119871

minus1119877 (119892119899+1) minus 119871

minus1(119860119899+1) 119899 ge 0

(34)

33 The Solution of the Two-Phase Inverse Stefan ProblemIn this section we introduce the solution of the two-phaseinverse Stefan problemWe split the two-phase inverse Stefanproblem into two problems The first is the determination of1199061(119909 119905) in domain1198631 and the moving interface 120585(119905) which issatisfying (7) (8) (10) and (13) Once the boundary 119909 = 120585(119905)and the heat flux (1205971199062(119909 119905)120597119909)|119909=120585(119905) have been obtained thesecond problem for determining the temperature 1199062(119909 119905) indomain1198632 is solved

The solution of 1199061(119909 119905) and moving interface 120585(119905) basedon HPM because the heat flux on the moving position isunknown in this two-phase inverse Stefan problem theHPMis used to compute 1199061(119909 119905) In this section the homotopyperturbation method is applied to solve the inverse problemWe make the homotopy map for (22)

1198671 (V1 119901) =1205972V11205971199092

minus120597211990610

1205971199092+119901(

120597211990610

1205971199092+

11205721

120597V119894

120597119905) (35)

the solution to this equation

1198671 (V1 119901) = 0 (36)

is sought in the form of a power series in 119901 and it satisfies

V1 =infin

sum

119894=01199011198941199061119894 (37)

where 119901 isin [0 1] by substituting (37) into (36) the followingequation can be obtained

infin

sum

119894=011990111989412059721199061119894

1205971199092=120597211990610

1205971199092minus119901

120597211990610

1205971199092+

11205721

infin

sum

119894=1

1205971199061119894minus1

120597119905 (38)

Next by comparing the same powers of parameter 119901 in (38)the following equation can be obtained

120597211990611

1205971199092=

11205721

12059711990610

120597119905minus120597211990610

1205971199092 (39)

12059721199061119894

1205971199092=

11205721

1205971199061119894minus1

120597119905for 119894 ge 2 (40)

Equations (39) and (40) must be supplemented by theboundary conditionsTherefore for (39) we set the followingboundary conditions

11990610 (0 119905) + 11990611 (0 119905) = 1205791 (119905) (41)

11990610 (120585 (119905) 119905) + 11990611 (120585 (119905) 119905) = 119906lowast (42)

and for (40) we set the conditions in the following (119894 ge 2)

1199061119894 (0 119905) = 0 (43)

1199061119894 (120585 (119905) 119905) = 0 (44)

The above conditions are selected So we give the 119899-orderapproximate solution

1119899 =119899

sum

119894=11199061119894 (45)

because 1199061119894 are determined by the unknown function 120585(119905)This function is derived in the form

120585 (119905) =

119898

sum

119894=1119888119894120595119894(119905) (46)

Thus we define a cost function 1198691 according to (10) and (17)

1198691 (1198881 119888119898) = int120585(0)

01199061119899 (119909 0) minus 1205931 (119909)

2119889119909

+int

120585(119905lowast

)

01199061119899 (119909 119905

lowast) minus 119906119879

1 (119909)2119889119909

(47)

Therefore 119888119894are chosen according to the minimum of (47)

The moving interface 120585(119905) and the solution of 1199061 in domain1198631 can be obtained

The boundary condition (13) on the boundary Γ3 and thecondition of temperature continuity (15) should be fulfilledfor 119899 ge 1 In this way while looking for the solution of thisproblem in domain 1198631 the initial approximation 11990611 will befound by solving the system of (39) with conditions (41) and(42) Conversely functions1199061119894 119894 = 1 2 are determined bythe recurrent solution of (40) with conditions (42) and (44)Consequently themoving interface 120585(119905) and the solution of1199061in domain1198631 can be obtained by the homotopy perturbationmethodwith optimization According to the Stefan condition(16) on the phase changemoving interface Γ5 the temperature1199062(119909 119905) in domain 1198632 can be recovered by IADM withoptimization method

The solution of 1199062(119909 119905) based on IADM with optimiza-tion in this problemwe consider the operator equations (27)where

119871 (1199062) =12059721199062

1205971199092

119877 (1199062) = minus11205722

1205971199062120597119905

119873 (1199062) = 0

120578 = 0

(48)

Mathematical Problems in Engineering 5

The inverse operator 119871minus1 is given by the following equation

119871minus1(1199062) = int

119909

119889

int

119909

120585(119905)

1199062 (119909 119905) 119889119909 119889119909 (49)

The boundary condition and the Stefan condition from (14)are used to obtain the following equation

119871minus1119871(

12059721199062

1205971199092) = 1199062 (119909 119905) minus 120579 (119905)

minus1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

(50)

Hence

119892lowast= 120579 (119905) +

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889) (51)

then the following recursive relation can be obtained

1198920 = 1199062 (119889 119905) = 120579 (119905)

1198921 =1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

+11205722

int

119909

119889

int

119909

120585(119905)

1205971198920120597119905

119889119909 119889119909

119892119899+2 =

11205722

int

119909

119889

int

119909

120585(119905)

120597119892119899minus1120597119905

119889119909 119889119909 119899 ge 0

(52)

because 1199061(119909 119905) and 120585(119905) are determined by the aboveHPM with optimization method (1205971199062(119909 119905)120597119909)|119909=120585(119905) can beobtained by (16) An approximation solution is expressed inthe form

1199062119899 =119899

sum

119894=0119892119894

119899 isin 119873 (53)

and in this inverse Stefan problem 120579(119905) is unknown on theboundary Γ4 and it is derived in the following form

120579 (119905) =

119898

sum

119894=1119902119894120601119894(119905) (54)

where 119902119894isin 119877 and 120601

119894(119905) are the basis functionThe coefficients

119902119894are selected to show minimal deviation of function (55)

from conditions (11) and (15) Thus 119902119894are chosen according

to the minimum of the following functional

1198692 (1199021 119902119898) = int119905lowast

01199062119899 (120585 (119905) 119905) minus 119906

lowast2119889119905

+int

119889

120585(0)1199062119899 (119909 0) minus 1205932 (119909)

2119889119909

(55)

where 1198692(1199021 119902119898) is a nonlinear function The optimiza-tion method plays a very important role in solving this two-phase inverse Stefan problem Therefore a modified spectral

DY conjugate gradientmethod is presented In Section 4 thisoptimization method is introduced

RemarkThe IADMwith optimizationmethod can be used tosolve the direct and inverse one-phase Stefan problem whichis designed in the literature [8 9] And this method is betterthan ADM with optimization method In Section 5 we givetwo examples of one-phase Stefan problem to illustrate theeffectiveness of the IADM with optimization method

4 A Modified Spectral DY ConjugateGradient Method

Conjugate gradient methods are very efficient for solvingthe unconstrained optimization problemmin

119909isin119877119899

119891(119909) Theiriterative formula is given as follows

119909119896+1 = 119909119896 +120572119896119889119896

119889119896=

minus119892119896 if 119896 = 1

minus119892119896+ 120573119896119889119896minus1

if 119896 ge 1

(56)

where step-size 120572119896is positive 119892

119896= nabla119891(119909

119896) and 120573

119896is a scalar

In addition 120572119896is a step length which is calculated by the line

search 120573119896is a very important factor in conjugate gradient

methods the FR PRP DY and CD methods [29] are usedfor choosing this scalar Among them the DY method isregarded as one of the efficient conjugate gradient methodsThe scalar 120573

119896is given by

120573DY119896

=

10038171003817100381710038171198921198961003817100381710038171003817

119889119879

119896minus1119910119896minus1 (57)

where 119910119896minus1 = 119892

119896minus 119892119896minus1 and sdot stands for Euclidean

norm Recently Birgin andMartłnez [30] presented a spectralconjugate gradient method which is combining conjugategradient method and spectral gradient method And thedirection 119889

119896is obtained by

119889119896= minus 120579119896119892119896+120573119896119889119896minus1

120573119896=(120579119896119910119896minus1 minus 119904119896minus1)

119879

119892119896

119889119879

119896minus1119910119896minus1

(58)

where 120579119896= 119904119879

119896minus1119904119896minus1119904119879

119896minus1119910119896minus1 is a parameter with 119904119896minus1 =

119909119896minus 119909119896minus1 and 120579119896 is regarded as spectral gradient However

according to [31] the spectral conjugate gradient method cannot guarantee producing the descent directions

Based on the above analysis in the following we describea modified spectral DY conjugate gradient method and givethe convergence of this method with the Wolfe type linesearch rules

41 The Modified Spectral DY Conjugate Gradient MethodThe new method is introduced

119889119896=

minus119892119896 if 119896 = 1

minus120579119896119892119896+ 120573119896119889119896minus1 if 119896 ge 1

(59)

6 Mathematical Problems in Engineering

where 120579119896= 1 + 119892

119896minus121198921198962 and 120573

119896= 120573

DY119896

This method iscalled the MDY method

Algorithm 1 Consider the following

Step 1 The following are given 1199090 isin 119877119899 120576 gt 0 and the max

iteration step119873max

Step 2 Set 119896 = 1 compute 1198891 = minus1198921 = minusnabla119891(1199090) if 1198921 le 120576then stop

Step 3 Find 120572119896according to Wolfe type line search rule

Step 4 Let 119909119896+1 = 119909119896+120572119896119889119896 and 119892119896+1 = 119892(119909119896+1) if 119892119896+1 le 120576

then stop

Step 5 Compute 119889119896+1 by (59)

Step 6 Set 119896 = 119896 + 1 go to Step 3

42 Convergence Analysis From the above the MDY con-jugate gradient method provides a descent direction for theobjective function In the following we give the convergenceof this algorithmwith theWolfe type line search rules Firstlywe introduce some assumptions

(i) The level setΩ = 119909 isin 119877119899| 119891(119909) le 119891(1199091) is bounded

(ii) In some neighborhood 119873 of Ω 119891(119909) is continuouslydifferentiable and its gradient is Lipschitz continuousnamely there exists a constant 119871 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus119910

1003817100381710038171003817 forall119909 119910 isin 119873 (60)

It is implied from the above assumptions that there exist twopositive constants 120573 and 120574 such that

119909 le 119861

1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120574

forall119909 isin Ω

(61)

TheWolfe type line search [29] is

119891 (119909119896) minus119891 (119909

119896+120572119896119889119896) ge 120575120572

2119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(62)

119892 (119909119896+120572119896119889119896)119879

119889119896ge minus 2120575120572

119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(63)

and the strong Wolfe line search is

119891 (119909119896+120572119896119889119896) le 119891 (119909

119896) + 120575120572

119896119892119879

119896119889119896 (64)

100381610038161003816100381610038161003816119892 (119909119896+120572119896119889119896)119879

119889119896

100381610038161003816100381610038161003816le10038161003816100381610038161003816120590119892119879

119896119889119896

10038161003816100381610038161003816 (65)

where 0 lt 120575 lt 120590 lt 1

Lemma 2 Let the sequences 119892119896 and 119889

119896 be generated by the

proposed algorithm (Algorithm 1) with Wolfe type line searchrules then 119892119879

119896119889119896lt 0 and

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (66)

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= 120573DY119896

[minus120579119896119889119896minus1 (119892

119879

119896minus119892119879

119896minus1) + 119892119879

119896119889119896minus1]

= 120573DY119896

[119892119879

119896minus1119889119896minus1 minus

1003817100381710038171003817119892119896minus11003817100381710038171003817

2

119889119896minus1119910119879

119896minus11003817100381710038171003817119892119896

1003817100381710038171003817

2]

le 120573DY119896119892119879

119896minus1119889119896minus1

(67)

If we choose 119896 = 1 then

119892119879

1 1198891 = minus10038171003817100381710038171198921

1003817100381710038171003817

2

lt 0 (68)

according to (65) we have

119910119879

1 1198891 ge (120590 minus 1) 119892119879

1 1198891 ge 0 (69)

So 120573DY2 ge 0 For 119896 gt 1 according to (67) we have

119892119879

119896119889119896lt 0 (70)

by induction So we obtain

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (71)

the proof is completed

Theorem 3 Let the sequences 119892119896 and 119889

119896 be generated

by the proposed algorithm (Algorithm 1) with Wolfe type linesearch rules then

119892119879

119896119889119896le minus119862 119892

1198962 (72)

where 119862 gt 0 is a constant

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= minus1003817100381710038171003817119892119896minus1

1003817100381710038171003817

2

+(119892119879

119896119889119896minus1

119889119879

119896minus1119910119896minus1minus 1) 1003817100381710038171003817119892119896

1003817100381710038171003817

2

le (119892119879

119896minus1119889119896minus1

119889119879

119896minus1119910119896minus1)1003817100381710038171003817119892119896

1003817100381710038171003817

2

(73)

From the Wolfe type line search rules (65) we can obtain

minus (120590 + 1) 119892119879119896minus1119889119896minus1 ge 119889

119879

119896minus1 (119892119896 minus119892119896minus1) gt 0 (74)

according to (73) and (74) we have

119892119879

119896119889119896le minus

1120590 + 1

10038171003817100381710038171198921198961003817100381710038171003817

2

(75)

we define 119862 = 1(120590 + 1) Therefore we can obtain inequality(72)

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

119910 = 119892lowast+ 119871minus1(120578) can be divided into two parts namely 1199101

and 1199102 Under this assumption the following equation canbe obtained

119910 = 1199101 +1199102 (33)

Accordingly a slight variation is proposed on the components1198920 and 1198921 The suggestion is that only the part 1199101 is assignedto the component 1198920 whereas the remaining part 1199102 iscombined with other terms given in (33) to define 1198921Consequently the recursive relation is written as follows

1198920 = 1199101

1198921 = 1199102 minus119871minus1119877 (1198920) minus 119871

minus1(1198600)

119892119899+2 = minus119871

minus1119877 (119892119899+1) minus 119871

minus1(119860119899+1) 119899 ge 0

(34)

33 The Solution of the Two-Phase Inverse Stefan ProblemIn this section we introduce the solution of the two-phaseinverse Stefan problemWe split the two-phase inverse Stefanproblem into two problems The first is the determination of1199061(119909 119905) in domain1198631 and the moving interface 120585(119905) which issatisfying (7) (8) (10) and (13) Once the boundary 119909 = 120585(119905)and the heat flux (1205971199062(119909 119905)120597119909)|119909=120585(119905) have been obtained thesecond problem for determining the temperature 1199062(119909 119905) indomain1198632 is solved

The solution of 1199061(119909 119905) and moving interface 120585(119905) basedon HPM because the heat flux on the moving position isunknown in this two-phase inverse Stefan problem theHPMis used to compute 1199061(119909 119905) In this section the homotopyperturbation method is applied to solve the inverse problemWe make the homotopy map for (22)

1198671 (V1 119901) =1205972V11205971199092

minus120597211990610

1205971199092+119901(

120597211990610

1205971199092+

11205721

120597V119894

120597119905) (35)

the solution to this equation

1198671 (V1 119901) = 0 (36)

is sought in the form of a power series in 119901 and it satisfies

V1 =infin

sum

119894=01199011198941199061119894 (37)

where 119901 isin [0 1] by substituting (37) into (36) the followingequation can be obtained

infin

sum

119894=011990111989412059721199061119894

1205971199092=120597211990610

1205971199092minus119901

120597211990610

1205971199092+

11205721

infin

sum

119894=1

1205971199061119894minus1

120597119905 (38)

Next by comparing the same powers of parameter 119901 in (38)the following equation can be obtained

120597211990611

1205971199092=

11205721

12059711990610

120597119905minus120597211990610

1205971199092 (39)

12059721199061119894

1205971199092=

11205721

1205971199061119894minus1

120597119905for 119894 ge 2 (40)

Equations (39) and (40) must be supplemented by theboundary conditionsTherefore for (39) we set the followingboundary conditions

11990610 (0 119905) + 11990611 (0 119905) = 1205791 (119905) (41)

11990610 (120585 (119905) 119905) + 11990611 (120585 (119905) 119905) = 119906lowast (42)

and for (40) we set the conditions in the following (119894 ge 2)

1199061119894 (0 119905) = 0 (43)

1199061119894 (120585 (119905) 119905) = 0 (44)

The above conditions are selected So we give the 119899-orderapproximate solution

1119899 =119899

sum

119894=11199061119894 (45)

because 1199061119894 are determined by the unknown function 120585(119905)This function is derived in the form

120585 (119905) =

119898

sum

119894=1119888119894120595119894(119905) (46)

Thus we define a cost function 1198691 according to (10) and (17)

1198691 (1198881 119888119898) = int120585(0)

01199061119899 (119909 0) minus 1205931 (119909)

2119889119909

+int

120585(119905lowast

)

01199061119899 (119909 119905

lowast) minus 119906119879

1 (119909)2119889119909

(47)

Therefore 119888119894are chosen according to the minimum of (47)

The moving interface 120585(119905) and the solution of 1199061 in domain1198631 can be obtained

The boundary condition (13) on the boundary Γ3 and thecondition of temperature continuity (15) should be fulfilledfor 119899 ge 1 In this way while looking for the solution of thisproblem in domain 1198631 the initial approximation 11990611 will befound by solving the system of (39) with conditions (41) and(42) Conversely functions1199061119894 119894 = 1 2 are determined bythe recurrent solution of (40) with conditions (42) and (44)Consequently themoving interface 120585(119905) and the solution of1199061in domain1198631 can be obtained by the homotopy perturbationmethodwith optimization According to the Stefan condition(16) on the phase changemoving interface Γ5 the temperature1199062(119909 119905) in domain 1198632 can be recovered by IADM withoptimization method

The solution of 1199062(119909 119905) based on IADM with optimiza-tion in this problemwe consider the operator equations (27)where

119871 (1199062) =12059721199062

1205971199092

119877 (1199062) = minus11205722

1205971199062120597119905

119873 (1199062) = 0

120578 = 0

(48)

Mathematical Problems in Engineering 5

The inverse operator 119871minus1 is given by the following equation

119871minus1(1199062) = int

119909

119889

int

119909

120585(119905)

1199062 (119909 119905) 119889119909 119889119909 (49)

The boundary condition and the Stefan condition from (14)are used to obtain the following equation

119871minus1119871(

12059721199062

1205971199092) = 1199062 (119909 119905) minus 120579 (119905)

minus1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

(50)

Hence

119892lowast= 120579 (119905) +

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889) (51)

then the following recursive relation can be obtained

1198920 = 1199062 (119889 119905) = 120579 (119905)

1198921 =1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

+11205722

int

119909

119889

int

119909

120585(119905)

1205971198920120597119905

119889119909 119889119909

119892119899+2 =

11205722

int

119909

119889

int

119909

120585(119905)

120597119892119899minus1120597119905

119889119909 119889119909 119899 ge 0

(52)

because 1199061(119909 119905) and 120585(119905) are determined by the aboveHPM with optimization method (1205971199062(119909 119905)120597119909)|119909=120585(119905) can beobtained by (16) An approximation solution is expressed inthe form

1199062119899 =119899

sum

119894=0119892119894

119899 isin 119873 (53)

and in this inverse Stefan problem 120579(119905) is unknown on theboundary Γ4 and it is derived in the following form

120579 (119905) =

119898

sum

119894=1119902119894120601119894(119905) (54)

where 119902119894isin 119877 and 120601

119894(119905) are the basis functionThe coefficients

119902119894are selected to show minimal deviation of function (55)

from conditions (11) and (15) Thus 119902119894are chosen according

to the minimum of the following functional

1198692 (1199021 119902119898) = int119905lowast

01199062119899 (120585 (119905) 119905) minus 119906

lowast2119889119905

+int

119889

120585(0)1199062119899 (119909 0) minus 1205932 (119909)

2119889119909

(55)

where 1198692(1199021 119902119898) is a nonlinear function The optimiza-tion method plays a very important role in solving this two-phase inverse Stefan problem Therefore a modified spectral

DY conjugate gradientmethod is presented In Section 4 thisoptimization method is introduced

RemarkThe IADMwith optimizationmethod can be used tosolve the direct and inverse one-phase Stefan problem whichis designed in the literature [8 9] And this method is betterthan ADM with optimization method In Section 5 we givetwo examples of one-phase Stefan problem to illustrate theeffectiveness of the IADM with optimization method

4 A Modified Spectral DY ConjugateGradient Method

Conjugate gradient methods are very efficient for solvingthe unconstrained optimization problemmin

119909isin119877119899

119891(119909) Theiriterative formula is given as follows

119909119896+1 = 119909119896 +120572119896119889119896

119889119896=

minus119892119896 if 119896 = 1

minus119892119896+ 120573119896119889119896minus1

if 119896 ge 1

(56)

where step-size 120572119896is positive 119892

119896= nabla119891(119909

119896) and 120573

119896is a scalar

In addition 120572119896is a step length which is calculated by the line

search 120573119896is a very important factor in conjugate gradient

methods the FR PRP DY and CD methods [29] are usedfor choosing this scalar Among them the DY method isregarded as one of the efficient conjugate gradient methodsThe scalar 120573

119896is given by

120573DY119896

=

10038171003817100381710038171198921198961003817100381710038171003817

119889119879

119896minus1119910119896minus1 (57)

where 119910119896minus1 = 119892

119896minus 119892119896minus1 and sdot stands for Euclidean

norm Recently Birgin andMartłnez [30] presented a spectralconjugate gradient method which is combining conjugategradient method and spectral gradient method And thedirection 119889

119896is obtained by

119889119896= minus 120579119896119892119896+120573119896119889119896minus1

120573119896=(120579119896119910119896minus1 minus 119904119896minus1)

119879

119892119896

119889119879

119896minus1119910119896minus1

(58)

where 120579119896= 119904119879

119896minus1119904119896minus1119904119879

119896minus1119910119896minus1 is a parameter with 119904119896minus1 =

119909119896minus 119909119896minus1 and 120579119896 is regarded as spectral gradient However

according to [31] the spectral conjugate gradient method cannot guarantee producing the descent directions

Based on the above analysis in the following we describea modified spectral DY conjugate gradient method and givethe convergence of this method with the Wolfe type linesearch rules

41 The Modified Spectral DY Conjugate Gradient MethodThe new method is introduced

119889119896=

minus119892119896 if 119896 = 1

minus120579119896119892119896+ 120573119896119889119896minus1 if 119896 ge 1

(59)

6 Mathematical Problems in Engineering

where 120579119896= 1 + 119892

119896minus121198921198962 and 120573

119896= 120573

DY119896

This method iscalled the MDY method

Algorithm 1 Consider the following

Step 1 The following are given 1199090 isin 119877119899 120576 gt 0 and the max

iteration step119873max

Step 2 Set 119896 = 1 compute 1198891 = minus1198921 = minusnabla119891(1199090) if 1198921 le 120576then stop

Step 3 Find 120572119896according to Wolfe type line search rule

Step 4 Let 119909119896+1 = 119909119896+120572119896119889119896 and 119892119896+1 = 119892(119909119896+1) if 119892119896+1 le 120576

then stop

Step 5 Compute 119889119896+1 by (59)

Step 6 Set 119896 = 119896 + 1 go to Step 3

42 Convergence Analysis From the above the MDY con-jugate gradient method provides a descent direction for theobjective function In the following we give the convergenceof this algorithmwith theWolfe type line search rules Firstlywe introduce some assumptions

(i) The level setΩ = 119909 isin 119877119899| 119891(119909) le 119891(1199091) is bounded

(ii) In some neighborhood 119873 of Ω 119891(119909) is continuouslydifferentiable and its gradient is Lipschitz continuousnamely there exists a constant 119871 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus119910

1003817100381710038171003817 forall119909 119910 isin 119873 (60)

It is implied from the above assumptions that there exist twopositive constants 120573 and 120574 such that

119909 le 119861

1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120574

forall119909 isin Ω

(61)

TheWolfe type line search [29] is

119891 (119909119896) minus119891 (119909

119896+120572119896119889119896) ge 120575120572

2119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(62)

119892 (119909119896+120572119896119889119896)119879

119889119896ge minus 2120575120572

119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(63)

and the strong Wolfe line search is

119891 (119909119896+120572119896119889119896) le 119891 (119909

119896) + 120575120572

119896119892119879

119896119889119896 (64)

100381610038161003816100381610038161003816119892 (119909119896+120572119896119889119896)119879

119889119896

100381610038161003816100381610038161003816le10038161003816100381610038161003816120590119892119879

119896119889119896

10038161003816100381610038161003816 (65)

where 0 lt 120575 lt 120590 lt 1

Lemma 2 Let the sequences 119892119896 and 119889

119896 be generated by the

proposed algorithm (Algorithm 1) with Wolfe type line searchrules then 119892119879

119896119889119896lt 0 and

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (66)

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= 120573DY119896

[minus120579119896119889119896minus1 (119892

119879

119896minus119892119879

119896minus1) + 119892119879

119896119889119896minus1]

= 120573DY119896

[119892119879

119896minus1119889119896minus1 minus

1003817100381710038171003817119892119896minus11003817100381710038171003817

2

119889119896minus1119910119879

119896minus11003817100381710038171003817119892119896

1003817100381710038171003817

2]

le 120573DY119896119892119879

119896minus1119889119896minus1

(67)

If we choose 119896 = 1 then

119892119879

1 1198891 = minus10038171003817100381710038171198921

1003817100381710038171003817

2

lt 0 (68)

according to (65) we have

119910119879

1 1198891 ge (120590 minus 1) 119892119879

1 1198891 ge 0 (69)

So 120573DY2 ge 0 For 119896 gt 1 according to (67) we have

119892119879

119896119889119896lt 0 (70)

by induction So we obtain

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (71)

the proof is completed

Theorem 3 Let the sequences 119892119896 and 119889

119896 be generated

by the proposed algorithm (Algorithm 1) with Wolfe type linesearch rules then

119892119879

119896119889119896le minus119862 119892

1198962 (72)

where 119862 gt 0 is a constant

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= minus1003817100381710038171003817119892119896minus1

1003817100381710038171003817

2

+(119892119879

119896119889119896minus1

119889119879

119896minus1119910119896minus1minus 1) 1003817100381710038171003817119892119896

1003817100381710038171003817

2

le (119892119879

119896minus1119889119896minus1

119889119879

119896minus1119910119896minus1)1003817100381710038171003817119892119896

1003817100381710038171003817

2

(73)

From the Wolfe type line search rules (65) we can obtain

minus (120590 + 1) 119892119879119896minus1119889119896minus1 ge 119889

119879

119896minus1 (119892119896 minus119892119896minus1) gt 0 (74)

according to (73) and (74) we have

119892119879

119896119889119896le minus

1120590 + 1

10038171003817100381710038171198921198961003817100381710038171003817

2

(75)

we define 119862 = 1(120590 + 1) Therefore we can obtain inequality(72)

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The inverse operator 119871minus1 is given by the following equation

119871minus1(1199062) = int

119909

119889

int

119909

120585(119905)

1199062 (119909 119905) 119889119909 119889119909 (49)

The boundary condition and the Stefan condition from (14)are used to obtain the following equation

119871minus1119871(

12059721199062

1205971199092) = 1199062 (119909 119905) minus 120579 (119905)

minus1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

(50)

Hence

119892lowast= 120579 (119905) +

1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889) (51)

then the following recursive relation can be obtained

1198920 = 1199062 (119889 119905) = 120579 (119905)

1198921 =1205971199062 (119909 119905)

120597119909

10038161003816100381610038161003816100381610038161003816119909=120585(119905)

(119909 minus 119889)

+11205722

int

119909

119889

int

119909

120585(119905)

1205971198920120597119905

119889119909 119889119909

119892119899+2 =

11205722

int

119909

119889

int

119909

120585(119905)

120597119892119899minus1120597119905

119889119909 119889119909 119899 ge 0

(52)

because 1199061(119909 119905) and 120585(119905) are determined by the aboveHPM with optimization method (1205971199062(119909 119905)120597119909)|119909=120585(119905) can beobtained by (16) An approximation solution is expressed inthe form

1199062119899 =119899

sum

119894=0119892119894

119899 isin 119873 (53)

and in this inverse Stefan problem 120579(119905) is unknown on theboundary Γ4 and it is derived in the following form

120579 (119905) =

119898

sum

119894=1119902119894120601119894(119905) (54)

where 119902119894isin 119877 and 120601

119894(119905) are the basis functionThe coefficients

119902119894are selected to show minimal deviation of function (55)

from conditions (11) and (15) Thus 119902119894are chosen according

to the minimum of the following functional

1198692 (1199021 119902119898) = int119905lowast

01199062119899 (120585 (119905) 119905) minus 119906

lowast2119889119905

+int

119889

120585(0)1199062119899 (119909 0) minus 1205932 (119909)

2119889119909

(55)

where 1198692(1199021 119902119898) is a nonlinear function The optimiza-tion method plays a very important role in solving this two-phase inverse Stefan problem Therefore a modified spectral

DY conjugate gradientmethod is presented In Section 4 thisoptimization method is introduced

RemarkThe IADMwith optimizationmethod can be used tosolve the direct and inverse one-phase Stefan problem whichis designed in the literature [8 9] And this method is betterthan ADM with optimization method In Section 5 we givetwo examples of one-phase Stefan problem to illustrate theeffectiveness of the IADM with optimization method

4 A Modified Spectral DY ConjugateGradient Method

Conjugate gradient methods are very efficient for solvingthe unconstrained optimization problemmin

119909isin119877119899

119891(119909) Theiriterative formula is given as follows

119909119896+1 = 119909119896 +120572119896119889119896

119889119896=

minus119892119896 if 119896 = 1

minus119892119896+ 120573119896119889119896minus1

if 119896 ge 1

(56)

where step-size 120572119896is positive 119892

119896= nabla119891(119909

119896) and 120573

119896is a scalar

In addition 120572119896is a step length which is calculated by the line

search 120573119896is a very important factor in conjugate gradient

methods the FR PRP DY and CD methods [29] are usedfor choosing this scalar Among them the DY method isregarded as one of the efficient conjugate gradient methodsThe scalar 120573

119896is given by

120573DY119896

=

10038171003817100381710038171198921198961003817100381710038171003817

119889119879

119896minus1119910119896minus1 (57)

where 119910119896minus1 = 119892

119896minus 119892119896minus1 and sdot stands for Euclidean

norm Recently Birgin andMartłnez [30] presented a spectralconjugate gradient method which is combining conjugategradient method and spectral gradient method And thedirection 119889

119896is obtained by

119889119896= minus 120579119896119892119896+120573119896119889119896minus1

120573119896=(120579119896119910119896minus1 minus 119904119896minus1)

119879

119892119896

119889119879

119896minus1119910119896minus1

(58)

where 120579119896= 119904119879

119896minus1119904119896minus1119904119879

119896minus1119910119896minus1 is a parameter with 119904119896minus1 =

119909119896minus 119909119896minus1 and 120579119896 is regarded as spectral gradient However

according to [31] the spectral conjugate gradient method cannot guarantee producing the descent directions

Based on the above analysis in the following we describea modified spectral DY conjugate gradient method and givethe convergence of this method with the Wolfe type linesearch rules

41 The Modified Spectral DY Conjugate Gradient MethodThe new method is introduced

119889119896=

minus119892119896 if 119896 = 1

minus120579119896119892119896+ 120573119896119889119896minus1 if 119896 ge 1

(59)

6 Mathematical Problems in Engineering

where 120579119896= 1 + 119892

119896minus121198921198962 and 120573

119896= 120573

DY119896

This method iscalled the MDY method

Algorithm 1 Consider the following

Step 1 The following are given 1199090 isin 119877119899 120576 gt 0 and the max

iteration step119873max

Step 2 Set 119896 = 1 compute 1198891 = minus1198921 = minusnabla119891(1199090) if 1198921 le 120576then stop

Step 3 Find 120572119896according to Wolfe type line search rule

Step 4 Let 119909119896+1 = 119909119896+120572119896119889119896 and 119892119896+1 = 119892(119909119896+1) if 119892119896+1 le 120576

then stop

Step 5 Compute 119889119896+1 by (59)

Step 6 Set 119896 = 119896 + 1 go to Step 3

42 Convergence Analysis From the above the MDY con-jugate gradient method provides a descent direction for theobjective function In the following we give the convergenceof this algorithmwith theWolfe type line search rules Firstlywe introduce some assumptions

(i) The level setΩ = 119909 isin 119877119899| 119891(119909) le 119891(1199091) is bounded

(ii) In some neighborhood 119873 of Ω 119891(119909) is continuouslydifferentiable and its gradient is Lipschitz continuousnamely there exists a constant 119871 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus119910

1003817100381710038171003817 forall119909 119910 isin 119873 (60)

It is implied from the above assumptions that there exist twopositive constants 120573 and 120574 such that

119909 le 119861

1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120574

forall119909 isin Ω

(61)

TheWolfe type line search [29] is

119891 (119909119896) minus119891 (119909

119896+120572119896119889119896) ge 120575120572

2119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(62)

119892 (119909119896+120572119896119889119896)119879

119889119896ge minus 2120575120572

119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(63)

and the strong Wolfe line search is

119891 (119909119896+120572119896119889119896) le 119891 (119909

119896) + 120575120572

119896119892119879

119896119889119896 (64)

100381610038161003816100381610038161003816119892 (119909119896+120572119896119889119896)119879

119889119896

100381610038161003816100381610038161003816le10038161003816100381610038161003816120590119892119879

119896119889119896

10038161003816100381610038161003816 (65)

where 0 lt 120575 lt 120590 lt 1

Lemma 2 Let the sequences 119892119896 and 119889

119896 be generated by the

proposed algorithm (Algorithm 1) with Wolfe type line searchrules then 119892119879

119896119889119896lt 0 and

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (66)

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= 120573DY119896

[minus120579119896119889119896minus1 (119892

119879

119896minus119892119879

119896minus1) + 119892119879

119896119889119896minus1]

= 120573DY119896

[119892119879

119896minus1119889119896minus1 minus

1003817100381710038171003817119892119896minus11003817100381710038171003817

2

119889119896minus1119910119879

119896minus11003817100381710038171003817119892119896

1003817100381710038171003817

2]

le 120573DY119896119892119879

119896minus1119889119896minus1

(67)

If we choose 119896 = 1 then

119892119879

1 1198891 = minus10038171003817100381710038171198921

1003817100381710038171003817

2

lt 0 (68)

according to (65) we have

119910119879

1 1198891 ge (120590 minus 1) 119892119879

1 1198891 ge 0 (69)

So 120573DY2 ge 0 For 119896 gt 1 according to (67) we have

119892119879

119896119889119896lt 0 (70)

by induction So we obtain

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (71)

the proof is completed

Theorem 3 Let the sequences 119892119896 and 119889

119896 be generated

by the proposed algorithm (Algorithm 1) with Wolfe type linesearch rules then

119892119879

119896119889119896le minus119862 119892

1198962 (72)

where 119862 gt 0 is a constant

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= minus1003817100381710038171003817119892119896minus1

1003817100381710038171003817

2

+(119892119879

119896119889119896minus1

119889119879

119896minus1119910119896minus1minus 1) 1003817100381710038171003817119892119896

1003817100381710038171003817

2

le (119892119879

119896minus1119889119896minus1

119889119879

119896minus1119910119896minus1)1003817100381710038171003817119892119896

1003817100381710038171003817

2

(73)

From the Wolfe type line search rules (65) we can obtain

minus (120590 + 1) 119892119879119896minus1119889119896minus1 ge 119889

119879

119896minus1 (119892119896 minus119892119896minus1) gt 0 (74)

according to (73) and (74) we have

119892119879

119896119889119896le minus

1120590 + 1

10038171003817100381710038171198921198961003817100381710038171003817

2

(75)

we define 119862 = 1(120590 + 1) Therefore we can obtain inequality(72)

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

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Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

where 120579119896= 1 + 119892

119896minus121198921198962 and 120573

119896= 120573

DY119896

This method iscalled the MDY method

Algorithm 1 Consider the following

Step 1 The following are given 1199090 isin 119877119899 120576 gt 0 and the max

iteration step119873max

Step 2 Set 119896 = 1 compute 1198891 = minus1198921 = minusnabla119891(1199090) if 1198921 le 120576then stop

Step 3 Find 120572119896according to Wolfe type line search rule

Step 4 Let 119909119896+1 = 119909119896+120572119896119889119896 and 119892119896+1 = 119892(119909119896+1) if 119892119896+1 le 120576

then stop

Step 5 Compute 119889119896+1 by (59)

Step 6 Set 119896 = 119896 + 1 go to Step 3

42 Convergence Analysis From the above the MDY con-jugate gradient method provides a descent direction for theobjective function In the following we give the convergenceof this algorithmwith theWolfe type line search rules Firstlywe introduce some assumptions

(i) The level setΩ = 119909 isin 119877119899| 119891(119909) le 119891(1199091) is bounded

(ii) In some neighborhood 119873 of Ω 119891(119909) is continuouslydifferentiable and its gradient is Lipschitz continuousnamely there exists a constant 119871 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 le 1198711003817100381710038171003817119909 minus119910

1003817100381710038171003817 forall119909 119910 isin 119873 (60)

It is implied from the above assumptions that there exist twopositive constants 120573 and 120574 such that

119909 le 119861

1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120574

forall119909 isin Ω

(61)

TheWolfe type line search [29] is

119891 (119909119896) minus119891 (119909

119896+120572119896119889119896) ge 120575120572

2119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(62)

119892 (119909119896+120572119896119889119896)119879

119889119896ge minus 2120575120572

119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(63)

and the strong Wolfe line search is

119891 (119909119896+120572119896119889119896) le 119891 (119909

119896) + 120575120572

119896119892119879

119896119889119896 (64)

100381610038161003816100381610038161003816119892 (119909119896+120572119896119889119896)119879

119889119896

100381610038161003816100381610038161003816le10038161003816100381610038161003816120590119892119879

119896119889119896

10038161003816100381610038161003816 (65)

where 0 lt 120575 lt 120590 lt 1

Lemma 2 Let the sequences 119892119896 and 119889

119896 be generated by the

proposed algorithm (Algorithm 1) with Wolfe type line searchrules then 119892119879

119896119889119896lt 0 and

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (66)

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= 120573DY119896

[minus120579119896119889119896minus1 (119892

119879

119896minus119892119879

119896minus1) + 119892119879

119896119889119896minus1]

= 120573DY119896

[119892119879

119896minus1119889119896minus1 minus

1003817100381710038171003817119892119896minus11003817100381710038171003817

2

119889119896minus1119910119879

119896minus11003817100381710038171003817119892119896

1003817100381710038171003817

2]

le 120573DY119896119892119879

119896minus1119889119896minus1

(67)

If we choose 119896 = 1 then

119892119879

1 1198891 = minus10038171003817100381710038171198921

1003817100381710038171003817

2

lt 0 (68)

according to (65) we have

119910119879

1 1198891 ge (120590 minus 1) 119892119879

1 1198891 ge 0 (69)

So 120573DY2 ge 0 For 119896 gt 1 according to (67) we have

119892119879

119896119889119896lt 0 (70)

by induction So we obtain

120573DY119896

le119892119879

119896119889119896

119892119879

119896minus1119889119896minus1 (71)

the proof is completed

Theorem 3 Let the sequences 119892119896 and 119889

119896 be generated

by the proposed algorithm (Algorithm 1) with Wolfe type linesearch rules then

119892119879

119896119889119896le minus119862 119892

1198962 (72)

where 119862 gt 0 is a constant

Proof According to (59) we have

119892119879

119896119889119896= minus 120579119896

10038171003817100381710038171198921198961003817100381710038171003817

2

+120573DY119896119892119879

119896119889119896minus1

= minus1003817100381710038171003817119892119896minus1

1003817100381710038171003817

2

+(119892119879

119896119889119896minus1

119889119879

119896minus1119910119896minus1minus 1) 1003817100381710038171003817119892119896

1003817100381710038171003817

2

le (119892119879

119896minus1119889119896minus1

119889119879

119896minus1119910119896minus1)1003817100381710038171003817119892119896

1003817100381710038171003817

2

(73)

From the Wolfe type line search rules (65) we can obtain

minus (120590 + 1) 119892119879119896minus1119889119896minus1 ge 119889

119879

119896minus1 (119892119896 minus119892119896minus1) gt 0 (74)

according to (73) and (74) we have

119892119879

119896119889119896le minus

1120590 + 1

10038171003817100381710038171198921198961003817100381710038171003817

2

(75)

we define 119862 = 1(120590 + 1) Therefore we can obtain inequality(72)

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Lemma 4 Suppose that assumptions (i) and (ii) hold let 119892119896

and 119889119896 be generated by the proposed algorithm (Algorithm 1)

with Wolfe type line search rules (62) and (63) then one has

sum

119896ge0

(119892119879

119896119889119896)2

10038171003817100381710038171198891198961003817100381710038171003817

2le infin (76)

Proof From assumption (ii) we have100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

minus119892 (119909119896)119879100381710038171003817100381710038171003817le 119871

10038171003817100381710038171205721198961198891198961003817100381710038171003817 (77)

so we can obtain100381710038171003817100381710038171003817119892 (119909119896+120572119896119889119896)119879

119889119896minus119892 (119909

119896)119879

119889119896

100381710038171003817100381710038171003817le 119871120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

(78)

From inequality (63) we obtain

2120590120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2

ge minus119892 (119909119896+120572119896119889119896)119879

119889119896 (79)

According to inequalities (78) and (79) we have

(2120590+119871) 120572119896

10038171003817100381710038171198891198961003817100381710038171003817

2ge minus119892119879

119896119889119896 (80)

So we can obtain

120572119896

10038171003817100381710038171198891198961003817100381710038171003817 ge

12120590 + 119871

(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) (81)

Owing to inequality (81) we have

infin

sum

119896=1(minus119892119879

119896119889119896

10038171003817100381710038171198891198961003817100381710038171003817

) le (2120590+119871)2infin

sum

119896=11205722119896

10038171003817100381710038171198891198961003817100381710038171003817

2

le (2120590+119871)2infin

sum

119896=1119891 (119909119896) minus119891 (119909

119896+1)

lt +infin

(82)

From (82) we can obtain inequality (76)

Theorem 5 Suppose that assumptions (i) and (ii) hold Let119892119896 and 119889

119896 be generated by the algorithm with Wolfe type

line search rules Then one has

lim inf119896rarrinfin

10038171003817100381710038171198921198961003817100381710038171003817 = 0 (83)

Proof By contradiction we suppose that there exists a posi-tive constant 120574 gt 0 such that

10038171003817100381710038171198921198961003817100381710038171003817 ge 120574 (84)

holds for all 119896 gt 1 From (59) we have

10038171003817100381710038171198891198961003817100381710038171003817

2= minus 2120579

119896119889119879

119896119892119896+ (120573

DY119896)2 1003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2 (85)

According to Lemma 4 and the above equation we have

10038171003817100381710038171198891198961003817100381710038171003817

2le (

119892119879

119896119889119896

119892119879

119896minus1119889119896minus1)

21003817100381710038171003817119889119896minus1

1003817100381710038171003817

2minus 120579

2119896

10038171003817100381710038171198921198961003817100381710038171003817

2minus 2120579119896119889119879

119896119892119896 (86)

Multiplying the above inequality by 1(119892119879119896119889119896)2 we have

10038171003817100381710038171198891198961003817100381710038171003817

2

(119892119879

119896119889119896)2 le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 minus(

120579119896

10038171003817100381710038171198921198961003817100381710038171003817

119892119879

119896119889119896

+1

10038171003817100381710038171198921198961003817100381710038171003817

)

2

+1

10038171003817100381710038171198921198961003817100381710038171003817

2le

1003817100381710038171003817119889119896minus11003817100381710038171003817

2

(119892119879

119896minus1119889119896minus1)2 +

11003817100381710038171003817119892119896

1003817100381710038171003817

2

le

119896

sum

119894=1

11003817100381710038171003817119892

2119894

1003817100381710038171003817

2le

11205742 119896

(87)

Then from inequality (83) we have

infin

sum

119896=1

(119892119879

119896119889119896)2

10038171003817100381710038171198921198961003817100381710038171003817

2ge 120574

2infin

sum

119896=1

1119896= infin (88)

which contradicts (76) Therefore (83) holds

5 Numerical Simulation Experiments

In order to illustrate the validity of the IADM with opti-mization method we introduce the one-phase inverse Stefanproblem in Example 1 From the literature [9] we know theinverse Stefan problem consists in the calculation of thetemperature distribution in the domain and in the recon-struction of the temperature distribution on the boundarywhen the position of the functionwhich describes themovinginterface is known And then in Example 2 we solve the two-phase inverse Stefan problem based on the hybrid methodwhich combines the Adomian decomposition method withthe homotopy perturbation method

Example 1 In this example the one-phase inverse Stefanproblem [9] is taken for the following values of parameters1205721 = 1 1198961 = 1 119906lowast = 1 119905lowast = 12 120581 = 11205721 1205931(119909) = exp(1205721119909)and 1199061198791 (119909) = exp(12057212 minus 119909) Then the exact solution of suchformulated one-phase Stefan problem can be found from thefollowing functions

1199061 (119909 119905) = exp (1205721119905 minus 119909)

1205791 (119905) = exp (1205721119905)

120585 (119905) = 1205721119905

(89)

The following equation

120601119894(119905) = exp (minus (119894 minus 1) 119905) 119894 = 1 2 119898 (90)

is taken as basis function The approximate solutions arecompared with the exact solution and the values of theabsolute errors are calculated from

120575ℎ= int

119905lowast

0[ℎ119890(119905) minus ℎ

119903(119905)] 119889119905

12

120575119906= intint

119863

[119906119890(119909 119905) minus 119906

119899(119909 119905)] 119889119909 119889119905

12

(91)

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

where ℎ(119905) isin 120579(119905) 120585(119905) (120579(119905) stands for the boundarycondition on Γ1 in the literature [9] and 120585(119905) is the movinginterface) 119863 is the domain of the temperature distribu-tion ℎ

119890(119905) is the exact value of function ℎ(119905) ℎ

119903(119905) is the

approximate value of function ℎ(119905) 119906119890(119909 119905) is the exact

distribution of the temperature in the domain and 119906119899(119909 119905)

is the reconstructed distribution of temperature Percentagerelative errors are calculated from

Δℎ=

120575ℎ

radicint119905lowast

0 [ℎ119890(119905)]

2119889119905

times 100

Δ119906=

120575119906

radicintint119863[119906119890(119909 119905)]

2119889119909 119889119905

times 100(92)

The comparison of the ADM and the IADM for recon-structing distribution of the temperature on boundary Γ1 inthe literature [9] is shown in Figure 3 for 119899 = 1 119899 = 2 and119898 = 2 119898 = 3 From Figure 3 the method based on theimproved Adomian decompositionmethod (IADM) is betterthan the Adomian decomposition method (ADM)

Example 2 This example is introduced to illustrate the effec-tiveness of the hybrid method which combines the Adomiandecomposition method with the homotopy perturbationmethod which is presented in the previous sections Here weconsider an example of the two-phase inverse Stefan problem[12] in which 1205721 = 52 1205722 = 54 1198961 = 6 1198962 = 2 120581 = 45119906lowast= 1 119905lowast = 1 119889 = 3 119904 = 32 and 1205931(119909) = 119890

(3minus2119909)101205932(119909) = 119890

(3minus2119909)5 1205791(119905) = 119890(119905+3)10 and 1199061198791 (119909) = 119890

(4minus2119909)10The exact solution of this inverse Stefan problem is describedby the following functions

1199061 (119909 119905) = 119890(119905minus2119909+3)10

1199062 (119909 119905) = 119890(119905minus2119909+3)5

120579 (119905) = 119890(119905minus3)5

120585 (119905) =(119905 + 3)

2

(93)

First we use HPM with optimization method to solve1199061(119909 119905) and 120585(119905) As the initial approximation 11990610 this fulfillsthe initial condition

11990610 (119909 119905) = 119890(3minus2119909)10

(94)

By using the function 11990610 in (39) and the conditions (41) and(42) the following system of equations can be obtained

120597211990611 (119909 119905)

1205971199092= minus

125119890(3minus2119909)10

(95)

With the conditions

11990611 (0 119905) = 119890310

(119890(11990510)minus1

)

11990611 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 1minus 119890(3minus2120585(119905))10(96)

this system of equations can be solved And the solution ofthis system is shown in the following

11990611 (119909 119905) = minus 119890310minus1199095

+1 minus 119890(3+119905)10

120585 (119905)119909 + 119890(119905+3)10

(97)

Next we designate the functions 1199061119894 for 119894 ge 2 in thefollowing

12059721199061119894

1205971199092=251205971199061119894minus1

120597119905 (98)

and the conditions are shown

1199061119894 (0 119905) = 0

1199061119894 (119909 119905)1003816100381610038161003816119909=120585(119905)

= 0(99)

By using (98) and the boundary conditions (99) we derive1199061119894(119909 119905) for 119894 ge 2 according to (45) the approximate solution1199061(119909 119905) is obtained Here we define basis functions accordingto (46)

120595119894(119905) = 119905

119894minus1119894 = 1 119898 (100)

We use the modified spectral DY (MDY) conjugate gradientmethod to solve function (47) The solution of moving inter-face 120585(119905) is shown in Figure 4 So the temperature 1199061(119909 119905) isderived in domain1198631 Values of the error in reconstruction ofthe position of the moving interface 120585(119905) and the temperaturedistribution 1199061(119909 119905) are shown in Table 1

In order to verify the validity of the modified spectralDY (MDY) conjugate gradient method we choose 119899 = 2119898 = 2 120576 = 10minus4 the initial point 1198880 = (22 25) and themax step number119873max = 100 The comparison results of DYand MDY are shown in Figure 5 and Table 2 Comparing toDY conjugate gradient method the MDY conjugate gradientmethod is slightly better

The moving interface 120585(119905) and 1199061(119909 119905) in domain 1198631are obtained by the homotopy perturbation method withoptimization Next with the help of condition (16) we candetermine the temperature 1199062(119909 119905) in domain 1198632 by theimproved Adomian decomposition method (IADM) withoptimization Without loss of generality we choose the basisfunctions

120601119894(119905) = 119905

119894minus1 119894 = 1 119898 (101)

The exact and the reconstructed distribution of thetemperature on boundary Γ4 are shown in Figure 6 fordifferent number of basis functions 120601

119894(119905) And the errors in

reconstruction of the temperature 120579(119905) (on boundary Γ4 inFigure 2) and the temperature distribution 1199062(119909 119905) are shownin Table 3The inversion results of the function 120579(119905)match theexact value very well

The measured temperature 1199061198791 (119909) at the final time 119905lowast onthe boundary Γ6 in Figure 2 is perturbed by the randomerror So in the next section we consider the influence ofmeasurement errors on the results We set the measured

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 1 Values of the error in reconstruction of the position of the moving interface and the temperature distribution 1199061(119909 119905)

HPM 120575120585

Δ120585

1205751199061

Δ1199061

119899 = 2 119898 = 2 0108712413045 061599738364 0116978666115 0109525848228119899 = 2 119898 = 3 0210642147214 119356205914 0299360015653 0280470260729119899 = 2 119898 = 3 0015061910569 008534533677 0102526697222 0096057215386

Exact

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

The ADM (m = 2 n = 1)The IADM (m = 2 n = 1)

(a)

0 01 02 03 04 05

08

1

12

14

16

18

2

t

(t)

ExactThe ADM (m = 2 n = 2)The IADM (m = 2 n = 2)

(b)

0 01 02 03 04 05

t

(t)

ExactThe ADM (m = 3 n = 2)The IADM (m = 3 n = 2)

09

1

11

12

13

14

15

16

17

18

(c)

Figure 3 The reconstructing distribution of the temperature on boundary (Γ1 in the literature [9]) for 119899 = 1 119899 = 2 and119898 = 2119898 = 3

temperature 1199061198791 (119909)with an error of 1 and 3We can obtainthe moving interface 120585(119905) and 120579(119905) the results are shown inFigures 7 and 8

In Figures 7 and 8 the exact and the reconstructeddistribution of the temperature on moving interface 120585(119905) and

the boundary Γ4 are shown and the errors in reconstructionof the function describing themoving interface and boundaryconditions are compiled in Table 4 The obtained recon-struction results illustrate that the functions 120585(119905) and 120579(119905)

are reconstructed very well From the results the hybrid

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 102 04 06 08

t

Exact

15

16

17

18

19

2

21

22120585(t)

HPM (n = 2 m = 2)

(a)

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

ExactHPM (n = 2 m = 3)

(b)

0 102 04 06 08

t

15

14

16

17

18

19

2

120585(t)

ExactHPM (n = 3 m = 2)

(c)

Figure 4 The position of moving interface 120585(119905)

Table 2 The comparison results of DY and MDY

Algorithm Iteration numbers Optimization valueDY 50 (152 048)MDY 23 (151 0501)

method with optimization is stable with the input errorsWhen the input data are contaminated by the errors the errorof the reconstructed boundary conditions does not exceed theerror of the input data

6 Conclusion

Based on the homotopy perturbation method (HPM) withthe improved Adomian decomposition method (IADM) ahybrid method with optimization is proposed to solve thetwo-phase inverse Stefan problem This problem is morerealistic from the practical point of view The simulationexperiment is used to verify this method Experimentalresults match the exact value very well Furthermore thisinvestigated method can also be used for solving one-phasedirect Stefan problems

Optimization method is very crucial in this paperso a modified spectral DY conjugate gradient method is

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 3 Values of the error in reconstruction of the temperature 120579(119905) on boundary Γ4 in Figure 2 and the temperature distribution 1199062(119909 119905)

IADM 120575120579(119905)

Δ120579(119905)

1205751199062(119909119905)

Δ1199062(119909119905)

119899 = 2 119898 = 2 00125536846 020525000280 0649823007 05444948151119899 = 2 119898 = 3 000907338187 014834781217 0717820233 06014705398

Table 4 Values of the error in reconstruction the moving interface 120585(119905) and the temperature 120579(119905) on boundary Γ4 in Figure 2

Perturbation error on the 1199061198791 (119909) 120575120585(119905)

Δ120585(119905)

120575120579(119905)

Δ120579(119905)

1 error 00968150593 0771949532 0032490326 074750267273 error 03725894970 29708217908 0061914613 14244652005

0 20 40 60 80 1000

002

004

006

008

01

012

014

016

018

02

Iteration times

Cos

t fun

ctio

n va

lue (

J)

DYMDY

Figure 5 The cost function 1198691 changes with the increasing ofiteration numbers

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

ExactIADM (n = 2 m = 2)IADM (n = 2 m = 3)

Figure 6 The reconstructing distribution of the temperature 120579(119905)on boundary Γ4 in Figure 2

Exact

0 102 04 06 08

t

15

16

17

18

19

2

21

22

120585(t)

HPM (n = 3 m = 2) with 1 input errorHPM (n = 3 m = 2) with 3 input error

Figure 7 The reconstructed position of moving interface 120585(119905)

Exact

054

056

058

06

062

064

066

068

0 102 04 06 08

t

120579(t)

IADM (n = 2 m = 2) with 1 input errorIADM (n = 2 m = 2) with 3 input error

Figure 8The reconstructing distribution of the temperature 120579(119905) onboundary Γ4 in Figure 2

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

presented Andwe give the convergence of thisMDYmethodSimulation experiment illustrates the validity of this opti-mization method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (60974091 61333006 and 61473074)and the Fundamental Research Funds for the Central Uni-versities (N120708001)

References

[1] A Fic A J Nowak and R Bialecki ldquoHeat transfer analysisof the continuous casting process by the front tracking BEMrdquoEngineering Analysis with Boundary Elements vol 24 no 3pp 215ndash223 2000

[2] C A Santos J A Spim andAGarcia ldquoMathematicalmodelingand optimization strategies (genetic algorithm and knowledgebase) applied to the continuous casting of steelrdquo EngineeringApplications of Artificial Intelligence vol 16 no 5-6 pp 511ndash5272003

[3] R Kamal A Hojatollah A R Jamal and P Kourosh ldquoApplica-tion of mesh-free methods for solving the inverse one dimen-sional Stefan problemrdquo Engineering Analysis with BoundaryElements vol 40 pp 1ndash21 2014

[4] D Slota ldquoDirect and inverse one-phase Stefan problem solvedby the variational iteration methodrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1139ndash1146 2007

[5] B T Johansson D Lesnic and T Reeve ldquoA method offundamental solutions for the one-dimensional inverse Stefanproblemrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol35 no 9 pp 4367ndash4378 2011

[6] D Slota ldquoThe application of the homotopy perturbationmethod to one-phase inverse Stefan problemrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 6 pp587ndash592 2010

[7] R Grzymkowski M Pleszczynski and D Slota ldquoComparingthe Adomian decomposition method and the Runge-Kuttamethod for solutions of the Stefan problemrdquo InternationalJournal of Computer Mathematics vol 83 no 4 pp 409ndash4172006

[8] R Grzymkowski and D Slota ldquoStefan problem solved byAdomian decomposition methodrdquo International Journal ofComputer Mathematics vol 82 no 7 pp 851ndash856 2005

[9] R Grzymkowski and D Slota ldquoOne-phase inverse Stefan prob-lem solved by adomian decomposition methodrdquo Computers ampMathematics with Applications vol 51 no 1 pp 33ndash40 2006

[10] E Hetmaniok D Słota R Wituła and A Zielonka ldquoCompari-son of the Adomian decomposition method and the variationaliteration method in solving the moving boundary problemrdquoComputers amp Mathematics with Applications vol 61 no 8 pp1931ndash1934 2011

[11] B T JohanssonD Lesnic andT Reeve ldquoAmeshlessmethod foran inverse two-phase one-dimensional linear Stefan problemrdquo

Inverse Problems in Science and Engineering vol 21 no 1 pp17ndash33 2013

[12] D Słota ldquoHomotopy perturbation method for solving the two-phase inverse stefan problemrdquo Numerical Heat Transfer Part AApplications vol 59 no 10 pp 755ndash768 2011

[13] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[14] J-H He ldquoAn elementary introduction to the homotopy pertur-bation methodrdquo Computers amp Mathematics with Applicationsvol 57 no 3 pp 410ndash412 2009

[15] C Chun H Jafari and Y-I Kim ldquoNumerical method for thewave and nonlinear diffusion equations with the homotopyperturbation methodrdquo Computers ampMathematics with Applica-tions vol 57 no 7 pp 1226ndash1231 2009

[16] A Yildirim ldquoAnalytical approach to fractional partial differen-tial equations in fluidmechanics bymeans of the homotopy per-turbation methodrdquo International Journal of Numerical Methodsfor Heat amp Fluid Flow vol 20 no 2 pp 186ndash200 2010

[17] E Hetmaniok I Nowak D Slota R Witula and A ZielonkaldquoSolution of the inverse heat conduction problem with neu-mann boundary condition by using the homotopy perturbationmethodrdquoThermal Science vol 17 no 3 pp 643ndash650 2013

[18] J Biazar and H Ghazvini ldquoNumerical solution for special non-linear Fredholm integral equation byHPMrdquoAppliedMathemat-ics and Computation vol 195 no 2 pp 681ndash687 2008

[19] X Zhang J Zhao J Liu and B Tang ldquoHomotopy perturbationmethod for two dimensional time-fractional wave equationrdquoApplied Mathematical Modelling vol 38 no 23 pp 5545ndash55522014

[20] M Ghasemi M Tavassoli Kajani and E Babolian ldquoApplicationof Hersquos homotopy perturbation method to nonlinear integro-differential equationsrdquo Applied Mathematics and Computationvol 188 no 1 pp 538ndash548 2007

[21] A Golbabai and M Javidi ldquoApplication of Hersquos homotopy per-turbationmethod for 119899 th-order integro-differential equationsrdquoAppliedMathematics and Computation vol 190 no 2 pp 1409ndash1416 2007

[22] J Biazar Z Ayati andM R Yaghouti ldquoHomotopy perturbationmethod for homogeneous Smoluchowskirsquos equationrdquo Numeri-cal Methods for Partial Differential Equations vol 26 no 5 pp1146ndash1153 2010

[23] A Alawneh K Al-Khaled and M Al-Towaiq ldquoReliable algo-rithms for solving integro-differential equations with applica-tionsrdquo International Journal of Computer Mathematics vol 87no 7 pp 1538ndash1554 2010

[24] H Aminikhah and J Biazar ldquoA new analytical method forsolving systems of Volterra integral equationsrdquo InternationalJournal of Computer Mathematics vol 87 no 5 pp 1142ndash11572010

[25] J Biazar B Ghanbari M G Porshokouhi and M G Por-shokouhi ldquoHersquos homotopy perturbation method a stronglypromising method for solving non-linear systems of the mixedVolterra-Fredholm integral equationsrdquo Computers amp Mathe-matics with Applications vol 61 no 4 pp 1016ndash1023 2011

[26] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press New York NY USA1983

[27] M Almazmumy F A Hendi H O Bakodah and H AlzumildquoRecent modifications of Adomian decomposition methodfor initial value problem in ordinary differential equationsrdquo

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

American Journal of Computational Mathematics vol 2 no 3pp 228ndash234 2012

[28] D Lesnic ldquoConvergence of Adomianrsquos decomposition methodperiodic temperaturesrdquo Computers amp Mathematics with Appli-cations vol 44 no 1-2 pp 13ndash24 2002

[29] W Cao K R Wang and Y L Wang ldquoGlobal convergence ofa modified spectral CD conjugate gradient methodrdquo Journal ofMathematical Research amp Exposition vol 31 pp 261ndash268 2011

[30] E G Birgin and J M Martłnez ldquoA spectral conjugate gradientmethod for unconstrained optimizationrdquo Applied Mathematicsand Optimization vol 43 no 2 pp 117ndash128 2001

[31] L Zhang W J Zhou and D H Li ldquoGlobal convergence ofa modified Fletcher-Reeves conjugate gradient method withArmijo-type line searchrdquo Numerische Mathematik vol 104 no4 pp 561ndash572 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of