research article a convenient adomian-pade …downloads.hindawi.com/journals/aaa/2014/601643.pdf ·...

Research ArticleA Convenient Adomian-Pade Technique forthe Nonlinear Oscillator Equation

Na Wei1 Cheng-Xi Liu2 Teng Hu2 and Fu-Cai You3

1 State Key Laboratory of Oil and Gas Reservoirs Geology and Exploration Southwest Petroleum University Chengdu 610500 China2 Key Laboratory of Numerical Simulation of Sichuan Province and College of Mathematics and Information ScienceNeijiang Normal University Neijiang 641112 China

3Department of Basic Sciences Shenyang Institute of Engineering Shenyang 110136 China

Correspondence should be addressed to Cheng-Xi Liu liuchengxi050317163com

Received 26 February 2014 Revised 12 March 2014 Accepted 12 March 2014 Published 9 April 2014

Academic Editor Dumitru Baleanu

Copyright copy 2014 Na Wei et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Very recently the convenient way to calculate the Adomian series was suggestedThis paper combines this technique and the Padeapproximation to develop some new iteration schemesThen the combinedmethod is applied to nonlinearmodels and the residualfunctions illustrate the accuracies and conveniences

1 Introduction

Analytical methods for nonlinear systems have caught muchattention due to their convenience for obtaining solutionsin real engineering problems One of the most often usedmethods is the Adomian decomposition method (ADM) [1]Also due to the rapid development of the computer sciencevarious modifications of these nonlinear analytical methodshave been proposed and have been extensively applied tovarious nonlinear systems [2ndash14]

Very recently for the ADM Duan [4ndash6] suggested aconvenient Adomian calculation scheme The method canhelp us get a higher accuracy and can hand higher orderapproximation problem due to its easier calculation of theAdomian series than the classical one [1] The technique hasbeen successfully extended to fractional differential equationsand boundary value problems

Recently Tsai and Chen [11] proposed a Laplace-Ado-mian-Pademethod (LAPM)Themethod holds the followingmerits (a) Laplace transform can be used to determine theinitial iteration value (b) the Pade technique is adopted toaccelerate the convergence

With Duan and Tsairsquos idea this paper suggests a novelapproximation scheme for the oscillating physical mecha-nism of the nonlinear models [15]

119889119879

119889119905= 119862119879 + 119863ℎ minus 120576119879

3 119879 (0) = 1

119889ℎ

119889119905= minus 119864119879 minus 119877

ℎℎ ℎ (0) = 1

(1)

where 119862119863 119864 and 119877ℎare physical constants 119879 describes the

temperature of the eastern equatorial Pacific sea surface andℎ is the thermocline depth anomaly

2 Preliminaries of the Adomian Series

Generally consider the following nonlinear equation

119871 [119906] + 119877 [119906] + 119873 [119906] = 119892 (119905) (2)

where 119871 is the highest derivative of 119906 for example 119898 order119877 is the remaining linear part containing the lower orderderivatives and119873 is the nonlinear operator

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 601643 5 pageshttpdxdoiorg1011552014601643

2 Abstract and Applied Analysis

Apply the inverse 119871minus1 of the linear operator 119871 in (2) andwe can obtain

119906 = 119906 (0) + 1199061015840(0) 119905 + sdot sdot sdot + 119906

(119898minus1)(0)

119905119898minus1

(119898 minus 1)

+ 119871minus1(119892 (119905)) minus 119871

minus1(119877 [119906]) minus 119871

minus1(119873 [119906])

(3)

Consider the basic idea of the Picard method

119906119899+1

= 119906 (0) + 1199061015840(0) 119905 + sdot sdot sdot + 119906

(119898minus1)(0)

119905119898minus1

(119898 minus 1)

+ 119871minus1(119892 (119905)) minus 119871

minus1(119877 [119906119899]) minus 119871

minus1(119873 [119906

119899])

(4)

and assume that

119906 =

infin

sum

119894=0

V119894 119906

119899=

119899

sum

119894=0

V119894 (5)

The classical ADM [1] supposes that the nonlinear term119873[119906] can be expanded approximately as

119873[119906] =

infin

sum

119899=0

119860119899 (6)

where 119860119899is calculated by

119860119899=1

119899

120597119899

120597120582119899[119873(

infin

sum

119896=0

V119896120582119896)]

120582=0

(7)

For example suminfin119899=0

119860119899is the Adomian series of 1198793 namely

1198600= V30

1198601= 3V20V1

1198602= 3V0V21+ 3V20V2

(8)

Duan et al [4ndash6] very recently suggested a convenientway to calculate the Adomian series as

119860119899=1

119899

119899minus1

sum

119896=0

(119896 + 1) V119896+1

119889119860119899minus1minus119896

119889V0

(9)

as well as the case of the119898-variable

119860119899=1

119899

119898

sum

119894=1

119899minus1

sum

119896=0

(119896 + 1) V119894119896+1120597119860119899minus1minus119896

120597V1198940

(10)

For the single variable case 119873[119906] = 119891(119906) the first threecomponents are listed as

1198601= V1

119889119891 (V0)

119889V0

1198602=1

2V1

21198892119891 (V0)

119889V02

+ V2

119889119891 (V0)

119889V0

1198603=1

6V1

31198893119891 (V0)

119889V03

+ V2V1

1198892119891 (V0)

119889V02

+ V3

119889119891 (V0)

119889V0

1198604=1

24V1

41198894119891 (V0)

119889V04

+1

2V2V1

21198893119891 (V0)

119889V03

+ V3V1

1198892119891 (V0)

119889V02

+1

2V2

21198892119891 (V0)

119889V02

+ V4

119889119891 (V0)

119889V0

(11)

And for the two-variable case 119873[119906] = 119891(1199061 1199062) the first

three components are listed as

1198601= V11

120597119891 (V10 V20)

120597V10

+ V21

120597119891 (V10 V20)

120597V20

1198602=1

2V11

21205972119891 (V10 V20)

120597V102

+ V11V21

1205972119891 (V10 V20)

120597V20120597V10

+ V12

120597119891 (V10 V20)

120597V10

+1

2V21

21205972119891 (V10 V20)

120597V202

+ V22

120597119891 (V10 V20)

120597V20

1198603=1

6V11

31205973119891 (V10 V20)

120597V103

+1

2V11

2V21

1205973119891 (V10 V20)

120597V20120597V102

+ V11V12

1205972119891 (V10 V20)

120597V102

+1

2V11V21

21205973119891 (V10 V20)

120597V202120597V10

+ V11V22

1205972119891 (V10 V20)

120597V20120597V10

+ V12V21

1205972119891 (V10 V20)

120597V20120597V10

+ V13

120597119891 (V10 V20)

120597V10

+1

6V21

31205973119891 (V10 V20)

120597V203

+ V21V22

1205972119891 (V10 V20)

120597V202

+ V23

120597119891 (V10 V20)

120597V20

(12)

The above formulae (9) spend less time deriving the 119860119899 On

the other hand this provides a possible tool to investigate thehigher order approximation solution

Abstract and Applied Analysis 3

3 Iteration Schemes Based on the ConvenientAdomian Series

Now we present our analytical schemes using the convenientAdomian series Laplace transform and Pade approximationWe adopt the steps in [16] Considering (2) we show thefollowing iteration schemes

(i) Take Laplace transform to both sides

[119871 [119906] + 119877 [119906]] + [119873 [119906]] = [119892 (119905)] (13)

We can have iteration formula (4) through inverse of Laplacetransform

minus1

119906 (119905) = 119891 (119905) + minus1[ (119904) [119873 [119906]]] (14)

where 119891(119905) and (119904) can be determined by calculation ofLaplace transform to 119871[119906] 119877[119906] and 119892(119905) The calculation of(119904) is similar to the determination of the Lagrangemultiplierof the variational iteration method in [17]

(ii) Through the Picard successive approximation wecan obtain the following iteration formula

119906119899+1

= 119891 (119905) + minus1[ (119904) [119873 [119906

119899]]] (15)

(iii) Let 119906119899= sum119899

119894=0V119894and apply the Adomian series to

expand the term119873[119906] as suminfin119894=0119860119894 Then the iteration

formula reads

V119899+1

= minus1[ (119904) [119860119899]]

V0= 119891 (119905)

(16)

where 119860119894are calculated by

119860119899=1

119899

119899minus1

sum

119896=0

(119896 + 1) V119896+1

119889119860119899minus1minus119896

119889V0

(17)

(iv) Employ the Pade technique to accelerate theconvergence of 119906

119899= sum119899

119894=0V119894

4 Applications of the Iteration Formulae

In this study we consider a reduced case where 119863 = 0 and0 lt 120576 ≪ 1 in (1) as follows

119889119879

119889119905= 119862119879 minus 120576119879

3 119862 = 1 119879 (0) = 1 (18)

In order to solve (18) with the Maple software applyLaplace transform 119871 to both sides firstly This step can fullyand optimally determine the initial iteration We can derive

V119899+1 (119905) = minus

minus1[120576 [119860

119899]

(119904 minus 119862)] 119899 ge 1

V0= minus1[119879 (0)

(119904 minus 119862)]

(19)

Formula (19

)Formula (

21

)

2

0

minus2

minus4

minus6

minus8

minus10

minus12

543210

gn

t

Figure 1 The comparisons of the approximate solutions using (19)and (21)

Setting 120576 = 000001 in the model (18) now we can obtainthe first few as

1198790= V0= e119905

1198791= V0+ V1= e119905 minus 00001 e2119905 sinh (119905)

(20)

Apply the Pade technique to 119879119899and denote the result as

1198751198791119899[119901119902]

We now can compare the accuracies of the differentversions of the Adomian decomposition methods

For example we can write out the classical Adomianformula for (18) as

V119899+1

(119905) = 119862int

119905

0

V119899119889120591 minus 120576int

119905

0

119860119899119889120591 119899 ge 1

V0= 119879 (0)

(21)

Also apply the Pade technique to 119879119899and denote the result as

1198751198792119899[119901119902]

Define the residual function as

119892119899= log

100381610038161003816100381610038161003816100381610038161003816

119889 (119875119879119894119899[119901119902])

119889119905minus 119862(119875119879

119894119899[119901

119902])

+ 120576(119875119879119894119899[119901

119902])

3100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2

(22)

Consider the same 119899 = 20 and 119901 = 119902 = 20 fromthe comparison illustrated through Figure 1 we can see that


80

70

60

50

40

30

20

10

Tn

543210

t

Figure 2 Analytical solution of (18) via (19)

the iteration formula (19) has a higher accuracy almost in theinterval [0 5]

As a result we decide to adopt the iteration formula(19) and give the numerical simulation of (18) in the caseof the higher order approximation The analytical solution isillustrated in Figure 2

The approximate solution is reliable from the erroranalysis of the iteration formula (19) in Figure 1

5 Conclusions

The approximate solution is compared with the nonlineartechniques in higher order iteration and the result shows thenew wayrsquos higher accuracy to calculate the Adomian seriesIn view of this point the comparison of different versionsof the Adomian method is possible The results show thatthe iteration formula fully using all the linear parts has ahigher accuracy It provides an efficient tool to select a suitablealgorithm when solving engineering problems

Conflict of Interests

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

Acknowledgments

This work is supported by the Open Fund of State Key Labo-ratory of Oil and Gas Geology and Exploration SouthwestPetroleum University (PLN1309) National Natural ScienceFoundation of China ldquoStudy on wellbore flow model inliquid-based whole process underbalanced drillingrdquo (GrantNo 51204140) the Scientific Research Fund of Sichuan

Provincial Education Department (4ZA0244) and the Pro-gram for Liaoning Excellent Talents in University underGrant no LJQ2011136

References

[1] G Adomian Solving Frontier Problems of Physics the Decom-position Method Kluwer Academic Publishers Boston MassUSA 1994

[2] S S Ray andRK Bera ldquoAn approximate solution of a nonlinearfractional differential equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 167 no1 pp 561ndash571 2005

[3] H Jafari and V Daftardar-Gejji ldquoRevised Adomian decom-position method for solving a system of nonlinear equationsrdquoApplied Mathematics and Computation vol 175 no 1 pp 1ndash72006

[4] J-S Duan ldquoRecurrence trianglefor Adomian polynomialsrdquoAppliedMathematics and Computation vol 216 no 4 pp 1235ndash1241 2010

[5] J-S Duan ldquoAn efficient algorithm for the multivariable Ado-mian polynomialsrdquoAppliedMathematics and Computation vol217 no 6 pp 2456ndash2467 2010

[6] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012

[7] G-C Wu ldquoAdomian decomposition method for non-smoothinitial value problemsrdquoMathematical and Computer Modellingvol 54 no 9-10 pp 2104ndash2108 2011

[8] V Daftardar-Gejji and H Jafari ldquoAdomian decomposition atool for solving a system of fractional differential equationsrdquoJournal of Mathematical Analysis and Applications vol 301 no2 pp 508ndash518 2005

[9] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

[10] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[11] P-Y Tsai and C-K Chen ldquoAn approximate analytic solutionof the nonlinear Riccati differential equationrdquo Journal of theFranklin Institute vol 347 no 10 pp 1850ndash1862 2010

[12] D Q Zeng and Y M Qin ldquoThe Laplace-Adomian-Padetechnique for the seepage flows with the Riemann-Liouvillederivativesrdquo Communications in Fractional Calculus vol 3 no1 pp 26ndash29 2012

[13] H Jafari C M Khalique and M Nazari ldquoApplication of theLaplace decompositionmethod for solving linear and nonlinearfractional diffusionwave equationsrdquo Applied Mathematics Let-ters vol 24 no 11 pp 1799ndash1805 2011

[14] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquoComputersampMathematics withApplications vol 66no 5 pp 838ndash843 2013

[15] J QMo andW-T Lin ldquoGeneralized variation iteration solutionof an atmosphere-ocean oscillator model for global climaterdquoJournal of Systems Science andComplexity vol 24 no 2 pp 271ndash276 2011


[16] Y Zeng ldquoThe Laplace-Adomian-Pade technique for the ENSOmodelrdquoMathematical Problems in Engineering vol 2013 ArticleID 954857 4 pages 2013

[17] G C Wu ldquoChallenge in the variational iteration methodmdasha new approach to identification of the Lagrange multipliersrdquoJournal of King Saud UniversitymdashScience vol 25 no 2 pp 175ndash178 2013

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Apply the inverse 119871minus1 of the linear operator 119871 in (2) andwe can obtain

119906 = 119906 (0) + 1199061015840(0) 119905 + sdot sdot sdot + 119906

(119898minus1)(0)

119905119898minus1

(119898 minus 1)

+ 119871minus1(119892 (119905)) minus 119871

minus1(119877 [119906]) minus 119871

minus1(119873 [119906])

(3)

Consider the basic idea of the Picard method

119906119899+1

= 119906 (0) + 1199061015840(0) 119905 + sdot sdot sdot + 119906

(119898minus1)(0)

119905119898minus1

(119898 minus 1)

+ 119871minus1(119892 (119905)) minus 119871

minus1(119877 [119906119899]) minus 119871

minus1(119873 [119906

119899])

(4)

and assume that

119906 =

infin

sum

119894=0

V119894 119906

119899=

119899

sum

119894=0

V119894 (5)

The classical ADM [1] supposes that the nonlinear term119873[119906] can be expanded approximately as

119873[119906] =

infin

sum

119899=0

119860119899 (6)

where 119860119899is calculated by

119860119899=1

119899

120597119899

120597120582119899[119873(

infin

sum

119896=0

V119896120582119896)]

120582=0

(7)

For example suminfin119899=0

119860119899is the Adomian series of 1198793 namely

1198600= V30

1198601= 3V20V1

1198602= 3V0V21+ 3V20V2

(8)

Duan et al [4ndash6] very recently suggested a convenientway to calculate the Adomian series as

119860119899=1

119899

119899minus1

sum

119896=0

(119896 + 1) V119896+1

119889119860119899minus1minus119896

119889V0

(9)

as well as the case of the119898-variable

119860119899=1

119899

119898

sum

119894=1

119899minus1

sum

119896=0

(119896 + 1) V119894119896+1120597119860119899minus1minus119896

120597V1198940

(10)

For the single variable case 119873[119906] = 119891(119906) the first threecomponents are listed as

1198601= V1

119889119891 (V0)

119889V0

1198602=1

2V1

21198892119891 (V0)

119889V02

+ V2

119889119891 (V0)

119889V0

1198603=1

6V1

31198893119891 (V0)

119889V03

+ V2V1

1198892119891 (V0)

119889V02

+ V3

119889119891 (V0)

119889V0

1198604=1

24V1

41198894119891 (V0)

119889V04

+1

2V2V1

21198893119891 (V0)

119889V03

+ V3V1

1198892119891 (V0)

119889V02

+1

2V2

21198892119891 (V0)

119889V02

+ V4

119889119891 (V0)

119889V0

(11)

And for the two-variable case 119873[119906] = 119891(1199061 1199062) the first

three components are listed as

1198601= V11

120597119891 (V10 V20)

120597V10

+ V21

120597119891 (V10 V20)

120597V20

1198602=1

2V11

21205972119891 (V10 V20)

120597V102

+ V11V21

1205972119891 (V10 V20)

120597V20120597V10

+ V12

120597119891 (V10 V20)

120597V10

+1

2V21

21205972119891 (V10 V20)

120597V202

+ V22

120597119891 (V10 V20)

120597V20

1198603=1

6V11

31205973119891 (V10 V20)

120597V103

+1

2V11

2V21

1205973119891 (V10 V20)

120597V20120597V102

+ V11V12

1205972119891 (V10 V20)

120597V102

+1

2V11V21

21205973119891 (V10 V20)

120597V202120597V10

+ V11V22

1205972119891 (V10 V20)

120597V20120597V10

+ V12V21

1205972119891 (V10 V20)

120597V20120597V10

+ V13

120597119891 (V10 V20)

120597V10

+1

6V21

31205973119891 (V10 V20)

120597V203

+ V21V22

1205972119891 (V10 V20)

120597V202

+ V23

120597119891 (V10 V20)

120597V20

(12)

The above formulae (9) spend less time deriving the 119860119899 On

the other hand this provides a possible tool to investigate thehigher order approximation solution





[119871 [119906] + 119877 [119906]] + [119873 [119906]] = [119892 (119905)] (13)


minus1

119906 (119905) = 119891 (119905) + minus1[ (119904) [119873 [119906]]] (14)



119906119899+1

= 119891 (119905) + minus1[ (119904) [119873 [119906

119899]]] (15)

(iii) Let 119906119899= sum119899



formula reads

V119899+1

= minus1[ (119904) [119860119899]]

V0= 119891 (119905)

(16)


119860119899=1

119899

119899minus1

sum

119896=0

(119896 + 1) V119896+1

119889119860119899minus1minus119896

119889V0

(17)


119899= sum119899

119894=0V119894



119889119879

119889119905= 119862119879 minus 120576119879

3 119862 = 1 119879 (0) = 1 (18)


V119899+1 (119905) = minus

minus1[120576 [119860

119899]

(119904 minus 119862)] 119899 ge 1

V0= minus1[119879 (0)

(119904 minus 119862)]

(19)

Formula (19

)Formula (

21

)

2

0

minus2

minus4

minus6

minus8

minus10

minus12

543210

gn

t



1198790= V0= e119905

1198791= V0+ V1= e119905 minus 00001 e2119905 sinh (119905)

(20)


1198751198791119899[119901119902]



V119899+1

(119905) = 119862int

119905

0

V119899119889120591 minus 120576int

119905

0

119860119899119889120591 119899 ge 1

V0= 119879 (0)

(21)


1198751198792119899[119901119902]


119892119899= log

100381610038161003816100381610038161003816100381610038161003816

119889 (119875119879119894119899[119901119902])

119889119905minus 119862(119875119879

119894119899[119901

119902])

+ 120576(119875119879119894119899[119901

119902])

3100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2

(22)



80

70

60

50

40

30

20

10

Tn

543210

t





5 Conclusions




Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













[119871 [119906] + 119877 [119906]] + [119873 [119906]] = [119892 (119905)] (13)


minus1

119906 (119905) = 119891 (119905) + minus1[ (119904) [119873 [119906]]] (14)



119906119899+1

= 119891 (119905) + minus1[ (119904) [119873 [119906

119899]]] (15)

(iii) Let 119906119899= sum119899



formula reads

V119899+1

= minus1[ (119904) [119860119899]]

V0= 119891 (119905)

(16)


119860119899=1

119899

119899minus1

sum

119896=0

(119896 + 1) V119896+1

119889119860119899minus1minus119896

119889V0

(17)


119899= sum119899

119894=0V119894



119889119879

119889119905= 119862119879 minus 120576119879

3 119862 = 1 119879 (0) = 1 (18)


V119899+1 (119905) = minus

minus1[120576 [119860

119899]

(119904 minus 119862)] 119899 ge 1

V0= minus1[119879 (0)

(119904 minus 119862)]

(19)

Formula (19

)Formula (

21

)

2

0

minus2

minus4

minus6

minus8

minus10

minus12

543210

gn

t



1198790= V0= e119905

1198791= V0+ V1= e119905 minus 00001 e2119905 sinh (119905)

(20)


1198751198791119899[119901119902]



V119899+1

(119905) = 119862int

119905

0

V119899119889120591 minus 120576int

119905

0

119860119899119889120591 119899 ge 1

V0= 119879 (0)

(21)


1198751198792119899[119901119902]


119892119899= log

100381610038161003816100381610038161003816100381610038161003816

119889 (119875119879119894119899[119901119902])

119889119905minus 119862(119875119879

119894119899[119901

119902])

+ 120576(119875119879119894119899[119901

119902])

3100381610038161003816100381610038161003816100381610038161003816

119894 = 1 2

(22)



80

70

60

50

40

30

20

10

Tn

543210

t





5 Conclusions




Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










80

70

60

50

40

30

20

10

Tn

543210

t





5 Conclusions




Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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