research article a new piecewise-spectral homotopy analysis method for solving chaotic...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 583193 13 pageshttpdxdoiorg1011552013583193

Research ArticleA New Piecewise-Spectral Homotopy Analysis Method forSolving Chaotic Systems of Initial Value Problems

H Saberi Nik1 Sohrab Effati1 Sandile S Motsa2 and Stanford Shateyi3

1 Department of Applied Mathematics Ferdowsi University of Mashhad Mashhad Iran2 School of Mathematics Computer Science and Statistics University of KwaZulu-Natal Private Bag X01 ScottsvillePietermaritzburg 3209 South Africa

3 Department of Mathematics University of Venda P Bag X5050 Thohoyandou 0950 South Africa

Correspondence should be addressed to Stanford Shateyi stanfordshateyiunivenacza

Received 30 September 2012 Revised 14 February 2013 Accepted 25 February 2013

Academic Editor Trung NguyenThoi

Copyright copy 2013 H Saberi Nik et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An accurate algorithm for solving initial value problems (IVPs) which are highly oscillatory is proposed The proposed method isbased on a novel technique of extending the standard spectral homotopy analysis method (SHAM) and adapting it to a sequence ofmultiple intervals In this new application the method is referred to as the piecewise spectral homotopy analysis method (PSHAM)The applicability of the proposed method is examined on the differential equation system modeling HIV infection of CD4+ Tcells and the Genesio-Tesi system which is known to be chaotic and highly oscillatory Also for the first time we present herea convergence proof for SHAM We treat in detail Legendre collocation and Chebyshev collocation The method is compared toMATLABrsquos ode45 inbuilt solver as a measure of accuracy and efficiency

1 Introduction

This paper introduces a new method for solving highlyoscillatory and chaotic initial value problems (IVPs) Thenew method extends for the first time the application ofthe spectral homotopy analysis method [1 2] to IVPs Thespectral homotopy analysis method was developed by Motsaet al [1 2] for solving nonlinear boundary value problems(BVPs) over finite intervals It has been successfully beenapplied to other BVPs arising mainly in fluid mechanics-related problems [3ndash6] In the previously cited applicationsthe SHAM method was applied to problems which possesssmooth solutions over small regions For rapidly oscillatingchaotic systems over very large regions the SHAM may notgive accurate resultsThe current work seeks to develop a newmethod that will be valid for rapidly changing solutions overall regions small medium- and large sized A simple way ofensuring the validity of the approximations for large intervalsand for all functions is to determine the solution in a sequenceof equal intervals which are subject to continuity conditionsat the end points of each interval

Recently in an effort to increase the radius of convergenceof some analytical methods of approximations multistage orpiecewise approximations have been developed for solvingIVPs over general intervalsThismultistage approach seeks toimplement the standard approximationmethodon sequencesof subintervals whose union makes up the domain of theunderlying problem The effect of this piece-wise (multistep)approach is to accelerate the convergence of the approximatesolution over a large region and to improve the accuracyof the parent approximate method of solution particularlyover rapidly oscillating regions Examples include the mul-tistage homotopy analysis method [7] piecewise homotopyperturbation methods [8] multistage Adomian decomposi-tion method [9 10] multistage differential transformationmethod [11 12] and multistage variational iteration method[13] Because these methods attempt to obtain analyticalsolutions at each interval they involve time-consuming andtedious computational operations and if too many smallintervals are considered as may be the case when dealingwith highly oscillatory systems the analytical integrationprocess will be too much to handle even with the use of

2 Mathematical Problems in Engineering

symbolic scientific software For this reason focus is nowshifted towards multistage methods which use numericalintegration techniques such as the piecewise iterationmethod[14] which uses spectral collocation methods

This work examines the applicability of a method thatblends the piecewise implementation technique with thespectral homotopy analysis method For this reason theproposed method is called the piecewise homotopy analysismethod (PSHAM) The validity of the method is testedagainst two initial value problems of practical importancenamely the mathematical models of HIV infection of CD4

+

T cells and the Genesio-Tesi system Chaotic dynamics ofthe Genesio-Tesi system were introduced in [15] Since thenthe system has been used as one of the benchmarks fortesting the validity of new and existing methods of solvingIVPs Approximate analytical or numerical methods thathave been used to find solutions of the Genesio-Tesi systeminclude the differential transformmethod [16] the piecewise-spectral parametric iteration method [14] variational itera-tion method (VIM) and the multistage VIM [17]

Mathematical models play a significant role in studyingepidemic and viral dynamics Viral models are valuable tohelp us improve the understanding of both diseases and var-ious drug therapy strategies against them In 1993 Perelsonet al [18] proposed an ODE model of cell-free viral spreadof human immunodeficiency virus (HIV) in a well-mixedcompartment such as the bloodstream This model consistsof four components uninfected healthy CD4

+ T cells latentlyinfected CD4

+ T cells actively infected CD4+ T cells and

free virus This model has been important in the field ofmathematical modelling of HIV infection and many othermodels have beenproposedwhich take themodel of Perelsonet al [18] as their inspiration For numerically solving amodelfor HIV infection of CD4

+ T cells Ongun [19] has appliedthe Laplace Adomian decomposition method Merdan hasused the homotopy perturbation method [20] and Merdanet al [21] have applied the Pade approximation and themodified variational iteration method [22] More recentlyYuzbasi [23] used the Bessel collocation method for solvingthe same problem It is worth noting that in all these studiesresults were given for solutions over small time intervalsThus it may be concluded that these methods only offera good approximation to the true solution in a very smallregions Liao [24] introduced the homotopy analysis method(HAM) which has convergence controlling parameters toimprove accuracy of series solutions and increase the regionof convergence The HAM was successfully used to solveseveral nonlinear IVPs [24ndash28] However even the standardHAM approach may not be suitable to resolve solutionsover very large intervals and for rapidly oscillating systemswhich show chaotic behaviour For practical purposes resultsobtained using the previously mentioned analytical methodsof approximation may not be very useful since in additionto the short-term behaviour of some systems long-termbehaviour of the solutions may be required and insights intochaotic systems may be sought The multistage approachsuch as the one discussed in this paper presents an importantstrategy of addressing the limitations of some of the previ-ously cited methods of solution

The rest of this paper is organized as follows In Section 2the basic description of the SHAM is presented Section 3presents existence and uniqueness of solution of SHAMthat give a guarantee of convergence of SHAM Section 4deals with the description and formulation of the piecewise-SHAM In Section 5 the numerical implementation of thePSHAM to the Genesio-Tesi system and HIV infection ofCD4+ T cells models is considered In Section 6 we present

the main results and discussion of the numerical simulationsand finally the conclusion is given in Section 7

2 Basic Idea behind the Spectral HomotopyAnalysis Method (SHAM)

For convenience of the interested reader wewill first present abrief description of the basic idea behind the standard SHAM[1 2] This will be followed by a description of the piecewiseversion of the SHAM algorithm which is suitable for solvinginitial value problems To this end we consider the initialvalue problem (IVP) of dimension 119899 given as

y (119905) = f (119905 y (119905)) y (1199050) = y0 (1)

y R 997888rarr R119899

f R timesR119899

997888rarr R119899

(2)

where the dot denotes differentiation with respect to 119905 Wemake the usual assumption that f is sufficiently smooth forlinearization techniques to be valid If y = (119910

1 1199102 119910

119899) we

can apply the SHAM by rewriting (1) as

L119903[119910119903(119905)] +F

119903[119910119903(119905)] = 120601

119903(119905) (3)

where L119903is a linear operator which is derived from the

entire linear part of (1) and F119903is the remaining nonlinear

component for 119903 = 1 2 119899The SHAM approach imports the conventional ideas of

the standard homotopy analysis method (HAM) by definingthe following zeroth-order deformation equations

(1 minus 119902)L119903[119884119903(119905 119902) minus 119910

1199030(119905)]

= 119902ℏ119903119867119903(119905) N

119903[119884119903(119905 119902)] minus 120601

119903(119905)

(4)

where 119902 isin [0 1] is an embedding parameter 119884119903(119905 119902) are

unknown functions and ℏ119903and 119867

119903(119905) are convergence con-

trolling parameters and functions respectivelyThenonlinearoperatorN

119903is defined as

N119903[119884119903(119905 119902)] = L

119903[119884119903(119905 119902)] +F

119903[119884119903(119905 119902)] (5)

Using the ideas of the standard HAM approach (see eg[24ndash28]) we differentiate the zeroth-order equations (4) 119898

times with respect to 119902 and then set 119902 = 0 and finallydivide the resulting equations by 119898 to obtain the followingequations which are referred to as the 119898th order (or higherorder) deformation equations

L119903[119910119903119898

(119905) minus 120594119898119910119903119898minus1

(119905)] = ℏ119903119867119903(119905) 119877119903119898minus1

(6)

Mathematical Problems in Engineering 3

where

119877119903119898minus1

=1

(119898 minus 1)

120597119898minus1

119884119903(119905 119902) minus 120601

119903(119905)

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

120594119898

= 0 119898 ⩽ 1

1 119898 gt 1

(7)

After obtaining solutions for (6) the approximate solu-tion for each 119910

119903(119905) is determined as the series solution

119910119903(119905) = 119910

1199030(119905) + 119910

1199031(119905) + 119910

1199032(119905) + sdot sdot sdot (8)

A HAM solution is said to be of order 119872 if the previousseries is truncated at119898 = 119872 that is if

119910119903(119905) =

119872

sum

119898=0

119910119903119898

(119905) (9)

It should be emphasized that the HAM and SHAMapproach will be equivalent if the same linear operator Lis used In the HAM implementation the linear operatoris usually chosen in such a way that the higher orderdeformation equations (6) can be solved analytically and theresulting solution conforms to the so-called rule of coefficientergodicity and a predefined rule of solution expression [24] Ifthese requirements are adhered to then equations of the form(6) constitute a system of decoupled differential equationswhich can easily be integrated especially with the use ofsymbolic scientific software such as MATLAB MAPLE andMathematica If the HAM solution does not converge to thetrue solution of the underlying problem the convergence con-trolling parameters ℏ

119903and119867

119903(119905)may be adjusted to improve

convergence In some cases even this intervention will not beenough to guarantee convergenceThe SHAMapproach seeksto remove all of the restriction of theHAMbymaking it moreflexible In the SHAM application the governing differentialequations are just separated into the linear and nonlinearcomponents as in (3)The linear operator is just chosen usingthe linear part of the equation The penalty for relaxing therestrictions is that the resulting higher order deformationequations will not be solvable analytically That is where theldquoSrdquo in the SHAM comes in Spectral collocation methodsare used to solve the higher order deformation equationsA suitable initial guess to start off the SHAM algorithm isobtained by solving the linear part of (3) subject to the giveninitial conditions that is we solve

L119903[1199101199030

(119905)] = 120601119903(119905) 119910

1199030(0) = 119910

0

119903 (10)

If (10) cannot be solved exactly the spectral collocationmethod is used as a means of solution The solution 119910

1199030(119905) of

(10) is then fed to (6) which is iteratively solved for119910119903119898

(119905) (for119898 = 1 2 3 119872)

3 Convergence Theorem of the SpectralHomotopy Analysis Method

In this section we give some definitions and theorems thatexplain the convergence properties of the SHAM from a

functional analysis framework [29] We remark that theNewtonrsquos iteration approach was used in [22] to address thenonlinearities whereas in this work we use the homotopyanalysismethod approach to decompose the nonlinear termsIn addition our proposed method is applied in a series ofsubintervals as opposed to the entire interval

We begin by giving a brief description of the propertiesof the Legendre polynomials followed by the existence anduniqueness properties of the solution of the SHAM and itrsquosconvergence properties

31 Properties of Shifted Legendre Polynomials The well-known Legendre polynomials are defined on the interval(minus1 1) and can be determined with the aid of the followingrecurrence formula [30]

T0(119909) = 1 T

1(119909) = 119909

T119895+1

(119909) =2119895 + 1

119895 + 1119909T119895(119909) minus

119895

119895 + 1T119895minus1

(119909) 119895 ge 1

(11)

In order to use these polynomials on the interval 119909 isin (0 119879)

we defined the so-called shifted Legendre polynomials byintroducing the change of variable 119909 = (2119905119879) minus 1 Let theshifted Legendre polynomials T

119895((2119905119879) minus 1) be denoted by

T119879119895

(119905)ThenT119879119895

(119905) can be generated by using the followingrecurrence relation

T119879119895+1

(119905) =2119895 + 1

119895 + 1(2119905

119879minus 1)T

119879119895(119905)

minus119895

119895 + 1T119879119895minus1

(119905) 119895 = 1 2

(12)

whereT1198790

(119905) = 1 andT1198791

(119905) = (2119905119879)minus1Theorthogonalitycondition is

int

119879

0

T119879119894

(119905)T119879119896

(119905) 119889119905 =119879

2119894 + 1120575119894119896 (13)

where 120575119894119896

is the Kronecker function Any function 119910(119905)square integrable in (0 119879) may be expressed in terms ofshifted Legendre polynomials as

119910 (119905) =

infin

sum

119895=0

119886119895T119879119895

(119905) (14)

where the coefficients 119886119895are given by

119886119895=

2119897 + 1

119879int

119879

0

119910 (119905)T119879119895

(119905) 119889119905 119895 = 0 1 2 (15)

In practice only the first (119873 + 1) terms of shifted Legendrepolynomials are considered Hence we can write

119910 (119905) =

119873

sum

119895=0

119886119895T119879119895

(119905) (16)

Now we turn to Legendre-Gauss interpolationWe denote by119905119873

119895 0 le 119895 le 119873 the nodes of the standard Legendre-Gauss


interpolation on the interval (minus1 1) The correspondingChristoffel numbers are 120596

119873

119895 0 le 119895 le 119873 The nodes of the

shifted Legendre-Gauss interpolation on the interval (0 119879)

are the zeros ofT119879119873+1

(119905) which are denoted by 119905119873

119879119895 0 le 119895 le

119873 Clearly 119905119873

119879119895= (1198792)(119905

119873

119895+1)The correspondingChristoffel

numbers are 120596119873

119879119895 = (1198792)120596

119873

119895 Let P

119873(0 119879) be the set of all

polynomials of degree at most 119873 Due to the property of thestandard Legendre-Gauss quadrature it follows that for anyΦ isin P

2119873+1(0 119879)

int

119879

0

Φ (119905) 119889119905 =119879

2int

1

minus1

Φ(119879

2(119905 + 1)) 119889119905

=119879

2

119873

sum

119895=0

120596119873

119895Φ(

119879

2(119905119873

119895+ 1))

=

119873

sum

119895=0

120596119873

119879119895Φ(119905119873

119879119895)

(17)

Definition 1 Let (119906 V)119879and ||V||

119879be the inner product and

the norm of space 1198712(0 119879) respectively We introduce the

following discrete inner product and norm

(119906 V)119879119873

=

119873

sum

119895=0

119906 (119905119873

119879119895) V (119905119873

119879119895) 120596119873

119879119895 V

119879119873= (V V)

12

119879119873

(18)

From (17) for anyΦ120595 isin P2119873+120582

(0 119879)

(Φ 120595)119879= (Φ 120595)

119879119873 (19)

where 120582 = minus1 0 1 for the Legendre-Gauss interpolationthe Legendre Gauss-Radau interpolation and the LegendreGauss-Lobatto integration respectively

Moreover for the Legendre-Gauss integration and theLegendre-Gauss-Radau integration

10038171003817100381710038171206011003817100381710038171003817119879

=10038171003817100381710038171206011003817100381710038171003817119879119873

120601 isin P119873(0 119879) (20)

For the Legendre-Gauss-Lobatto integration ||120601||119879

= ||120601||119879119873

usually But formostly used orthogonal systems in [0 119879] theyare equivalent namely for certain positive constants 119888

119897and 1198882

1198881

10038171003817100381710038171206011003817100381710038171003817119879

le10038171003817100381710038171206011003817100381710038171003817119879119873

le 1198882

10038171003817100381710038171206011003817100381710038171003817119879

(21)

32 Existence and Uniqueness of the SHAM Solution Weconsider the initial value problem (IVP) of dimension 119899 givenas

y (119905) = f (119905 y (119905)) y (1199050) = y0

y R 997888rarr R119899

f R timesR119899

997888rarr R119899

(22)

where the dot denotes differentiation with respect to 119905 Wemake the usual assumption that f is sufficiently smooth forlinearization techniques to be valid rewriting (22) as

L [119910 (119905)] +N [119910 (119905)] = 120601 (119905) (23)

where L is a linear operator which is derived from theentire linear part of (22) and N is the remaining nonlinearcomponent

Let us define the nonlinear operatorN and the sequence119884119899infin

119899=0as

N [119910 (119905)] =

infin

sum

119896=0

119873119896(1199100 1199101 119910

119896) (24)

1198840= 1199100

1198841= 1199100+ 1199101

119884119899= 1199100+ 1199101+ 1199102+ sdot sdot sdot + 119910

119899

(25)

Using the ideas of the HAM approach [24] we obtain thefollowing equation which is referred to as the 119898th order (orhigher order) deformation equation

L [119910119898(119905) minus 120594

119898119910119898minus1

(119905)] = ℏ119867 (119905) 119877119898[119910119898minus1

(119905)] (26)

subject to the initial condition

119910119898(0) = 0 (27)

where

119877119898(119898minus1

) = L [119910119898minus1

] +N119898minus1

[1199100 1199101 119910

119898minus1]

minus (1 minus 120594119898) 120601 (119905)

(28)

Therefore

L [1199101(119905)] = ℏ119867 (119905) L [119910

0] +N

0minus 120601 (119905)

L [1199102(119905) minus 119910

1(119905)] = ℏ119867 (119905) L [119910

1] +N

1

L [1199103(119905) minus 119910

2(119905)] = ℏ119867 (119905) L [119910

2] +N

2

L [119910119898(119905) minus 119910

119898minus1(119905)] = ℏ119867 (119905) L [119910

119898minus1] +N

119898minus1

(29)

Upon summing these equations we have

L [119910119898(119905)] = ℏ119867 (119905)

119898minus1

sum

119896=0

L [119910119896] +

119898minus1

sum

119896=0

N119896minus 120601 (119905) (30)

and from (25) we obtain

L [119884119898(119905) minus 119884

119898minus1(119905)]

= ℏ119867 (119905) L [119884119898minus1

] +N [119884119898minus1

] minus 120601 (119905)

(31)


119884119898(0) = 0 (32)


Consequently the collocation method is based on a solution119884119873(119905) isin P

119873+1(0 119879) for (31) such that

L [119884119873

119898(119905119873

119879119896) minus 119884119873

119898minus1(119905119873

119879119896)]

= ℏ119867119873

(119905119873

119879119896) L [119884

119873

119898minus1(119905119873

119879119896)]

(33)

+ N [119884119873

119898minus1(119905119873

119879k)] minus 120601119873

(119905119873

119879119896) (34)


119884119873

119898(0) = 0 (35)

Definition 2 Amapping 119891 of space 1198712(0 119879) into itself is saidto satisfy a Lipschitz condition with Lipschitz constant 120574 if forany 119911 and 119911

lowast1003816100381610038161003816119891 (119911 119905) minus 119891 (119911

lowast

119905)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(36)

If this condition is satisfied with a Lipschitz constant 120574 suchthat 0 le 120574 lt 1 then 119891 is called a contraction mapping

Theorem 3 (Banachrsquos fixed point theorem) Assume that K isa nonempty closed set in a Banach space V and further that119879

119870 rarr 119870 is a contractive mapping with contractivity constant120574 0 le 120574 lt 1 Then the following results hold

(i) There exists a unique 119884 isin 119870 such that 119884 = 119879(119884)(ii) For any 119884

0isin 119870 the sequence 119884

119899infin

119899=0sub 119870 defined by

119884119899+1

= 119879(119884119899) converges to 119884 [29]

Theorem 4 (existence and uniqueness of solution) Assumethat f(119905 y(119905)) in the initial value problem (IVP) (22) satisfiescondition of (36) then (33) has a unique solution

Proof From (33) we have

L [119884119873

119898(119905119873

119879119896)]

= (1 + ℏ119867119873

(119905119873

119879119896))L [119884

119873

119898minus1(119905119873

119879119896)]

+ ℏ119867119873

(119905119873

119879119896) N [119884

119873

119898minus1(119905119873

119879119896)] minus120601

119873

(119905119873

119879119896)

0 le 119896 le 119873 119898 ge 1

119884119873

119898(0) = 0 119898 ge 0

(37)

Since 119891(119905 119911) satisfies the Lipschitz-continuous conditionthen there exists a constant 120574 ge 0 such that

1003816100381610038161003816119891 (119905 119911) minus 119891 (119905 119911

lowast

)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(38)

for all 119905 isin [0 119879] and all 119911 and 119911lowast

Now for the problem (23) we choose

119871 [119884 (119905)] =119889

119889119905119884 + 120572 (119905) 119884

119873 [119884 (119905)] = minus120572 (119905) 119884 minus 119891 (119905 119884)

(39)

and 120601(119905) equiv 0 where 120572(119905) is an arbitrary analytic function

Let 119873119898(119905) = 119884

119873

119898(119905) minus 119884

119873

119898minus1(119905) then we have from (37) that

L [119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

timesL [119884119873

119898minus1(119905119873

119879119896) minus 119884119873

119898minus2(119905119873

119879119896)]

+ ℏ119867(119905119873

119879119896)

times N [119884119873

119898minus1(119905119873

119879119896)] minusN [119884

119873

119898minus2(119905119873

119879119896)]

0 le 119896 le 119873 119898 ge 1

(40)

or according to the definitions of 119871[119884(119905)] and119873[119884(119905)]

119889

119889119905[119873

119898(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898

= (1 + ℏ119867(119905119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898minus1

minus ℏ119867(119905119873

119879119896) 119891 (119905

119873

119879119896 119884119873

119898minus1(119905119873

119879119896)) + 120572 (119905

119873

119879119896) 119884119873

119898minus1

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896)) minus 120572 (119905

119873

119879119896) 119884119873

119898minus2

0 le 119896 le 119873 119898 ge 1

(41)

from where

119889

119889119905[119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)]

+ 120572 (119905119873

119879119896) 119873

119898minus1minus 120572 (119905

119873

119879119896) 119873

119898minus ℏ119867(119905

119873

119879119896)

times 119891 (119905119873

119879119896 119884119873

119898minus1(119905119873

119879119896))

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896))

0 le 119896 le 119873 119898 ge 1

(42)

It is obvious that 119873119898(0) = 0 clearly

(119873

119898(119879))

2

= 2(119873

119898119889

119889119905(119873

119898))

119879

le 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905(119873

119898)

10038171003817100381710038171003817100381710038171003817119879

(43)

Furthermore for any 119905 isin [0 119879]

(119873

119898(119905))

2

= (119873

119898(119879))

2

minus int

119879

119905

119889

119889119909(119873

119898(119909))

2

d119909

le (119873

119898(119879))

2

+ 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119909119873

119898

10038171003817100381710038171003817100381710038171003817119879

(44)

Integrating the previously mentioned with respect to 119905 yieldsthat

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

le 119879(119873

119898(119879))

2

+ 2119879100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(45)


from where

(119873

119898(119879))

2

ge1

119879

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

minus 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(46)

Using (46) and (43) we have

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le 4119879

10038171003817100381710038171003817100381710038171003817

119889

d119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

119898 ge 0 (47)

Let 120572 = max119905isin[0119879]

|120572(119905)| and119867 = max119905isin[0119879]

|119867(119905)| Thereforea combination of (21) (36) and (43) results in

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

le

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879119873

le (1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879119873

+ 119888120574 |ℏ|119867100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

+ 120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879119873+ 120572

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

le 1198882(1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+ 1198882119862120574 |ℏ|119867

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879+ 1198882120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

+ 1198882120572100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(48)

Using (47) and (48) results in

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+41198882119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(49)

where 119888 is a positive constant Then by (47) with 119898 minus 1instead of 119898 and multiplying the resulting inequality by(1198882(1 + |ℏ|119867))(1 minus 4119888

2120572119879) ge 0 we have

1198882(1 + |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

(50)

Subtracting (50) from (49) after simplifying we obtain

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

1198882(1 + |ℏ|119867) + 4119888

2119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879 (51)

Therefore if (1198882(1 + |ℏ|119867) + 4119888

2119879(120572 + 119888120574|ℏ|119867))(1 minus 4119888

2120572119879) le

120573 lt 1 then ||119873

119898||119879

rarr 0 as 119898 rarr infin Accordingto Theorem 3 it implies the existence and uniqueness ofsolution of (33)

4 Piecewise-Spectral HomotopyAnalysis Method

It is worth noting that the SHAM method described previ-ously is ideally suited for boundary value problems whose

solutions do not rapidly change in behaviour or oscillate oversmall regions of the domain of the governing problem TheSHAM solution can thus be considered to be local in natureandmay not be suitable for initial value problems at very largevalues of the independent variable 119905 A simpleway of ensuringthe validity of the approximations for large 119905 is to determinethe solution in a sequence of equal intervals which are subjectto continuity conditions at the end points of each interval Toextend this solution over the interval Λ = [119905

0 119905119865] we divide

the intervalΛ into subintervalsΛ119894= [119905119894minus1

119905119894] 119894 = 1 2 3 119865

where 1199050le 1199051le sdot sdot sdot le 119905

119865 We solve (6) in each subintervalΛ119894 Let 119910

1

119903(119905) be the solution of (6) in the first subinterval

[1199050 1199051] and let 119910119894

119903(119905) be the solutions in the subintervals Λ

119894

for 2 le 119894 le 119865 The initial conditions used in obtaining thesolutions in the subinterval Λ

119894(2 le 119894 le 119865) are obtained from

the initial conditions of the subinterval Λ119894minus1

Thus we solve

L119903[119910119894

119903119898(119905) minus 120594

119898119910119894

119903119898minus1(119905)] = ℏ

119903119867119903(119905) 119877119894

119903119898minus1 119905 isin [119905

119894minus1

119905119894

]

(52)


119910119894

119903119898(119905119894minus1

) = 0 (53)

The initial approximations for solving (8) are obtained assolutions of the system

L [119910119894

1199030(119905)] = 120601

119903(119905) 119905 isin [119905

119894minus1

119905119894

] (54)


119910119894

1199030(119905119894minus1

) = 119910119894minus1

119903(119905119894minus1

) (55)

The Legendre spectral collocation method is then appliedto solve (52)ndash(54) on each interval [119905119894minus1 119905119894] Before applyingthe spectral method it is convenient to transform the region[119905119894minus1

119905119894] to the interval [minus1 1] on which the spectral method

is defined This can be achieved by using the linear transfor-mation

119905 =

(119905119894minus 119905119894minus1

) 119909

2+

(119905119894+ 119905119894minus1

)

2

(56)

in each interval [119905119894minus1

119905119894] (for 119894 = 1 119865) After the

transformation the interval [119905119894minus1

119905119894] is discretized using

the Legendre-Gauss-Lobatto (LGL) nodes These points 119909119894119895

119895 = 0 1 119873 are unevenly distributed on [minus1 1] and aredefined by 119909

119894

0= minus1 119909119894

119873= 1 and for 1 le 119895 le 119873 minus 1 119909119894

119895are

the zeros of 119873 the derivative of the Legendre polynomial of

degree119873 119871119873

The Legendre spectral differentiation matrix119863 is used toapproximate the derivatives of the unknown variables 119910119894

119903119898(119905)

at the collocation points as the matrix vector product

119889119910119894

119903119898

119889119905=

119873

sum

119896=0

D119895119896119910119894

119903119898(119909119894

119896) = DY119894

119903119898 119895 = 0 1 119873 (57)

where D = 2119863(119905119894minus 119905

iminus1) and Y119894

119903119898= [119910119894

119903119898(119909119894

0) 119910119894

119903119898(119909119894

2)

119910119894

119903119898(119909119894

119873)]119879 is the vector function at the collocation points


119909119894

119895 The matrix 119863 is of size (119873 + 1) times (119873 + 1) and its entries

are defined [31 32] as

119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)

Applying the the Legendre spectral collocationmethod in(52)ndash(55) gives

AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)

where A is an (119873 + 1) times (119873 + 1) matrix that comes fromtransforming the linear operator L using the derivativematrix119863Φ

119903 andR119894

119903119898minus1are (119873+1)times1 vectors corresponding

to the functions120601119903(119905) and119877

119894

119903119898minus1 respectively when evaluated

at the collocation points We remark that throughout thispaper for simplicity we have set the auxiliary function119867119903(119905) appearing in (52) to be 1 Thus starting from the

initial approximation Y1198941199030

which is the solution of (59) therecurrence formula

Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)

can be used to obtain the solution 119910119894

119903(119905) using (9) in the

interval [119905119894 119905119894minus1

] The solution approximating 119910119903(119905) in the

entire interval [1199050 119905119865] is given by

119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)

It must be noted that when 119865 = 1 the proposed piecewise-spectral homotopy analysis method (PSHAM) becomesequivalent to the original SHAM algorithm

5 Numerical Experiments

To demonstrate the applicability of the proposed piecewise-spectral homotopy analysis method (PSHAM) algorithm asan appropriate tool for solving nonlinear IVPs we apply theproposed algorithm to the Genesio-Tesi system [15] and anonlinear initial value problem that models dynamics of HIVInfection of CD4

+ T Cells [18] Both these examples exhibitsolutions which are chaotic and highly oscillatory over smallregions Thus the standard SHAM approach would not besuitable as a method of solving these problems The resultsobtained by the piecewise SLM are compared with the resultsof the standard SLM approach and Runge-Kutta-generatedresults

51 Gensio-Tesi System TheGenesio-Tesi system [15] consid-ered in this paper is given as follows

1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)

In this example the parameters used in the SHAM andPSHAM algorithms are

L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[

D minus119868 OO D minus119868

119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894

1119898minus1minus119895

]]]]

]

(64)

With these definitions the PSHAM algorithm gives

AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)

Because the right-hand side of (65) is known the solution caneasily be obtained by using methods for solving linear systemof equations

52 HIV Infection of 1198621198634+ T Cells [18] model Of particular

interest to us is the model in [18] which is given by thefollowing system of differential equations

1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)

with the initial conditions

1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910

3denote the concentration

of susceptible CD4+ T cells infected CD4

+ T cells and freeHIV virus particles in the blood respectively To explain


Table 1 Numerical comparison between PSHAM and ode45 for the Genesio-Tesi system

1199051199101(119905) 119910

2(119905) 119910

3(119905)

PSHAM ode45 PSHAM ode45 PSHAM ode45

10 0549279 0549279 minus0794566 minus0794566 minus2165635 minus2165635


30 minus1358402 minus1358402 1702324 1702324 7595114 7595114

40 minus0234737 minus0234737 minus5138959 minus5138959 1042740 1042740

50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)

Figure 1 Comparison between the PSHAM (solid line) and ode45 results for the Genesio-Tesi system

the parameters we note that 119904 is the source of CD4+ T

cells from precursors 120572 120573 and 120574 are natural turnover ratesof uninfected T cells infected T cells and virus particlesrespectively and 119879max is the maximum level of CD4

+ T cellsconcentration in the body Because of the viral burden on the

HIV infected T cells we assume that 120572 le 120573 The logisticgrowth of the healthy CD4

+ T cells is now described by119903lowast1199101(1 minus ((119910

1+ 1199102)(119879max))) and proliferation of infected

CD4+ T cells is neglected The term 119896119910

21199103describes the

incidence of HIV infection of health CD4+ T cells where


minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)

minus10 minus5 0 5 10minus4

minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)

Figure 2 Phase portraits of the Genesio-Tesi system using the PSHAM approach for 119888 = 6 119887 = 292 and 119886 = 12

Table 2 Numerical comparison between PSHAM and ode45 for the HIV infection of CD4+ T cells model

119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482

119896 gt 0 is the infection rate Each of the infected CD4+ T cell

is assumed to produce 1198731virus particles during its life time

including any of its daughter cells [21 33] Throughout thispaper we set

119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889

119889119905minus (119903lowastminus 120572) 0 0

0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)

Figure 3 Comparison between the PSHAM (solid line) and ode45 results for the HIV infection model

A = [

[

D minus 119868 (119903lowastminus 120572) O O

O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)


6 Results and Discussion

In this section we present the results of the implementation ofthe piecewise spectral homotopy analysis method (PSHAM)described in the previous section on the Genesio-Tesi andthe HIV Infection of CD4

+ T cells model Unless otherwisespecified the results presented in this paper were generatedusing 119872 = 10 order of the PSHAM approximation with119873 = 20 collocation points in each [119905

119894minus1 119905119894] intervalThewidth

of each interval was taken to be 01We remark that the valuesof119873 and119872were chosen through numerical experimentation


0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)

Figure 4 Phase portraits of the HIV model using the PSHAM approach

by testing a few values that give converging results for thegoverning parameters of the problems under considerationThe convergence controlling auxiliary parameter used ingenerating all the results was chosen to be ℏ

119903= minus1 We

remark that the accuracy of the homotopy analysis derivativemethods such as the one discussed in this paper can beimproved by seeking an optimal value of ℏ

119903using different

techniques For instance in the traditional homotopy analysismethod Liao [24] proposed that the optimal value of ℏ wasthe one located in the flat region of the so-called ℏ-curvesMore recently Marinca and Herisanu [34] suggested the useof the minimum of the square of the residuals to obtain theoptimal values of ℏ A detailed review of the various methodsfor obtaining the optimal ℏ can be found in [35] Since thePSHAM approach requires the implementation of the SHAMin numerous intervals it may not be practical to seek anoptimal value of ℏ in every interval Thus for illustrationpurposes ℏ

119903= minus1 was used in this paper and it was found

that this value gives accurate results for all the parametersconsidered in this paper In order to assess the accuracy andperformance of the proposed PSHAM approach the presentresults are compared with those obtained with the MATLABinbuilt solver ode45 The ode45 solver integrates a system of

ordinary differential equations using explicit 4119905ℎamp5119905ℎRunge-Kutta (45) formula the Dormand-Prince pair [36]

Table 1 gives a comparison between the present PSHAMapproximate results and the numerically generated ode45for all the governing dependent variables of the Genesio-Tesisystem at selected values of time 119905 It can be seen from thetable that there is good agreement between the two results

Figure 1 gives the graphical profiles of the classes1199101(119905) 1199102(119905) 1199103(119905) in the time interval [0 40] The initial con-

ditions used to generate the graphs are 1199101(0) = 02 119910

2(0) =

minus03 and 1199103(0) = 01 The PSHAM results are plotted

against the ode45 results (small dots) It can be seen from thefigure that there is good agreement between the two results

In Figure 2 we give the various phase portraits of theGenesio-Tesi problem using the PSHAM method The phaseportraits are qualitatively similar to those obtained in [12]wherein the multistage differential transform method wasused to solve the same problem Numerical simulations usingode45 yield exactly the same results

In Table 2 we give a comparison between the presentPSHAM approximate results and the ode45 for all thegoverning dependent variables of the HIV infection modelat selected values of time 119905 This was done with the standard


parameter values given in (68) and initial values 1199101(0) =

01 1199102(0) = 0 and 119910

3(0) = 01 For this problem in order

to get accurate results the order of the PSHAM used was119872 = 25 We observe again that there is good agreementbetween the two results This confirms the validity of theproposed PSHAMapproach as a potentialmethod for solvinginitial value problems including thosewith chaotic behaviour

Figure 3 gives a comparison between the PSHAM andthe ode45 results (small dots) for the variation of the classes1199101(119905) 1199102(119905) 1199103(119905) in the time interval [0 100] Again excellent

agreement between the two results is observed The graphicillustrations depicted in Figure 3 indicate that our results arequalitatively similar to other results from the literature (seeeg [12 33]) In particular we observe that for the selectedparameters the numerical simulations show the existence ofperiodic solutions Because of this periodic nature of thesolutions it would be difficult to resolve the solutions for verylarge time intervals using the standard SHAM approach

Figure 4 is an illustration of the phase portraits of theHIV infection model problem generated using the PSHAMmethod We observe that the graphical results in Figure 4are qualitatively similar with themultistage differential trans-form method results of [12]

7 Conclusion

In this paper we presented a new application of the spectralhomotopy analysis method in solving a class of nonlinear dif-ferential equations whose solutions show chaotic behaviourThe proposed method (referred to as the piecewise-spectralhomotopy analysis method (PSHAM)) of solution extendsfor the first time the application of the SHAM to initial valueproblems The PSHAM accelerates the convergence of theSHAMsolution over a large region and improves the accuracyof the method The approximate PSHAM numerical resultswere compared with results generated using the MATLABode45 solver and excellent agreement was obtained Thisconfirms the validity of the proposed PSHAM approachas a suitable method for solving a wide variety of initialvalue problems in practical applications including those withchaotic behaviour

References

[1] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homotopy analysismethod for solving a nonlinear second orderBVPrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 9 pp 2293ndash2302 2010

[2] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010

[3] Z GMakukula P Sibanda and S S Motsa ldquoA novel numericaltechnique for two-dimensional laminar flow between twomoving porous wallsrdquo Mathematical Problems in Engineeringvol 2010 Article ID 528956 15 pages 2010

[4] S S Motsa G T Marewo P Sibanda and S Shateyi ldquoAnimproved spectral homotopy analysis method for solving

boundary layer problemsrdquo Boundary Value Problems vol 2011article 3 2011

[5] S Shateyi and S S Motsa ldquoA new approach for the solutionof three-dimensional magnetohydrodynamic rotating flow overa shrinking sheetrdquo Mathematical Problems in Engineering vol2010 Article ID 586340 15 pages 2010

[6] P Sibanda S Motsa and Z Makukula ldquoA spectral-homotopyanalysis method for heat transfer flow of a third grade fluidbetween parallel platesrdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 22 no 1 pp 4ndash23 2012

[7] A K Alomari M S M Noorani and R Nazar ldquoAdaptationof homotopy analysis method for the numericanalytic solutionof Chen systemrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 pp 2336ndash2346 2009

[8] J I Ramos ldquoPiecewise homotopy methods for nonlinearordinary differential equationsrdquoAppliedMathematics and Com-putation vol 198 no 1 pp 92ndash116 2008

[9] O Abdulaziz N F M Noor I Hashim and M S M NooranildquoFurther accuracy tests onAdomian decompositionmethod forchaotic systemsrdquo Chaos Solitons and Fractals vol 36 no 5 pp1405ndash1411 2008

[10] M M Al-Sawalha M S M Noorani and I Hashim ldquoOnaccuracy of Adomian decomposition method for hyperchaoticRossler systemrdquo Chaos Solitons and Fractals vol 40 no 4 pp1801ndash1807 2009

[11] Z M Odibat C Bertelle M A Aziz-Alaoui and G H EDuchamp ldquoA multi-step differential transform method andapplication to non-chaotic or chaotic systemsrdquo Computers ampMathematics with Applications vol 59 no 4 pp 1462ndash14722010

[12] A Gokdogana A Yildirim and M Merdan ldquoSolving afractional order model of HIV infection of CD4+ T-cellsrdquoMathematical and Computer Modelling vol 54 no 9-10 pp2132ndash2138 2011

[13] M Merdan A Gokdogan and V S Erturk ldquoAn approximatesolution of a model for HIV infection of CD4+ T-cellsrdquo IranianJournal of Science and Technology A vol 35 no 1 pp 9ndash12 2011

[14] A Ghorbani and J Saberi-Nadjafi ldquoA piecewise-spectral para-metric iteration method for solving the nonlinear chaoticGenesio systemrdquo Mathematical and Computer Modelling vol54 no 1-2 pp 131ndash139 2011

[15] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquoAutomaticavol 28 no 3 pp 531ndash548 1992

[16] M R Faieghi and H Delavari ldquoChaos in fractional-orderGenesio-Tesi system and its synchronizationrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp731ndash741 2012

[17] S M Goh M S M Noorani and I Hashim ldquoEfficacy ofvariational iteration method for chaotic Genesio systemmdashclassical and multistage approachrdquo Chaos Solitons and Fractalsvol 40 no 5 pp 2152ndash2159 2009

[18] A S Perelson D E Kirschner and R De Boer ldquoDynamics ofHIV infection of CD4+ T-cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[19] M Y Ongun ldquoThe Laplace Adomian Decomposition Methodfor solving a model for HIV infection of CD4+ T-cellsrdquoMathematical and ComputerModelling vol 53 no 5-6 pp 597ndash603 2011

[20] M Merdan ldquoHomotopy perturbation method for solving amodel for hiv infection of CD4+ T-cellsrdquo Istanbul CommerceUniversity Journal of Science vol 12 pp 39ndash52 2007


[21] M Merdan A Gokdogan and A Yildirim ldquoOn the numericalsolution of the model for HIV infection of CD4+ T-cellsrdquoComputers amp Mathematics with Applications vol 62 no 1 pp118ndash123 2011

[22] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011

[23] S Yuzbasi ldquoA numerical approach to solve the model for HIVinfection of CD4+ T-cellsrdquoAppliedMathematicalModelling vol36 no 12 pp 5876ndash5890 2012

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC 2003

[25] A S Bataineh M S M Noorani and I Hashim ldquoSeriessolution of the multispecies Lotka-Volterra equations by meansof the homotopy analysis methodrdquo Differential Equations andNonlinear Mechanics vol 2008 Article ID 816787 14 pages2008

[26] A S Bataineh M S M Noorani and I Hashim ldquoHomotopyanalysis method for singular IVPs of Emden-Fowler typerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 4 pp 1121ndash1131 2009

[27] I Hashim O Abdulaziz and S Momani ldquoHomotopy analysismethod for fractional IVPsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 3 pp 674ndash6842009

[28] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008

[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework Springer 3rd edition 2009

[30] B-YGuo andPYKuo SpectralMethods andTheirApplicationsWorld Scientific Singapore 1998

[31] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Berlin Germany1988

[32] BDWelfertARemark on Pseudospectral DifferentiationMatri-ces Department of Mathematics Arizona State UniversityTempe Ariz USA 1992

[33] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T-cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006

[34] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008

[35] S J LiaoTheHomotopy Analysis Method in Nonlinear Differen-tial Equations Springer Heidelberg Germany 2012

[36] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


symbolic scientific software For this reason focus is nowshifted towards multistage methods which use numericalintegration techniques such as the piecewise iterationmethod[14] which uses spectral collocation methods

This work examines the applicability of a method thatblends the piecewise implementation technique with thespectral homotopy analysis method For this reason theproposed method is called the piecewise homotopy analysismethod (PSHAM) The validity of the method is testedagainst two initial value problems of practical importancenamely the mathematical models of HIV infection of CD4

+

T cells and the Genesio-Tesi system Chaotic dynamics ofthe Genesio-Tesi system were introduced in [15] Since thenthe system has been used as one of the benchmarks fortesting the validity of new and existing methods of solvingIVPs Approximate analytical or numerical methods thathave been used to find solutions of the Genesio-Tesi systeminclude the differential transformmethod [16] the piecewise-spectral parametric iteration method [14] variational itera-tion method (VIM) and the multistage VIM [17]

Mathematical models play a significant role in studyingepidemic and viral dynamics Viral models are valuable tohelp us improve the understanding of both diseases and var-ious drug therapy strategies against them In 1993 Perelsonet al [18] proposed an ODE model of cell-free viral spreadof human immunodeficiency virus (HIV) in a well-mixedcompartment such as the bloodstream This model consistsof four components uninfected healthy CD4

+ T cells latentlyinfected CD4

+ T cells actively infected CD4+ T cells and

free virus This model has been important in the field ofmathematical modelling of HIV infection and many othermodels have beenproposedwhich take themodel of Perelsonet al [18] as their inspiration For numerically solving amodelfor HIV infection of CD4

+ T cells Ongun [19] has appliedthe Laplace Adomian decomposition method Merdan hasused the homotopy perturbation method [20] and Merdanet al [21] have applied the Pade approximation and themodified variational iteration method [22] More recentlyYuzbasi [23] used the Bessel collocation method for solvingthe same problem It is worth noting that in all these studiesresults were given for solutions over small time intervalsThus it may be concluded that these methods only offera good approximation to the true solution in a very smallregions Liao [24] introduced the homotopy analysis method(HAM) which has convergence controlling parameters toimprove accuracy of series solutions and increase the regionof convergence The HAM was successfully used to solveseveral nonlinear IVPs [24ndash28] However even the standardHAM approach may not be suitable to resolve solutionsover very large intervals and for rapidly oscillating systemswhich show chaotic behaviour For practical purposes resultsobtained using the previously mentioned analytical methodsof approximation may not be very useful since in additionto the short-term behaviour of some systems long-termbehaviour of the solutions may be required and insights intochaotic systems may be sought The multistage approachsuch as the one discussed in this paper presents an importantstrategy of addressing the limitations of some of the previ-ously cited methods of solution

The rest of this paper is organized as follows In Section 2the basic description of the SHAM is presented Section 3presents existence and uniqueness of solution of SHAMthat give a guarantee of convergence of SHAM Section 4deals with the description and formulation of the piecewise-SHAM In Section 5 the numerical implementation of thePSHAM to the Genesio-Tesi system and HIV infection ofCD4+ T cells models is considered In Section 6 we present

the main results and discussion of the numerical simulationsand finally the conclusion is given in Section 7

2 Basic Idea behind the Spectral HomotopyAnalysis Method (SHAM)

For convenience of the interested reader wewill first present abrief description of the basic idea behind the standard SHAM[1 2] This will be followed by a description of the piecewiseversion of the SHAM algorithm which is suitable for solvinginitial value problems To this end we consider the initialvalue problem (IVP) of dimension 119899 given as

y (119905) = f (119905 y (119905)) y (1199050) = y0 (1)

y R 997888rarr R119899

f R timesR119899

997888rarr R119899

(2)

where the dot denotes differentiation with respect to 119905 Wemake the usual assumption that f is sufficiently smooth forlinearization techniques to be valid If y = (119910

1 1199102 119910

119899) we

can apply the SHAM by rewriting (1) as

L119903[119910119903(119905)] +F

119903[119910119903(119905)] = 120601

119903(119905) (3)

where L119903is a linear operator which is derived from the

entire linear part of (1) and F119903is the remaining nonlinear

component for 119903 = 1 2 119899The SHAM approach imports the conventional ideas of

the standard homotopy analysis method (HAM) by definingthe following zeroth-order deformation equations

(1 minus 119902)L119903[119884119903(119905 119902) minus 119910

1199030(119905)]

= 119902ℏ119903119867119903(119905) N

119903[119884119903(119905 119902)] minus 120601

119903(119905)

(4)

where 119902 isin [0 1] is an embedding parameter 119884119903(119905 119902) are

unknown functions and ℏ119903and 119867

119903(119905) are convergence con-

trolling parameters and functions respectivelyThenonlinearoperatorN

119903is defined as

N119903[119884119903(119905 119902)] = L

119903[119884119903(119905 119902)] +F

119903[119884119903(119905 119902)] (5)

Using the ideas of the standard HAM approach (see eg[24ndash28]) we differentiate the zeroth-order equations (4) 119898

times with respect to 119902 and then set 119902 = 0 and finallydivide the resulting equations by 119898 to obtain the followingequations which are referred to as the 119898th order (or higherorder) deformation equations

L119903[119910119903119898

(119905) minus 120594119898119910119903119898minus1

(119905)] = ℏ119903119867119903(119905) 119877119903119898minus1

(6)


where

119877119903119898minus1

=1

(119898 minus 1)

120597119898minus1

119884119903(119905 119902) minus 120601

119903(119905)

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

120594119898

= 0 119898 ⩽ 1

1 119898 gt 1

(7)



119910119903(119905) = 119910

1199030(119905) + 119910

1199031(119905) + 119910

1199032(119905) + sdot sdot sdot (8)


119910119903(119905) =

119872

sum

119898=0

119910119903119898

(119905) (9)


119903and119867



L119903[1199101199030

(119905)] = 120601119903(119905) 119910

1199030(0) = 119910

0

119903 (10)


1199030(119905) of


(119905) (for119898 = 1 2 3 119872)






T0(119909) = 1 T

1(119909) = 119909

T119895+1

(119909) =2119895 + 1

119895 + 1119909T119895(119909) minus

119895

119895 + 1T119895minus1

(119909) 119895 ge 1

(11)




T119879119895

(119905)ThenT119879119895


T119879119895+1

(119905) =2119895 + 1

119895 + 1(2119905

119879minus 1)T

119879119895(119905)

minus119895

119895 + 1T119879119895minus1

(119905) 119895 = 1 2

(12)

whereT1198790

(119905) = 1 andT1198791


int

119879

0

T119879119894

(119905)T119879119896

(119905) 119889119905 =119879

2119894 + 1120575119894119896 (13)

where 120575119894119896


119910 (119905) =

infin

sum

119895=0

119886119895T119879119895

(119905) (14)


119886119895=

2119897 + 1

119879int

119879

0

119910 (119905)T119879119895

(119905) 119889119905 119895 = 0 1 2 (15)


119910 (119905) =

119873

sum

119895=0

119886119895T119879119895

(119905) (16)





119873





119879119895 0 le 119895 le

119873 Clearly 119905119873

119879119895= (1198792)(119905

119873



119879119895 = (1198792)120596

119873

119895 Let P

119873(0 119879) be the set of all


2119873+1(0 119879)

int

119879

0

Φ (119905) 119889119905 =119879

2int

1

minus1

Φ(119879

2(119905 + 1)) 119889119905

=119879

2

119873

sum

119895=0

120596119873

119895Φ(

119879

2(119905119873

119895+ 1))

=

119873

sum

119895=0

120596119873

119879119895Φ(119905119873

119879119895)

(17)





(119906 V)119879119873

=

119873

sum

119895=0

119906 (119905119873

119879119895) V (119905119873

119879119895) 120596119873

119879119895 V

119879119873= (V V)

12

119879119873

(18)


(0 119879)

(Φ 120595)119879= (Φ 120595)

119879119873 (19)



10038171003817100381710038171206011003817100381710038171003817119879

=10038171003817100381710038171206011003817100381710038171003817119879119873

120601 isin P119873(0 119879) (20)


= ||120601||119879119873


119897and 1198882

1198881

10038171003817100381710038171206011003817100381710038171003817119879

le10038171003817100381710038171206011003817100381710038171003817119879119873

le 1198882

10038171003817100381710038171206011003817100381710038171003817119879

(21)


y (119905) = f (119905 y (119905)) y (1199050) = y0

y R 997888rarr R119899

f R timesR119899

997888rarr R119899

(22)


L [119910 (119905)] +N [119910 (119905)] = 120601 (119905) (23)



119899=0as

N [119910 (119905)] =

infin

sum

119896=0

119873119896(1199100 1199101 119910

119896) (24)

1198840= 1199100

1198841= 1199100+ 1199101

119884119899= 1199100+ 1199101+ 1199102+ sdot sdot sdot + 119910

119899

(25)


L [119910119898(119905) minus 120594

119898119910119898minus1

(119905)] = ℏ119867 (119905) 119877119898[119910119898minus1

(119905)] (26)


119910119898(0) = 0 (27)

where

119877119898(119898minus1

) = L [119910119898minus1

] +N119898minus1

[1199100 1199101 119910

119898minus1]

minus (1 minus 120594119898) 120601 (119905)

(28)

Therefore

L [1199101(119905)] = ℏ119867 (119905) L [119910

0] +N

0minus 120601 (119905)

L [1199102(119905) minus 119910

1(119905)] = ℏ119867 (119905) L [119910

1] +N

1

L [1199103(119905) minus 119910

2(119905)] = ℏ119867 (119905) L [119910

2] +N

2

L [119910119898(119905) minus 119910

119898minus1(119905)] = ℏ119867 (119905) L [119910

119898minus1] +N

119898minus1

(29)


L [119910119898(119905)] = ℏ119867 (119905)

119898minus1

sum

119896=0

L [119910119896] +

119898minus1

sum

119896=0

N119896minus 120601 (119905) (30)


L [119884119898(119905) minus 119884

119898minus1(119905)]

= ℏ119867 (119905) L [119884119898minus1

] +N [119884119898minus1

] minus 120601 (119905)

(31)


119884119898(0) = 0 (32)



119873+1(0 119879) for (31) such that

L [119884119873

119898(119905119873

119879119896) minus 119884119873

119898minus1(119905119873

119879119896)]

= ℏ119867119873

(119905119873

119879119896) L [119884

119873

119898minus1(119905119873

119879119896)]

(33)

+ N [119884119873

119898minus1(119905119873

119879k)] minus 120601119873

(119905119873

119879119896) (34)


119884119873

119898(0) = 0 (35)


lowast1003816100381610038161003816119891 (119911 119905) minus 119891 (119911

lowast

119905)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(36)






119899infin


119884119899+1

= 119879(119884119899) converges to 119884 [29]



L [119884119873

119898(119905119873

119879119896)]

= (1 + ℏ119867119873

(119905119873

119879119896))L [119884

119873

119898minus1(119905119873

119879119896)]

+ ℏ119867119873

(119905119873

119879119896) N [119884

119873

119898minus1(119905119873

119879119896)] minus120601

119873

(119905119873

119879119896)

0 le 119896 le 119873 119898 ge 1

119884119873

119898(0) = 0 119898 ge 0

(37)


1003816100381610038161003816119891 (119905 119911) minus 119891 (119905 119911

lowast

)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(38)



119871 [119884 (119905)] =119889

119889119905119884 + 120572 (119905) 119884

119873 [119884 (119905)] = minus120572 (119905) 119884 minus 119891 (119905 119884)

(39)


Let 119873119898(119905) = 119884

119873

119898(119905) minus 119884

119873


L [119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

timesL [119884119873

119898minus1(119905119873

119879119896) minus 119884119873

119898minus2(119905119873

119879119896)]

+ ℏ119867(119905119873

119879119896)

times N [119884119873

119898minus1(119905119873

119879119896)] minusN [119884

119873

119898minus2(119905119873

119879119896)]

0 le 119896 le 119873 119898 ge 1

(40)


119889

119889119905[119873

119898(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898

= (1 + ℏ119867(119905119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898minus1

minus ℏ119867(119905119873

119879119896) 119891 (119905

119873

119879119896 119884119873

119898minus1(119905119873

119879119896)) + 120572 (119905

119873

119879119896) 119884119873

119898minus1

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896)) minus 120572 (119905

119873

119879119896) 119884119873

119898minus2

0 le 119896 le 119873 119898 ge 1

(41)

from where

119889

119889119905[119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)]

+ 120572 (119905119873

119879119896) 119873

119898minus1minus 120572 (119905

119873

119879119896) 119873

119898minus ℏ119867(119905

119873

119879119896)

times 119891 (119905119873

119879119896 119884119873

119898minus1(119905119873

119879119896))

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896))

0 le 119896 le 119873 119898 ge 1

(42)


(119873

119898(119879))

2

= 2(119873

119898119889

119889119905(119873

119898))

119879

le 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905(119873

119898)

10038171003817100381710038171003817100381710038171003817119879

(43)


(119873

119898(119905))

2

= (119873

119898(119879))

2

minus int

119879

119905

119889

119889119909(119873

119898(119909))

2

d119909

le (119873

119898(119879))

2

+ 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119909119873

119898

10038171003817100381710038171003817100381710038171003817119879

(44)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

le 119879(119873

119898(119879))

2

+ 2119879100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(45)


from where

(119873

119898(119879))

2

ge1

119879

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

minus 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(46)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le 4119879

10038171003817100381710038171003817100381710038171003817

119889

d119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

119898 ge 0 (47)

Let 120572 = max119905isin[0119879]

|120572(119905)| and119867 = max119905isin[0119879]


10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

le

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879119873

le (1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879119873

+ 119888120574 |ℏ|119867100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

+ 120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879119873+ 120572

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

le 1198882(1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+ 1198882119862120574 |ℏ|119867

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879+ 1198882120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

+ 1198882120572100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(48)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+41198882119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(49)


2120572119879) ge 0 we have

1198882(1 + |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

(50)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

1198882(1 + |ℏ|119867) + 4119888

2119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879 (51)

Therefore if (1198882(1 + |ℏ|119867) + 4119888

2119879(120572 + 119888120574|ℏ|119867))(1 minus 4119888

2120572119879) le

120573 lt 1 then ||119873

119898||119879





0 119905119865] we divide


119905119894] 119894 = 1 2 3 119865



1


[1199050 1199051] and let 119910119894


119894




Thus we solve

L119903[119910119894

119903119898(119905) minus 120594

119898119910119894

119903119898minus1(119905)] = ℏ

119903119867119903(119905) 119877119894

119903119898minus1 119905 isin [119905

119894minus1

119905119894

]

(52)


119910119894

119903119898(119905119894minus1

) = 0 (53)


L [119910119894

1199030(119905)] = 120601

119903(119905) 119905 isin [119905

119894minus1

119905119894

] (54)


119910119894

1199030(119905119894minus1

) = 119910119894minus1

119903(119905119894minus1

) (55)




119905 =

(119905119894minus 119905119894minus1

) 119909

2+

(119905119894+ 119905119894minus1

)

2

(56)


119905119894] (for 119894 = 1 119865) After the





119894

0= minus1 119909119894


119895are


degree119873 119871119873


119903119898(119905)


119889119910119894

119903119898

119889119905=

119873

sum

119896=0

D119895119896119910119894

119903119898(119909119894

119896) = DY119894

119903119898 119895 = 0 1 119873 (57)

where D = 2119863(119905119894minus 119905


119903119898= [119910119894

119903119898(119909119894

0) 119910119894

119903119898(119909119894

2)

119910119894

119903119898(119909119894



119909119894



119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)


AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)


119903 andR119894



119894





Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)


119903(119905) using (9) in the

interval [119905119894 119905119894minus1



119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)






1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)


L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[


119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894


]]]]

]

(64)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)




1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)


1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910






1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119877119903119898minus1

=1

(119898 minus 1)

120597119898minus1

119884119903(119905 119902) minus 120601

119903(119905)

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

120594119898

= 0 119898 ⩽ 1

1 119898 gt 1

(7)



119910119903(119905) = 119910

1199030(119905) + 119910

1199031(119905) + 119910

1199032(119905) + sdot sdot sdot (8)


119910119903(119905) =

119872

sum

119898=0

119910119903119898

(119905) (9)


119903and119867



L119903[1199101199030

(119905)] = 120601119903(119905) 119910

1199030(0) = 119910

0

119903 (10)


1199030(119905) of


(119905) (for119898 = 1 2 3 119872)






T0(119909) = 1 T

1(119909) = 119909

T119895+1

(119909) =2119895 + 1

119895 + 1119909T119895(119909) minus

119895

119895 + 1T119895minus1

(119909) 119895 ge 1

(11)




T119879119895

(119905)ThenT119879119895


T119879119895+1

(119905) =2119895 + 1

119895 + 1(2119905

119879minus 1)T

119879119895(119905)

minus119895

119895 + 1T119879119895minus1

(119905) 119895 = 1 2

(12)

whereT1198790

(119905) = 1 andT1198791


int

119879

0

T119879119894

(119905)T119879119896

(119905) 119889119905 =119879

2119894 + 1120575119894119896 (13)

where 120575119894119896


119910 (119905) =

infin

sum

119895=0

119886119895T119879119895

(119905) (14)


119886119895=

2119897 + 1

119879int

119879

0

119910 (119905)T119879119895

(119905) 119889119905 119895 = 0 1 2 (15)


119910 (119905) =

119873

sum

119895=0

119886119895T119879119895

(119905) (16)





119873





119879119895 0 le 119895 le

119873 Clearly 119905119873

119879119895= (1198792)(119905

119873



119879119895 = (1198792)120596

119873

119895 Let P

119873(0 119879) be the set of all


2119873+1(0 119879)

int

119879

0

Φ (119905) 119889119905 =119879

2int

1

minus1

Φ(119879

2(119905 + 1)) 119889119905

=119879

2

119873

sum

119895=0

120596119873

119895Φ(

119879

2(119905119873

119895+ 1))

=

119873

sum

119895=0

120596119873

119879119895Φ(119905119873

119879119895)

(17)





(119906 V)119879119873

=

119873

sum

119895=0

119906 (119905119873

119879119895) V (119905119873

119879119895) 120596119873

119879119895 V

119879119873= (V V)

12

119879119873

(18)


(0 119879)

(Φ 120595)119879= (Φ 120595)

119879119873 (19)



10038171003817100381710038171206011003817100381710038171003817119879

=10038171003817100381710038171206011003817100381710038171003817119879119873

120601 isin P119873(0 119879) (20)


= ||120601||119879119873


119897and 1198882

1198881

10038171003817100381710038171206011003817100381710038171003817119879

le10038171003817100381710038171206011003817100381710038171003817119879119873

le 1198882

10038171003817100381710038171206011003817100381710038171003817119879

(21)


y (119905) = f (119905 y (119905)) y (1199050) = y0

y R 997888rarr R119899

f R timesR119899

997888rarr R119899

(22)


L [119910 (119905)] +N [119910 (119905)] = 120601 (119905) (23)



119899=0as

N [119910 (119905)] =

infin

sum

119896=0

119873119896(1199100 1199101 119910

119896) (24)

1198840= 1199100

1198841= 1199100+ 1199101

119884119899= 1199100+ 1199101+ 1199102+ sdot sdot sdot + 119910

119899

(25)


L [119910119898(119905) minus 120594

119898119910119898minus1

(119905)] = ℏ119867 (119905) 119877119898[119910119898minus1

(119905)] (26)


119910119898(0) = 0 (27)

where

119877119898(119898minus1

) = L [119910119898minus1

] +N119898minus1

[1199100 1199101 119910

119898minus1]

minus (1 minus 120594119898) 120601 (119905)

(28)

Therefore

L [1199101(119905)] = ℏ119867 (119905) L [119910

0] +N

0minus 120601 (119905)

L [1199102(119905) minus 119910

1(119905)] = ℏ119867 (119905) L [119910

1] +N

1

L [1199103(119905) minus 119910

2(119905)] = ℏ119867 (119905) L [119910

2] +N

2

L [119910119898(119905) minus 119910

119898minus1(119905)] = ℏ119867 (119905) L [119910

119898minus1] +N

119898minus1

(29)


L [119910119898(119905)] = ℏ119867 (119905)

119898minus1

sum

119896=0

L [119910119896] +

119898minus1

sum

119896=0

N119896minus 120601 (119905) (30)


L [119884119898(119905) minus 119884

119898minus1(119905)]

= ℏ119867 (119905) L [119884119898minus1

] +N [119884119898minus1

] minus 120601 (119905)

(31)


119884119898(0) = 0 (32)



119873+1(0 119879) for (31) such that

L [119884119873

119898(119905119873

119879119896) minus 119884119873

119898minus1(119905119873

119879119896)]

= ℏ119867119873

(119905119873

119879119896) L [119884

119873

119898minus1(119905119873

119879119896)]

(33)

+ N [119884119873

119898minus1(119905119873

119879k)] minus 120601119873

(119905119873

119879119896) (34)


119884119873

119898(0) = 0 (35)


lowast1003816100381610038161003816119891 (119911 119905) minus 119891 (119911

lowast

119905)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(36)






119899infin


119884119899+1

= 119879(119884119899) converges to 119884 [29]



L [119884119873

119898(119905119873

119879119896)]

= (1 + ℏ119867119873

(119905119873

119879119896))L [119884

119873

119898minus1(119905119873

119879119896)]

+ ℏ119867119873

(119905119873

119879119896) N [119884

119873

119898minus1(119905119873

119879119896)] minus120601

119873

(119905119873

119879119896)

0 le 119896 le 119873 119898 ge 1

119884119873

119898(0) = 0 119898 ge 0

(37)


1003816100381610038161003816119891 (119905 119911) minus 119891 (119905 119911

lowast

)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(38)



119871 [119884 (119905)] =119889

119889119905119884 + 120572 (119905) 119884

119873 [119884 (119905)] = minus120572 (119905) 119884 minus 119891 (119905 119884)

(39)


Let 119873119898(119905) = 119884

119873

119898(119905) minus 119884

119873


L [119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

timesL [119884119873

119898minus1(119905119873

119879119896) minus 119884119873

119898minus2(119905119873

119879119896)]

+ ℏ119867(119905119873

119879119896)

times N [119884119873

119898minus1(119905119873

119879119896)] minusN [119884

119873

119898minus2(119905119873

119879119896)]

0 le 119896 le 119873 119898 ge 1

(40)


119889

119889119905[119873

119898(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898

= (1 + ℏ119867(119905119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898minus1

minus ℏ119867(119905119873

119879119896) 119891 (119905

119873

119879119896 119884119873

119898minus1(119905119873

119879119896)) + 120572 (119905

119873

119879119896) 119884119873

119898minus1

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896)) minus 120572 (119905

119873

119879119896) 119884119873

119898minus2

0 le 119896 le 119873 119898 ge 1

(41)

from where

119889

119889119905[119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)]

+ 120572 (119905119873

119879119896) 119873

119898minus1minus 120572 (119905

119873

119879119896) 119873

119898minus ℏ119867(119905

119873

119879119896)

times 119891 (119905119873

119879119896 119884119873

119898minus1(119905119873

119879119896))

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896))

0 le 119896 le 119873 119898 ge 1

(42)


(119873

119898(119879))

2

= 2(119873

119898119889

119889119905(119873

119898))

119879

le 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905(119873

119898)

10038171003817100381710038171003817100381710038171003817119879

(43)


(119873

119898(119905))

2

= (119873

119898(119879))

2

minus int

119879

119905

119889

119889119909(119873

119898(119909))

2

d119909

le (119873

119898(119879))

2

+ 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119909119873

119898

10038171003817100381710038171003817100381710038171003817119879

(44)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

le 119879(119873

119898(119879))

2

+ 2119879100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(45)


from where

(119873

119898(119879))

2

ge1

119879

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

minus 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(46)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le 4119879

10038171003817100381710038171003817100381710038171003817

119889

d119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

119898 ge 0 (47)

Let 120572 = max119905isin[0119879]

|120572(119905)| and119867 = max119905isin[0119879]


10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

le

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879119873

le (1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879119873

+ 119888120574 |ℏ|119867100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

+ 120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879119873+ 120572

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

le 1198882(1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+ 1198882119862120574 |ℏ|119867

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879+ 1198882120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

+ 1198882120572100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(48)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+41198882119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(49)


2120572119879) ge 0 we have

1198882(1 + |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

(50)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

1198882(1 + |ℏ|119867) + 4119888

2119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879 (51)

Therefore if (1198882(1 + |ℏ|119867) + 4119888

2119879(120572 + 119888120574|ℏ|119867))(1 minus 4119888

2120572119879) le

120573 lt 1 then ||119873

119898||119879





0 119905119865] we divide


119905119894] 119894 = 1 2 3 119865



1


[1199050 1199051] and let 119910119894


119894




Thus we solve

L119903[119910119894

119903119898(119905) minus 120594

119898119910119894

119903119898minus1(119905)] = ℏ

119903119867119903(119905) 119877119894

119903119898minus1 119905 isin [119905

119894minus1

119905119894

]

(52)


119910119894

119903119898(119905119894minus1

) = 0 (53)


L [119910119894

1199030(119905)] = 120601

119903(119905) 119905 isin [119905

119894minus1

119905119894

] (54)


119910119894

1199030(119905119894minus1

) = 119910119894minus1

119903(119905119894minus1

) (55)




119905 =

(119905119894minus 119905119894minus1

) 119909

2+

(119905119894+ 119905119894minus1

)

2

(56)


119905119894] (for 119894 = 1 119865) After the





119894

0= minus1 119909119894


119895are


degree119873 119871119873


119903119898(119905)


119889119910119894

119903119898

119889119905=

119873

sum

119896=0

D119895119896119910119894

119903119898(119909119894

119896) = DY119894

119903119898 119895 = 0 1 119873 (57)

where D = 2119863(119905119894minus 119905


119903119898= [119910119894

119903119898(119909119894

0) 119910119894

119903119898(119909119894

2)

119910119894

119903119898(119909119894



119909119894



119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)


AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)


119903 andR119894



119894





Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)


119903(119905) using (9) in the

interval [119905119894 119905119894minus1



119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)






1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)


L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[


119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894


]]]]

]

(64)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)




1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)


1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910






1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873





119879119895 0 le 119895 le

119873 Clearly 119905119873

119879119895= (1198792)(119905

119873



119879119895 = (1198792)120596

119873

119895 Let P

119873(0 119879) be the set of all


2119873+1(0 119879)

int

119879

0

Φ (119905) 119889119905 =119879

2int

1

minus1

Φ(119879

2(119905 + 1)) 119889119905

=119879

2

119873

sum

119895=0

120596119873

119895Φ(

119879

2(119905119873

119895+ 1))

=

119873

sum

119895=0

120596119873

119879119895Φ(119905119873

119879119895)

(17)





(119906 V)119879119873

=

119873

sum

119895=0

119906 (119905119873

119879119895) V (119905119873

119879119895) 120596119873

119879119895 V

119879119873= (V V)

12

119879119873

(18)


(0 119879)

(Φ 120595)119879= (Φ 120595)

119879119873 (19)



10038171003817100381710038171206011003817100381710038171003817119879

=10038171003817100381710038171206011003817100381710038171003817119879119873

120601 isin P119873(0 119879) (20)


= ||120601||119879119873


119897and 1198882

1198881

10038171003817100381710038171206011003817100381710038171003817119879

le10038171003817100381710038171206011003817100381710038171003817119879119873

le 1198882

10038171003817100381710038171206011003817100381710038171003817119879

(21)


y (119905) = f (119905 y (119905)) y (1199050) = y0

y R 997888rarr R119899

f R timesR119899

997888rarr R119899

(22)


L [119910 (119905)] +N [119910 (119905)] = 120601 (119905) (23)



119899=0as

N [119910 (119905)] =

infin

sum

119896=0

119873119896(1199100 1199101 119910

119896) (24)

1198840= 1199100

1198841= 1199100+ 1199101

119884119899= 1199100+ 1199101+ 1199102+ sdot sdot sdot + 119910

119899

(25)


L [119910119898(119905) minus 120594

119898119910119898minus1

(119905)] = ℏ119867 (119905) 119877119898[119910119898minus1

(119905)] (26)


119910119898(0) = 0 (27)

where

119877119898(119898minus1

) = L [119910119898minus1

] +N119898minus1

[1199100 1199101 119910

119898minus1]

minus (1 minus 120594119898) 120601 (119905)

(28)

Therefore

L [1199101(119905)] = ℏ119867 (119905) L [119910

0] +N

0minus 120601 (119905)

L [1199102(119905) minus 119910

1(119905)] = ℏ119867 (119905) L [119910

1] +N

1

L [1199103(119905) minus 119910

2(119905)] = ℏ119867 (119905) L [119910

2] +N

2

L [119910119898(119905) minus 119910

119898minus1(119905)] = ℏ119867 (119905) L [119910

119898minus1] +N

119898minus1

(29)


L [119910119898(119905)] = ℏ119867 (119905)

119898minus1

sum

119896=0

L [119910119896] +

119898minus1

sum

119896=0

N119896minus 120601 (119905) (30)


L [119884119898(119905) minus 119884

119898minus1(119905)]

= ℏ119867 (119905) L [119884119898minus1

] +N [119884119898minus1

] minus 120601 (119905)

(31)


119884119898(0) = 0 (32)



119873+1(0 119879) for (31) such that

L [119884119873

119898(119905119873

119879119896) minus 119884119873

119898minus1(119905119873

119879119896)]

= ℏ119867119873

(119905119873

119879119896) L [119884

119873

119898minus1(119905119873

119879119896)]

(33)

+ N [119884119873

119898minus1(119905119873

119879k)] minus 120601119873

(119905119873

119879119896) (34)


119884119873

119898(0) = 0 (35)


lowast1003816100381610038161003816119891 (119911 119905) minus 119891 (119911

lowast

119905)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(36)






119899infin


119884119899+1

= 119879(119884119899) converges to 119884 [29]



L [119884119873

119898(119905119873

119879119896)]

= (1 + ℏ119867119873

(119905119873

119879119896))L [119884

119873

119898minus1(119905119873

119879119896)]

+ ℏ119867119873

(119905119873

119879119896) N [119884

119873

119898minus1(119905119873

119879119896)] minus120601

119873

(119905119873

119879119896)

0 le 119896 le 119873 119898 ge 1

119884119873

119898(0) = 0 119898 ge 0

(37)


1003816100381610038161003816119891 (119905 119911) minus 119891 (119905 119911

lowast

)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(38)



119871 [119884 (119905)] =119889

119889119905119884 + 120572 (119905) 119884

119873 [119884 (119905)] = minus120572 (119905) 119884 minus 119891 (119905 119884)

(39)


Let 119873119898(119905) = 119884

119873

119898(119905) minus 119884

119873


L [119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

timesL [119884119873

119898minus1(119905119873

119879119896) minus 119884119873

119898minus2(119905119873

119879119896)]

+ ℏ119867(119905119873

119879119896)

times N [119884119873

119898minus1(119905119873

119879119896)] minusN [119884

119873

119898minus2(119905119873

119879119896)]

0 le 119896 le 119873 119898 ge 1

(40)


119889

119889119905[119873

119898(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898

= (1 + ℏ119867(119905119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898minus1

minus ℏ119867(119905119873

119879119896) 119891 (119905

119873

119879119896 119884119873

119898minus1(119905119873

119879119896)) + 120572 (119905

119873

119879119896) 119884119873

119898minus1

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896)) minus 120572 (119905

119873

119879119896) 119884119873

119898minus2

0 le 119896 le 119873 119898 ge 1

(41)

from where

119889

119889119905[119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)]

+ 120572 (119905119873

119879119896) 119873

119898minus1minus 120572 (119905

119873

119879119896) 119873

119898minus ℏ119867(119905

119873

119879119896)

times 119891 (119905119873

119879119896 119884119873

119898minus1(119905119873

119879119896))

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896))

0 le 119896 le 119873 119898 ge 1

(42)


(119873

119898(119879))

2

= 2(119873

119898119889

119889119905(119873

119898))

119879

le 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905(119873

119898)

10038171003817100381710038171003817100381710038171003817119879

(43)


(119873

119898(119905))

2

= (119873

119898(119879))

2

minus int

119879

119905

119889

119889119909(119873

119898(119909))

2

d119909

le (119873

119898(119879))

2

+ 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119909119873

119898

10038171003817100381710038171003817100381710038171003817119879

(44)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

le 119879(119873

119898(119879))

2

+ 2119879100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(45)


from where

(119873

119898(119879))

2

ge1

119879

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

minus 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(46)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le 4119879

10038171003817100381710038171003817100381710038171003817

119889

d119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

119898 ge 0 (47)

Let 120572 = max119905isin[0119879]

|120572(119905)| and119867 = max119905isin[0119879]


10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

le

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879119873

le (1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879119873

+ 119888120574 |ℏ|119867100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

+ 120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879119873+ 120572

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

le 1198882(1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+ 1198882119862120574 |ℏ|119867

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879+ 1198882120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

+ 1198882120572100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(48)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+41198882119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(49)


2120572119879) ge 0 we have

1198882(1 + |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

(50)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

1198882(1 + |ℏ|119867) + 4119888

2119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879 (51)

Therefore if (1198882(1 + |ℏ|119867) + 4119888

2119879(120572 + 119888120574|ℏ|119867))(1 minus 4119888

2120572119879) le

120573 lt 1 then ||119873

119898||119879





0 119905119865] we divide


119905119894] 119894 = 1 2 3 119865



1


[1199050 1199051] and let 119910119894


119894




Thus we solve

L119903[119910119894

119903119898(119905) minus 120594

119898119910119894

119903119898minus1(119905)] = ℏ

119903119867119903(119905) 119877119894

119903119898minus1 119905 isin [119905

119894minus1

119905119894

]

(52)


119910119894

119903119898(119905119894minus1

) = 0 (53)


L [119910119894

1199030(119905)] = 120601

119903(119905) 119905 isin [119905

119894minus1

119905119894

] (54)


119910119894

1199030(119905119894minus1

) = 119910119894minus1

119903(119905119894minus1

) (55)




119905 =

(119905119894minus 119905119894minus1

) 119909

2+

(119905119894+ 119905119894minus1

)

2

(56)


119905119894] (for 119894 = 1 119865) After the





119894

0= minus1 119909119894


119895are


degree119873 119871119873


119903119898(119905)


119889119910119894

119903119898

119889119905=

119873

sum

119896=0

D119895119896119910119894

119903119898(119909119894

119896) = DY119894

119903119898 119895 = 0 1 119873 (57)

where D = 2119863(119905119894minus 119905


119903119898= [119910119894

119903119898(119909119894

0) 119910119894

119903119898(119909119894

2)

119910119894

119903119898(119909119894



119909119894



119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)


AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)


119903 andR119894



119894





Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)


119903(119905) using (9) in the

interval [119905119894 119905119894minus1



119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)






1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)


L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[


119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894


]]]]

]

(64)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)




1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)


1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910






1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873+1(0 119879) for (31) such that

L [119884119873

119898(119905119873

119879119896) minus 119884119873

119898minus1(119905119873

119879119896)]

= ℏ119867119873

(119905119873

119879119896) L [119884

119873

119898minus1(119905119873

119879119896)]

(33)

+ N [119884119873

119898minus1(119905119873

119879k)] minus 120601119873

(119905119873

119879119896) (34)


119884119873

119898(0) = 0 (35)


lowast1003816100381610038161003816119891 (119911 119905) minus 119891 (119911

lowast

119905)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(36)






119899infin


119884119899+1

= 119879(119884119899) converges to 119884 [29]



L [119884119873

119898(119905119873

119879119896)]

= (1 + ℏ119867119873

(119905119873

119879119896))L [119884

119873

119898minus1(119905119873

119879119896)]

+ ℏ119867119873

(119905119873

119879119896) N [119884

119873

119898minus1(119905119873

119879119896)] minus120601

119873

(119905119873

119879119896)

0 le 119896 le 119873 119898 ge 1

119884119873

119898(0) = 0 119898 ge 0

(37)


1003816100381610038161003816119891 (119905 119911) minus 119891 (119905 119911

lowast

)1003816100381610038161003816le 120574

1003816100381610038161003816119911 minus 119911lowast1003816100381610038161003816

(38)



119871 [119884 (119905)] =119889

119889119905119884 + 120572 (119905) 119884

119873 [119884 (119905)] = minus120572 (119905) 119884 minus 119891 (119905 119884)

(39)


Let 119873119898(119905) = 119884

119873

119898(119905) minus 119884

119873


L [119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

timesL [119884119873

119898minus1(119905119873

119879119896) minus 119884119873

119898minus2(119905119873

119879119896)]

+ ℏ119867(119905119873

119879119896)

times N [119884119873

119898minus1(119905119873

119879119896)] minusN [119884

119873

119898minus2(119905119873

119879119896)]

0 le 119896 le 119873 119898 ge 1

(40)


119889

119889119905[119873

119898(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898

= (1 + ℏ119867(119905119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)] + 120572 (119905

119873

119879119896) 119873

119898minus1

minus ℏ119867(119905119873

119879119896) 119891 (119905

119873

119879119896 119884119873

119898minus1(119905119873

119879119896)) + 120572 (119905

119873

119879119896) 119884119873

119898minus1

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896)) minus 120572 (119905

119873

119879119896) 119884119873

119898minus2

0 le 119896 le 119873 119898 ge 1

(41)

from where

119889

119889119905[119873

119898(119905119873

119879119896)] = (1 + ℏ119867(119905

119873

119879119896))

119889

119889119905[119873

119898minus1(119905119873

119879119896)]

+ 120572 (119905119873

119879119896) 119873

119898minus1minus 120572 (119905

119873

119879119896) 119873

119898minus ℏ119867(119905

119873

119879119896)

times 119891 (119905119873

119879119896 119884119873

119898minus1(119905119873

119879119896))

minus119891 (119905119873

119879119896 119884119873

119898minus2(119905119873

119879119896))

0 le 119896 le 119873 119898 ge 1

(42)


(119873

119898(119879))

2

= 2(119873

119898119889

119889119905(119873

119898))

119879

le 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905(119873

119898)

10038171003817100381710038171003817100381710038171003817119879

(43)


(119873

119898(119905))

2

= (119873

119898(119879))

2

minus int

119879

119905

119889

119889119909(119873

119898(119909))

2

d119909

le (119873

119898(119879))

2

+ 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119909119873

119898

10038171003817100381710038171003817100381710038171003817119879

(44)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

le 119879(119873

119898(119879))

2

+ 2119879100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(45)


from where

(119873

119898(119879))

2

ge1

119879

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

minus 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(46)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le 4119879

10038171003817100381710038171003817100381710038171003817

119889

d119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

119898 ge 0 (47)

Let 120572 = max119905isin[0119879]

|120572(119905)| and119867 = max119905isin[0119879]


10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

le

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879119873

le (1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879119873

+ 119888120574 |ℏ|119867100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

+ 120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879119873+ 120572

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

le 1198882(1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+ 1198882119862120574 |ℏ|119867

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879+ 1198882120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

+ 1198882120572100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(48)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+41198882119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(49)


2120572119879) ge 0 we have

1198882(1 + |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

(50)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

1198882(1 + |ℏ|119867) + 4119888

2119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879 (51)

Therefore if (1198882(1 + |ℏ|119867) + 4119888

2119879(120572 + 119888120574|ℏ|119867))(1 minus 4119888

2120572119879) le

120573 lt 1 then ||119873

119898||119879





0 119905119865] we divide


119905119894] 119894 = 1 2 3 119865



1


[1199050 1199051] and let 119910119894


119894




Thus we solve

L119903[119910119894

119903119898(119905) minus 120594

119898119910119894

119903119898minus1(119905)] = ℏ

119903119867119903(119905) 119877119894

119903119898minus1 119905 isin [119905

119894minus1

119905119894

]

(52)


119910119894

119903119898(119905119894minus1

) = 0 (53)


L [119910119894

1199030(119905)] = 120601

119903(119905) 119905 isin [119905

119894minus1

119905119894

] (54)


119910119894

1199030(119905119894minus1

) = 119910119894minus1

119903(119905119894minus1

) (55)




119905 =

(119905119894minus 119905119894minus1

) 119909

2+

(119905119894+ 119905119894minus1

)

2

(56)


119905119894] (for 119894 = 1 119865) After the





119894

0= minus1 119909119894


119895are


degree119873 119871119873


119903119898(119905)


119889119910119894

119903119898

119889119905=

119873

sum

119896=0

D119895119896119910119894

119903119898(119909119894

119896) = DY119894

119903119898 119895 = 0 1 119873 (57)

where D = 2119863(119905119894minus 119905


119903119898= [119910119894

119903119898(119909119894

0) 119910119894

119903119898(119909119894

2)

119910119894

119903119898(119909119894



119909119894



119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)


AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)


119903 andR119894



119894





Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)


119903(119905) using (9) in the

interval [119905119894 119905119894minus1



119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)






1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)


L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[


119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894


]]]]

]

(64)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)




1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)


1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910






1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










from where

(119873

119898(119879))

2

ge1

119879

100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817

2

119879

minus 2100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

(46)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le 4119879

10038171003817100381710038171003817100381710038171003817

119889

d119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

119898 ge 0 (47)

Let 120572 = max119905isin[0119879]

|120572(119905)| and119867 = max119905isin[0119879]


10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879

le

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898

10038171003817100381710038171003817100381710038171003817119879119873

le (1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879119873

+ 119888120574 |ℏ|119867100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

+ 120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879119873+ 120572

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879119873

le 1198882(1 + |ℏ|119867)

10038171003817100381710038171003817100381710038171003817

119889

119889119905119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+ 1198882119862120574 |ℏ|119867

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879+ 1198882120572100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879

+ 1198882120572100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(48)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

+41198882119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879

(49)


2120572119879) ge 0 we have

1198882(1 + |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879le

41198882119879 (1 + |ℏ|119867)

1 minus 41198882120572119879

10038171003817100381710038171003817100381710038171003817

dd119905

119873

119898minus1

10038171003817100381710038171003817100381710038171003817119879

(50)


100381710038171003817100381710038171003817119873

119898

100381710038171003817100381710038171003817119879le

1198882(1 + |ℏ|119867) + 4119888

2119879 (120572 + 119888120574 |ℏ|119867)

1 minus 41198882120572119879

100381710038171003817100381710038171003817119873

119898minus1

100381710038171003817100381710038171003817119879 (51)

Therefore if (1198882(1 + |ℏ|119867) + 4119888

2119879(120572 + 119888120574|ℏ|119867))(1 minus 4119888

2120572119879) le

120573 lt 1 then ||119873

119898||119879





0 119905119865] we divide


119905119894] 119894 = 1 2 3 119865



1


[1199050 1199051] and let 119910119894


119894




Thus we solve

L119903[119910119894

119903119898(119905) minus 120594

119898119910119894

119903119898minus1(119905)] = ℏ

119903119867119903(119905) 119877119894

119903119898minus1 119905 isin [119905

119894minus1

119905119894

]

(52)


119910119894

119903119898(119905119894minus1

) = 0 (53)


L [119910119894

1199030(119905)] = 120601

119903(119905) 119905 isin [119905

119894minus1

119905119894

] (54)


119910119894

1199030(119905119894minus1

) = 119910119894minus1

119903(119905119894minus1

) (55)




119905 =

(119905119894minus 119905119894minus1

) 119909

2+

(119905119894+ 119905119894minus1

)

2

(56)


119905119894] (for 119894 = 1 119865) After the





119894

0= minus1 119909119894


119895are


degree119873 119871119873


119903119898(119905)


119889119910119894

119903119898

119889119905=

119873

sum

119896=0

D119895119896119910119894

119903119898(119909119894

119896) = DY119894

119903119898 119895 = 0 1 119873 (57)

where D = 2119863(119905119894minus 119905


119903119898= [119910119894

119903119898(119909119894

0) 119910119894

119903119898(119909119894

2)

119910119894

119903119898(119909119894



119909119894



119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)


AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)


119903 andR119894



119894





Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)


119903(119905) using (9) in the

interval [119905119894 119905119894minus1



119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)






1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)


L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[


119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894


]]]]

]

(64)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)




1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)


1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910






1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119909119894



119863119895119896

=

minus119873(119873 + 1)

4 if 119895 = 119896 = 0

119873 (119873 + 1)

4 if 119895 = 119896 = 119873

119871119873(119905119895)

119871119873(119905119896)

1

119905119895minus 119905119896

if 119895 = 119896

0 otherwise

(58)


AY1198941199030

= Φ119903 Y119894

1199030(119909119894minus1

119873) = Y119894minus1

119903(119909119894minus1

119873) (59)

A [Y119894119903119898

minus 120594119898Y119894119903119898minus1

] = ℏ119903R119894119903119898minus1

Y119894119903119898

(119909119894minus1

119873) = 0

(60)


119903 andR119894



119894





Y119894119903119898

= 120594119898Y119894119903119898minus1

+ ℏ119903Aminus1R119894119903119898minus1

(61)


119903(119905) using (9) in the

interval [119905119894 119905119894minus1



119910119903(119905) =

1199101

119903(119905) 119905 isin [119905

0 1199051]

1199102

119903(119905) 119905 isin [119905

1 1199052]

119910119865

119903(119905) 119905 isin [119905

119865minus1 119905119865]

(62)






1198891199101

119889119905= 1199102 1199101(0) = 119910

0

1

1198891199102

119889119905= 1199103 1199102(0) = 119910

0

2

1198891199103

119889119905= minus119888119910

1minus 1198871199102minus 1198861199103+ 1199102

1 1199103(0) = 119910

0

3

(63)


L119903=

[[[[[

[

119889

119889119905minus1 0

0119889

119889119905minus1

119888 119887119889

119889119905+ 119886

]]]]]

]

A = [

[


119888119868 119887119868 D + 119886119868

]

]

F = [

[

0

0

minus1199102

1

]

]

120601 = [

[

0

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1=

[[[[

[

0

0

minus

119898minus1

sum

119895=0

119910119894

1119895119910119894


]]]]

]

(64)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903Q119894119903119898minus1

(65)




1198891199101

119889119905= 119904 minus 120572119910

1+ 119903lowast

1199101(1 minus

1199101+ 1199102

119879max) minus 119896119910

11199103

1198891199102

119889119905= 11989611991011199103minus 1205731199102

1198891199103

119889119905= 11987311205731199102minus 1205741199103

(66)


1199101(0) = 119903

1 1199102(0) = 119903

2 1199103(0) = 119903

3 (67)

In this model 1199101 1199102 and 119910






1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199051199101(119905) 119910

2(119905) 119910

3(119905)




30 minus1358402 minus1358402 1702324 1702324 7595114 7595114


50 5405530 5405530 0923128 0923128 minus9945397 minus9945397



0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

4

5

6

(119905)

1199101(119905)

(a)

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

(119905)

1199102(119905)

(b)

0 5 10 15 20 25 30 35 40minus15

minus10

minus5

0

5

10

15

(119905)

1199103(119905)

(c)












minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus10 0 10

minus5

0

51199102(119905)

1199103(119905)

(a)


minus2

0

2

4

6

1199101(119905)

1199103(119905)

(b)

minus5 0 5

minus2

0

2

4

6

1199102(119905)

1199101(119905)

(c)



119905119879(119905) 119868(119905) 119881(119905)


10 57875453 57875453 82860772 82860772 66880364 66880364

20 30569897 30569897 77163784 77163784 75017130 75017130

30 4297717 4297717 77528130 77528130 98392328 98392328

40 1926140 1926140 50715995 50715995 69741875 69741875

50 7475706 7475706 36955303 36955303 48163241 48163241

60 17514776 17514776 57019159 57019159 64631285 64631285

70 5072849 5072849 59399657 59399657 76895482 76895482




119904 = 01 120572 = 002 120573 = 03 119903lowast

= 3 120574 = 24

119896 = 00027 119879max = 1500 1198731= 10

(68)


L119903=

[[[[[

[

119889


0119889

119889119905+ 120573 0

0 minus1198731120573

119889

119889119905+ 120574

]]]]]

]


0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

500

1000

1500

(119905)

1199101(119905)

(a)

0 20 40 60 80 1000

200

400

600

800

1000

1200

(119905)

1199102(119905)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

(119905)

1199103(119905)

(c)


A = [

[


O D + 120573119868 OO minus119873

1120573 D + 120574119868

]

]

F =

[[[

[

1199031199101(1199101+ 1199102

119879max) + 119896119910

11199103

minus11989611991011199103

0

]]]

]

120601 = [

[

119904

0

0

]

]

119877119894

119903119898minus1= L119903[119910119894

119903119898minus1] + 119876119894

119903119898minus1

119876119894

119903119898minus1

=

[[[[[[[

[

119903lowast

119898minus1

sum

119895=0

119910119894

1119895(119910119894

1119898minus1minus119895+ 119910119894

2119898minus1minus119895) + 119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


minus119896

119898minus1

sum

119895=0

119910119894

1119895119910119894


0

]]]]]]]

]

(69)


AY119894119903119898

= (120594119898+ ℏ119903)AY119894119903119898minus1

+ ℏ119903[Q119894119903119898minus1

minus (1 minus 120594119898)Φ]

(70)








0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 15000

500

1000

1500

1199102(119905)

1199101(119905)

(a)

0

500

1000

1500

0 500 1000 15001199101(119905)

1199103(119905)

(b)

0 500 1000 15000

500

1000

1500

1199102(119905)

1199103(119905)

(c)



119903= minus1 We










2(0) =







01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











01 1199102(0) = 0 and 119910






7 Conclusion


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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