Research Article A Comparison between Adomian s...
5
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 943232, 4 pages http://dx.doi.org/10.1155/2013/943232 Research Article A Comparison between Adomian’s Polynomials and He’s Polynomials for Nonlinear Functional Equations Hossein Jafari, 1,2 Saber Ghasempoor, 1 and Chaudry Masood Khalique 2 1 Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran 2 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa Correspondence should be addressed to Hossein Jafari; [email protected] Received 20 March 2013; Revised 11 May 2013; Accepted 2 June 2013 Academic Editor: Mufid Abudiab Copyright © 2013 Hossein Jafari et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We will compare the standard Adomian decomposition method and the homotopy perturbation method applied to obtain the solution of nonlinear functional equations. We prove analytically that the two methods are equivalent for solving nonlinear functional equations. In Ghorbani (2009), Ghorbani presented a new definition which he called as He’s polynomials. In this paper, we also show that He’s polynomials are only the Adomian polynomials. 1. Introduction e Adomian decomposition method (ADM) and the homo- topy perturbation method (HPM) are two powerful methods which consider the approximate solution of a nonlinear equa- tion as an infinite series usually converging to the accurate solution. ese methods have been used in obtaining analytic and approximate solutions to a wide class of linear and non- linear, differential, and integral equations. ¨ Ozis ¸ and Yıldırım compared Adomian’s method and He’s homotopy perturbation method [1] for solving certain non- linear problems. Li also has shown that the ADM and HPM for solving nonlinear equations are equivalent [2]. In [3], Ghorbani has presented a definition which he called it as He’s polynomials. Consider the following nonlinear functional equation: = + () , (1) where is a nonlinear operator from Hilbert space to , is an unknown function, and is a known function in . We are looking for a solution of (1) belonging to . We will suppose that (1) admits a unique solution. If (1) does not possess a unique solution, the ADM and HPM will give a solution among many (possible) other solutions. However, relatively few papers deal with the comparison of these methods with other existing techniques. In [4], a useful comparison between the decomposition method and the per- turbation method showed the efficiency of the decomposition method compared to the tedious work required by the pertur- bation techniques. In [5], the advantage of the decomposition method over the Picard’s method has been emphasized. Sadat has shown that the Adomian decomposition method and perturbation method are closely related and lead to the same solution in many heat conduction problems [6]. In [7, 8] the HPM has compared with Liao’s homotopy analysis method and showed the HPM is special case of HAM, and the advantage of the HAM over the HPM has been emphasized. In this paper, we want to prove that He’s polynomials are only Adomian’s polynomials. We will also show that the standard Adomian decomposition method and the standard HPM are equivalent when applied for solving nonlinear functional equations. 2. Adomian’s Decomposition Method (ADM) Let us consider the nonlinear equation (1) which can be written in the following canonical form: = + () . (2)
Transcript of Research Article A Comparison between Adomian s...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 943232 4 pageshttpdxdoiorg1011552013943232
Research ArticleA Comparison between Adomianrsquos Polynomials and HersquosPolynomials for Nonlinear Functional Equations
Hossein Jafari12 Saber Ghasempoor1 and Chaudry Masood Khalique2
1 Department of Mathematics University of Mazandaran PO Box 47416-95447 Babolsar Iran2 International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical SciencesNorth-West University Mafikeng Campus Mmabatho 2735 South Africa
Correspondence should be addressed to Hossein Jafari jafariumzacir
Received 20 March 2013 Revised 11 May 2013 Accepted 2 June 2013
Academic Editor Mufid Abudiab
Copyright copy 2013 Hossein Jafari 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
We will compare the standard Adomian decomposition method and the homotopy perturbation method applied to obtain thesolution of nonlinear functional equations We prove analytically that the two methods are equivalent for solving nonlinearfunctional equations In Ghorbani (2009) Ghorbani presented a new definition which he called as Hersquos polynomials In this paperwe also show that Hersquos polynomials are only the Adomian polynomials
1 Introduction
TheAdomian decompositionmethod (ADM) and the homo-topy perturbation method (HPM) are two powerful methodswhich consider the approximate solution of a nonlinear equa-tion as an infinite series usually converging to the accuratesolutionThesemethods have been used in obtaining analyticand approximate solutions to a wide class of linear and non-linear differential and integral equations
Ozis and Yıldırım compared Adomianrsquos method and Hersquoshomotopy perturbation method [1] for solving certain non-linear problems Li also has shown that the ADM and HPMfor solving nonlinear equations are equivalent [2] In [3]Ghorbani has presented a definition which he called it as Hersquospolynomials
Consider the following nonlinear functional equation
119906 = 119891 + 119873 (119906) (1)
where 119873 is a nonlinear operator from Hilbert space 119867 to119867 119906 is an unknown function and 119891 is a known functionin 119867 We are looking for a solution 119906 of (1) belonging to 119867We will suppose that (1) admits a unique solution If (1) doesnot possess a unique solution the ADM and HPM will givea solution among many (possible) other solutions Howeverrelatively few papers deal with the comparison of these
methods with other existing techniques In [4] a usefulcomparison between the decompositionmethod and the per-turbationmethod showed the efficiency of the decompositionmethod compared to the tediouswork required by the pertur-bation techniques In [5] the advantage of the decompositionmethod over the Picardrsquos method has been emphasized Sadathas shown that the Adomian decomposition method andperturbation method are closely related and lead to the samesolution in many heat conduction problems [6] In [7 8] theHPM has compared with Liaorsquos homotopy analysis methodand showed the HPM is special case of HAM and theadvantage of the HAM over the HPM has been emphasized
In this paper we want to prove that Hersquos polynomialsare only Adomianrsquos polynomials We will also show that thestandard Adomian decomposition method and the standardHPM are equivalent when applied for solving nonlinearfunctional equations
2 Adomianrsquos Decomposition Method (ADM)
Let us consider the nonlinear equation (1) which can bewritten in the following canonical form
119906 = 119891 + 119873 (119906) (2)
2 Mathematical Problems in Engineering
The standardADMconsists of representing the solution of (1)as a series
119906 (119909) =
infin
sum
119894=0
119906119894(119909) (3)
and the nonlinear function as the decomposed form
119873(119906 (119909)) =
infin
sum
119894=0
119860119894 (4)
where 119860119899 119899 = 0 1 2 are the Adomian polynomials of
1199060 1199061 119906
119899given by [9 10]
119860119899=
1
119899
119889119899
119889119901119899[119873(
119899
sum
119894=0
119906119894119901119894
)]
119901=0
(5)
Substituting (3) and (4) into (1) yieldsinfin
sum
119894=0
119906119894(119909) = 119891 +
infin
sum
119894=0
119860119894 (6)
The convergence of the series in (6) gives the desired relation
1199060= 119891
119906119899+1
= 119860119899 119899 = 0 1 2
(7)
It should be pointed out that 1198600depends only on 119906
0 1198601
depends only on 1199060and11990611198602depends only on 119906
01199061 and119906
2
and so onTheAdomian technique is very simple in its princi-plesThe difficulties consist in proving the convergence of theintroduced series
3 Homotopy Perturbation Method (HPM)
This is a basic idea of homotopy method which is to con-tinuously deform a simple problem easy to solve into thedifficult problem under study
In this section we apply the homotopy perturbationmethod [11ndash13] to the discussed problem To illustrate thehomotopy perturbation method (HPM) we consider (1) as
119871 (V) = V (119909) minus 119891 (119909) minus 119873 (V) = 0 (8)
with solution 119906(119909) The basic idea of the HPM is to constructa homotopy119867(V 119901) 119877 times [0 1] rarr 119877 which satisfies
H (V 119901) = (1 minus 119901) 119865 (V) + 119901119871 (V) = 0 (9)
where 119865(V) is a proper function with known solution whichcan be obtained easily The embedding parameter 119901 mono-tonically increases from 0 to 1 as the trivial problem 119865(V) = 0
is continuously transformed to the original problem V minus 119891 minus
119873(V) = 0 FromH(V 119901) = 0 we have119867(V 0) = 119865(V) = 0 and119867(V 1) = V minus 119891 minus 119873(V) = 0
It is better to take 119865(V) as a deformation of 119871(V) Forexample in (9) 119865(V) = Vminus119891(119909) By selecting 119865(V) = Vminus119891(119909)we can define another convex homotopyH(V 119901) by
H (V 119901) = V (119909) minus 119891 (119909) minus 119901119873 (V) = 0 (10)
The embedding parameter 119901 isin (0 1] can be considered as anexpanding parameter [14 15] The HPM uses the embeddingparameter 119901 as a ldquosmall parameterrdquo and writes the solution of(10) as a power series of 119901 that is
V = V0+ V1119901 + V21199012
+ sdot sdot sdot (11)
Setting 119901 = 1 results in the approximate solution of (10)
119906 = lim119901rarr1
V = V0+ V1+ V2+ sdot sdot sdot (12)
Substituting (11) into (10) and equating the terms withidentical powers of 119901 we can obtain a series of equations ofthe following form
1199010 V0minus 119891 (119909) = 0
1199011 V1minus 119867 (V
0) = 0
1199012 V2minus 119867 (V
0 V1) = 0
1199013 V3minus 119867 (V
0 V1 V2) = 0
(13)
where119867(V0 V1 V
119895) depend upon V
0 V1 V
119895 In view of
(10) to determine119867(V0 V1 V
119895) we use [16]
119867(V0 V1 V
119895) =
1
119895
120597119895
120597119901119895119873(
119895
sum
119894=0
V119894119901119894
)
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(14)
It is obvious that the system of nonlinear equations in (13) iseasy to solve and the components V
119894 119894 ge 0 of the homotopy
perturbation method can be completely determined and theseries solutions are thus entirely determined For the conver-gence of the previous method we refer the reader to the workof He [12 17 18]
4 Equivalence between ADM and HPM
In this section we prove that the HPM and the ADM givesame solution for solving nonlinear functional equationsWealso show that the He polynomials are like the Adomianpolynomials In [3] Ghorbani has presented the followingdefinition
Definition 1 (see [3]) The He polynomials are defined asfollows
119867119899(V0 V
119899) =
1
119899
120597119899
120597119901119899119873(
119899
sum
119894=0
V119894119901119894
)
1003816100381610038161003816100381610038161003816100381610038161003816119901=0
119899 = 0 1 2
(15)
Note 1 Comparison between (5) and (15) has shown that theHe polynomials are only Adomianrsquos polynomials and it iscalculated like Adomianrsquos polynomials
Mathematical Problems in Engineering 3
Theorem 2 Suppose that nonlinear function 119873(119906) and theparameterized representation of V are V(119901) = sum
infin
119894=0V119894119901119894 where
119901 is a parameter then we have
120597119899
119873(V (119901))
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(suminfin
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(16)
Proof (see [3 19]) In Theorem 3 we prove that the He poly-nomials are the Adomian polynomials
Theorem 3 The He polynomials which are given by (15) arethe Adomian polynomials
Proof From Taylorrsquos expansion of119873(V) we have
119873(V) = 119873 (V0) + 119873
1015840
(V0) (V minus V
0)
+
1
2
11987310158401015840
(V0) (V minus V
0)2
+ sdot sdot sdot
(17)
substituting (11) in (17) and expanding it in terms of 119901 leadsto
119873(
infin
sum
119894=0
V119894119901119894
) = 119873(V0) + 119873
1015840
(V0) (V1119901 + V21199012
+ sdot sdot sdot )
+
1
2
11987310158401015840
(V0) (V1119901 + V21199012
+ sdot sdot sdot )
2
+ sdot sdot sdot
= 119873 (V0) + 119873
1015840
(V0) V1119901
+ (1198731015840
(V0) V2+
1
2
11987310158401015840
(V0) V21)1199012
+ sdot sdot sdot
= 1198670+ 1198671119901 + 119867
21199012
+ sdot sdot sdot
(18)where119867
119894 119894 = 0 1 2 depends only on V
0 V1 V
119894
In order to obtain119867119899 we give 119899-order derivative of both
sides of (18) with respect to 119901 and let 119901 = 0 that is
120597119899
119873(V (119901))
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
suminfin
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
(19)
According toTheorem 2
120597119899
119873(suminfin
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
120597119899
suminfin
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
sum119899
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
= 119899119867119899
(20)
We know that 119867119894
just depends on V0 V1 V
119894so
(120597119899
sum119899
119894=0119867119894119901119894
)120597119901119899
|119901=0
= 119899119867119899 Substituting (20) in (19) leads
us to find119867119894in the following form
119867119899=
1
119899
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(21)
which is called for the first time by Ghorbani as the Hepolynomials [3]
Theorem 4 The homotopy perturbation method for solvingnonlinear functional equations is the Adomian decompositionmethod with the homotopyH(V 119901) given by
H (V 119901) = V minus 119891 (119909) minus 119901119873 (V) (22)
Proof Substituting (11) and (18) into (10) and equating theterms with the identical powers of 119901 we have
H (V 119901) =infin
sum
119894=0
V119894119901119894
minus 119891 (119909) minus 119901
infin
sum
119894=0
119867119894119901119894
= 0
H (V 119901) = V0minus 119891 (119909) +
infin
sum
119894=0
(V119894+1
minus 119867119894) 119901119894+1
= 0
(23)
1199010 V0minus 119891 (119909) = 0
119901119899+1 V119899+1
minus 119867119899= 0 119899 = 0 1 2
(24)
From (24) we have
V0= 119891 (119909)
V119899+1
= 119867119899 119899 = 0 1 2
(25)
According to Theorem 3 we have119867119899= 119860119899 Substituting (25)
in (11) leads us to
V = V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119891 (119909) + 1198600119901 + 119860
11199012
+ sdot sdot sdot
(26)
so
lim119901rarr1
V = 119891 (119909) + 1198601+ 1198602+ sdot sdot sdot
= 119891 (119909) +
infin
sum
119894=0
119860119894=
infin
sum
119894=0
119906119894= 119906
(27)
Therefore by letting
H (V 119901) = V minus 119891 (119909) minus 119901119873 (V) (28)
we observe that the power series V0+ V1119901 + V
21199012
+ sdot sdot sdot
corresponds to the solution of the equation H(V 119901) = V minus119891(119909) minus 119901119873(V) = 0 and becomes the approximate solution of(1) if 119901 rarr 1 This shows that the homotopy perturbationmethod is the Adomian decomposition method with thehomotopy H(V 119901) given by (28) The proof of Theorem 4 iscompleted
These two approaches give the same equations for high-order approximations This is mainly because the Taylorseries of a given function is unique which is a basic theoryin calculus Thus nothing is new in Ghorbanirsquos definitionexcept the new name ldquoHersquos polynomialrdquo He just employed theearly ideas of ADM
4 Mathematical Problems in Engineering
Example 5 (see [20]) Consider the following nonlinear Vol-terra integral equation
119910 (119909) = 119909 + int
119909
0
1199102
(119905) 119889119905 (29)
with the exact solution 119910(119909) = tan119909
We apply standard ADM and HPM For applying stan-dard HPM we construct following homotopy
H (119906 119901) = 119906 (119909) minus 119909 minus 119901int
119909
0
[119906 (119905)]2
119889119905 = 0 (30)
In view of (13) we have
1199010 V0(119909) minus 119909 = 0
119901119899 V119899+1
(119909) minus int
119909
0
119867(V0 V1 V
119899) 119889119905 = 0 119899 ge 0
(31)
Now if we apply ADM for solving (29) substituting (3) and(4) in (29) leads to
infin
sum
119894=0
119906119894(119909) = 119909 + int
119909
0
infin
sum
119894=0
119860119894119889119905 (32)
In view of (7) we have following recursive formula
1199060(119909) = 119909
119906119899+1
(119909) = int
119909
0
119860119899119889119905 119899 ge 0
(33)
According to Theorem 3 we have 119860119899= 119867(V
0 V1 V
119899) By
solving (31) and (33) we have
119906 (119909) =
infin
sum
119894=0
119906119894(119909) = lim
119901rarr1
V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119909 +
1199093
3
+
21199095
15
+
171199097
315
+
621199099
2835
+ sdot sdot sdot = tan119909
(34)
5 Conclusion
It has been shown that the standard HPM provides exactlythe same solutions as the standard Adomian decompositionmethod for solving functional equations It has been provedthat Hersquos polynomials are only Adomianrsquos polynomials withdifferent name
References
[1] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008
[2] J-L Li ldquoAdomianrsquos decomposition method and homotopyperturbationmethod in solving nonlinear equationsrdquo Journal ofComputational and Applied Mathematics vol 228 no 1 pp168ndash173 2009
[3] A Ghorbani ldquoBeyond Adomian polynomials he polynomialsrdquoChaos Solitons and Fractals vol 39 no 3 pp 1486ndash1492 2009
[4] N Bellomo and R A Monaco ldquoComparison between Ado-mianrsquos decompositionmethods and perturbation techniques foronlinear random differential equationsrdquo Journal of Mathemati-cal Analysis and Applications vol 110 pp 495ndash502 1985
[5] R Rach ldquoOn the Adomian (decomposition) method andcomparisons with Picardrsquos methodrdquo Journal of MathematicalAnalysis and Applications vol 128 no 2 pp 480ndash483 1987
[6] H Sadat ldquoEquivalence between the Adomian decompositionmethod and a perturbationmethodrdquoPhysica Scripta vol 82 no4 Article ID 045004 2010
[7] S Liang and D J Jeffrey ldquoComparison of homotopy analysismethod and homotopy perturbation method through an evo-lution equationrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 12 pp 4057ndash4064 2009
[8] M Sajid and T Hayat ldquoComparison of HAM and HPM meth-ods in nonlinear heat conduction and convection equationsrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2296ndash2301 2008
[9] GAdomian YCherruault andKAbbaoui ldquoAnonperturbativeanalytical solution of immune response with time-delays andpossible generalizationrdquo Mathematical and Computer Mod-elling vol 24 no 10 pp 89ndash96 1996
[10] K Abbaoui and Y Cherruault ldquoConvergence of Adomianrsquosmethod applied to differential equationsrdquo Computers amp Mathe-matics with Applications vol 28 no 5 pp 103ndash109 1994
[11] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[12] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[13] J-H He ldquoNew interpretation of homotopy perturbationmethodrdquo International Journal of Modern Physics B vol 20 no18 pp 2561ndash2568 2006
[14] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995
[15] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981
[16] H Jafari and S Momani ldquoSolving fractional diffusion and waveequations bymodified homotopy perturbationmethodrdquoPhysicsLetters A vol 370 no 5-6 pp 388ndash396 2007
[17] H Jafari M Alipour and H Tajadodi ldquoConvergence of homo-topy perturbation method for solving integral equationsrdquo ThaiJournal of Mathematics vol 8 no 3 pp 511ndash520 2010
[18] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] Y Zhu Q Chang and S Wu ldquoA new algorithm for calculatingAdomian polynomialsrdquoAppliedMathematics and Computationvol 169 no 1 pp 402ndash416 2005
[20] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer 1st edition 2011
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2 Mathematical Problems in Engineering
The standardADMconsists of representing the solution of (1)as a series
119906 (119909) =
infin
sum
119894=0
119906119894(119909) (3)
and the nonlinear function as the decomposed form
119873(119906 (119909)) =
infin
sum
119894=0
119860119894 (4)
where 119860119899 119899 = 0 1 2 are the Adomian polynomials of
1199060 1199061 119906
119899given by [9 10]
119860119899=
1
119899
119889119899
119889119901119899[119873(
119899
sum
119894=0
119906119894119901119894
)]
119901=0
(5)
Substituting (3) and (4) into (1) yieldsinfin
sum
119894=0
119906119894(119909) = 119891 +
infin
sum
119894=0
119860119894 (6)
The convergence of the series in (6) gives the desired relation
1199060= 119891
119906119899+1
= 119860119899 119899 = 0 1 2
(7)
It should be pointed out that 1198600depends only on 119906
0 1198601
depends only on 1199060and11990611198602depends only on 119906
01199061 and119906
2
and so onTheAdomian technique is very simple in its princi-plesThe difficulties consist in proving the convergence of theintroduced series
3 Homotopy Perturbation Method (HPM)
This is a basic idea of homotopy method which is to con-tinuously deform a simple problem easy to solve into thedifficult problem under study
In this section we apply the homotopy perturbationmethod [11ndash13] to the discussed problem To illustrate thehomotopy perturbation method (HPM) we consider (1) as
119871 (V) = V (119909) minus 119891 (119909) minus 119873 (V) = 0 (8)
with solution 119906(119909) The basic idea of the HPM is to constructa homotopy119867(V 119901) 119877 times [0 1] rarr 119877 which satisfies
H (V 119901) = (1 minus 119901) 119865 (V) + 119901119871 (V) = 0 (9)
where 119865(V) is a proper function with known solution whichcan be obtained easily The embedding parameter 119901 mono-tonically increases from 0 to 1 as the trivial problem 119865(V) = 0
is continuously transformed to the original problem V minus 119891 minus
119873(V) = 0 FromH(V 119901) = 0 we have119867(V 0) = 119865(V) = 0 and119867(V 1) = V minus 119891 minus 119873(V) = 0
It is better to take 119865(V) as a deformation of 119871(V) Forexample in (9) 119865(V) = Vminus119891(119909) By selecting 119865(V) = Vminus119891(119909)we can define another convex homotopyH(V 119901) by
H (V 119901) = V (119909) minus 119891 (119909) minus 119901119873 (V) = 0 (10)
The embedding parameter 119901 isin (0 1] can be considered as anexpanding parameter [14 15] The HPM uses the embeddingparameter 119901 as a ldquosmall parameterrdquo and writes the solution of(10) as a power series of 119901 that is
V = V0+ V1119901 + V21199012
+ sdot sdot sdot (11)
Setting 119901 = 1 results in the approximate solution of (10)
119906 = lim119901rarr1
V = V0+ V1+ V2+ sdot sdot sdot (12)
Substituting (11) into (10) and equating the terms withidentical powers of 119901 we can obtain a series of equations ofthe following form
1199010 V0minus 119891 (119909) = 0
1199011 V1minus 119867 (V
0) = 0
1199012 V2minus 119867 (V
0 V1) = 0
1199013 V3minus 119867 (V
0 V1 V2) = 0
(13)
where119867(V0 V1 V
119895) depend upon V
0 V1 V
119895 In view of
(10) to determine119867(V0 V1 V
119895) we use [16]
119867(V0 V1 V
119895) =
1
119895
120597119895
120597119901119895119873(
119895
sum
119894=0
V119894119901119894
)
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(14)
It is obvious that the system of nonlinear equations in (13) iseasy to solve and the components V
119894 119894 ge 0 of the homotopy
perturbation method can be completely determined and theseries solutions are thus entirely determined For the conver-gence of the previous method we refer the reader to the workof He [12 17 18]
4 Equivalence between ADM and HPM
In this section we prove that the HPM and the ADM givesame solution for solving nonlinear functional equationsWealso show that the He polynomials are like the Adomianpolynomials In [3] Ghorbani has presented the followingdefinition
Definition 1 (see [3]) The He polynomials are defined asfollows
119867119899(V0 V
119899) =
1
119899
120597119899
120597119901119899119873(
119899
sum
119894=0
V119894119901119894
)
1003816100381610038161003816100381610038161003816100381610038161003816119901=0
119899 = 0 1 2
(15)
Note 1 Comparison between (5) and (15) has shown that theHe polynomials are only Adomianrsquos polynomials and it iscalculated like Adomianrsquos polynomials
Mathematical Problems in Engineering 3
Theorem 2 Suppose that nonlinear function 119873(119906) and theparameterized representation of V are V(119901) = sum
infin
119894=0V119894119901119894 where
119901 is a parameter then we have
120597119899
119873(V (119901))
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(suminfin
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(16)
Proof (see [3 19]) In Theorem 3 we prove that the He poly-nomials are the Adomian polynomials
Theorem 3 The He polynomials which are given by (15) arethe Adomian polynomials
Proof From Taylorrsquos expansion of119873(V) we have
119873(V) = 119873 (V0) + 119873
1015840
(V0) (V minus V
0)
+
1
2
11987310158401015840
(V0) (V minus V
0)2
+ sdot sdot sdot
(17)
substituting (11) in (17) and expanding it in terms of 119901 leadsto
119873(
infin
sum
119894=0
V119894119901119894
) = 119873(V0) + 119873
1015840
(V0) (V1119901 + V21199012
+ sdot sdot sdot )
+
1
2
11987310158401015840
(V0) (V1119901 + V21199012
+ sdot sdot sdot )
2
+ sdot sdot sdot
= 119873 (V0) + 119873
1015840
(V0) V1119901
+ (1198731015840
(V0) V2+
1
2
11987310158401015840
(V0) V21)1199012
+ sdot sdot sdot
= 1198670+ 1198671119901 + 119867
21199012
+ sdot sdot sdot
(18)where119867
119894 119894 = 0 1 2 depends only on V
0 V1 V
119894
In order to obtain119867119899 we give 119899-order derivative of both
sides of (18) with respect to 119901 and let 119901 = 0 that is
120597119899
119873(V (119901))
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
suminfin
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
(19)
According toTheorem 2
120597119899
119873(suminfin
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
120597119899
suminfin
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
sum119899
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
= 119899119867119899
(20)
We know that 119867119894
just depends on V0 V1 V
119894so
(120597119899
sum119899
119894=0119867119894119901119894
)120597119901119899
|119901=0
= 119899119867119899 Substituting (20) in (19) leads
us to find119867119894in the following form
119867119899=
1
119899
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(21)
which is called for the first time by Ghorbani as the Hepolynomials [3]
Theorem 4 The homotopy perturbation method for solvingnonlinear functional equations is the Adomian decompositionmethod with the homotopyH(V 119901) given by
H (V 119901) = V minus 119891 (119909) minus 119901119873 (V) (22)
Proof Substituting (11) and (18) into (10) and equating theterms with the identical powers of 119901 we have
H (V 119901) =infin
sum
119894=0
V119894119901119894
minus 119891 (119909) minus 119901
infin
sum
119894=0
119867119894119901119894
= 0
H (V 119901) = V0minus 119891 (119909) +
infin
sum
119894=0
(V119894+1
minus 119867119894) 119901119894+1
= 0
(23)
1199010 V0minus 119891 (119909) = 0
119901119899+1 V119899+1
minus 119867119899= 0 119899 = 0 1 2
(24)
From (24) we have
V0= 119891 (119909)
V119899+1
= 119867119899 119899 = 0 1 2
(25)
According to Theorem 3 we have119867119899= 119860119899 Substituting (25)
in (11) leads us to
V = V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119891 (119909) + 1198600119901 + 119860
11199012
+ sdot sdot sdot
(26)
so
lim119901rarr1
V = 119891 (119909) + 1198601+ 1198602+ sdot sdot sdot
= 119891 (119909) +
infin
sum
119894=0
119860119894=
infin
sum
119894=0
119906119894= 119906
(27)
Therefore by letting
H (V 119901) = V minus 119891 (119909) minus 119901119873 (V) (28)
we observe that the power series V0+ V1119901 + V
21199012
+ sdot sdot sdot
corresponds to the solution of the equation H(V 119901) = V minus119891(119909) minus 119901119873(V) = 0 and becomes the approximate solution of(1) if 119901 rarr 1 This shows that the homotopy perturbationmethod is the Adomian decomposition method with thehomotopy H(V 119901) given by (28) The proof of Theorem 4 iscompleted
These two approaches give the same equations for high-order approximations This is mainly because the Taylorseries of a given function is unique which is a basic theoryin calculus Thus nothing is new in Ghorbanirsquos definitionexcept the new name ldquoHersquos polynomialrdquo He just employed theearly ideas of ADM
4 Mathematical Problems in Engineering
Example 5 (see [20]) Consider the following nonlinear Vol-terra integral equation
119910 (119909) = 119909 + int
119909
0
1199102
(119905) 119889119905 (29)
with the exact solution 119910(119909) = tan119909
We apply standard ADM and HPM For applying stan-dard HPM we construct following homotopy
H (119906 119901) = 119906 (119909) minus 119909 minus 119901int
119909
0
[119906 (119905)]2
119889119905 = 0 (30)
In view of (13) we have
1199010 V0(119909) minus 119909 = 0
119901119899 V119899+1
(119909) minus int
119909
0
119867(V0 V1 V
119899) 119889119905 = 0 119899 ge 0
(31)
Now if we apply ADM for solving (29) substituting (3) and(4) in (29) leads to
infin
sum
119894=0
119906119894(119909) = 119909 + int
119909
0
infin
sum
119894=0
119860119894119889119905 (32)
In view of (7) we have following recursive formula
1199060(119909) = 119909
119906119899+1
(119909) = int
119909
0
119860119899119889119905 119899 ge 0
(33)
According to Theorem 3 we have 119860119899= 119867(V
0 V1 V
119899) By
solving (31) and (33) we have
119906 (119909) =
infin
sum
119894=0
119906119894(119909) = lim
119901rarr1
V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119909 +
1199093
3
+
21199095
15
+
171199097
315
+
621199099
2835
+ sdot sdot sdot = tan119909
(34)
5 Conclusion
It has been shown that the standard HPM provides exactlythe same solutions as the standard Adomian decompositionmethod for solving functional equations It has been provedthat Hersquos polynomials are only Adomianrsquos polynomials withdifferent name
References
[1] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008
[2] J-L Li ldquoAdomianrsquos decomposition method and homotopyperturbationmethod in solving nonlinear equationsrdquo Journal ofComputational and Applied Mathematics vol 228 no 1 pp168ndash173 2009
[3] A Ghorbani ldquoBeyond Adomian polynomials he polynomialsrdquoChaos Solitons and Fractals vol 39 no 3 pp 1486ndash1492 2009
[4] N Bellomo and R A Monaco ldquoComparison between Ado-mianrsquos decompositionmethods and perturbation techniques foronlinear random differential equationsrdquo Journal of Mathemati-cal Analysis and Applications vol 110 pp 495ndash502 1985
[5] R Rach ldquoOn the Adomian (decomposition) method andcomparisons with Picardrsquos methodrdquo Journal of MathematicalAnalysis and Applications vol 128 no 2 pp 480ndash483 1987
[6] H Sadat ldquoEquivalence between the Adomian decompositionmethod and a perturbationmethodrdquoPhysica Scripta vol 82 no4 Article ID 045004 2010
[7] S Liang and D J Jeffrey ldquoComparison of homotopy analysismethod and homotopy perturbation method through an evo-lution equationrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 12 pp 4057ndash4064 2009
[8] M Sajid and T Hayat ldquoComparison of HAM and HPM meth-ods in nonlinear heat conduction and convection equationsrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2296ndash2301 2008
[9] GAdomian YCherruault andKAbbaoui ldquoAnonperturbativeanalytical solution of immune response with time-delays andpossible generalizationrdquo Mathematical and Computer Mod-elling vol 24 no 10 pp 89ndash96 1996
[10] K Abbaoui and Y Cherruault ldquoConvergence of Adomianrsquosmethod applied to differential equationsrdquo Computers amp Mathe-matics with Applications vol 28 no 5 pp 103ndash109 1994
[11] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[12] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[13] J-H He ldquoNew interpretation of homotopy perturbationmethodrdquo International Journal of Modern Physics B vol 20 no18 pp 2561ndash2568 2006
[14] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995
[15] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981
[16] H Jafari and S Momani ldquoSolving fractional diffusion and waveequations bymodified homotopy perturbationmethodrdquoPhysicsLetters A vol 370 no 5-6 pp 388ndash396 2007
[17] H Jafari M Alipour and H Tajadodi ldquoConvergence of homo-topy perturbation method for solving integral equationsrdquo ThaiJournal of Mathematics vol 8 no 3 pp 511ndash520 2010
[18] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] Y Zhu Q Chang and S Wu ldquoA new algorithm for calculatingAdomian polynomialsrdquoAppliedMathematics and Computationvol 169 no 1 pp 402ndash416 2005
[20] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer 1st edition 2011
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Theorem 2 Suppose that nonlinear function 119873(119906) and theparameterized representation of V are V(119901) = sum
infin
119894=0V119894119901119894 where
119901 is a parameter then we have
120597119899
119873(V (119901))
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(suminfin
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(16)
Proof (see [3 19]) In Theorem 3 we prove that the He poly-nomials are the Adomian polynomials
Theorem 3 The He polynomials which are given by (15) arethe Adomian polynomials
Proof From Taylorrsquos expansion of119873(V) we have
119873(V) = 119873 (V0) + 119873
1015840
(V0) (V minus V
0)
+
1
2
11987310158401015840
(V0) (V minus V
0)2
+ sdot sdot sdot
(17)
substituting (11) in (17) and expanding it in terms of 119901 leadsto
119873(
infin
sum
119894=0
V119894119901119894
) = 119873(V0) + 119873
1015840
(V0) (V1119901 + V21199012
+ sdot sdot sdot )
+
1
2
11987310158401015840
(V0) (V1119901 + V21199012
+ sdot sdot sdot )
2
+ sdot sdot sdot
= 119873 (V0) + 119873
1015840
(V0) V1119901
+ (1198731015840
(V0) V2+
1
2
11987310158401015840
(V0) V21)1199012
+ sdot sdot sdot
= 1198670+ 1198671119901 + 119867
21199012
+ sdot sdot sdot
(18)where119867
119894 119894 = 0 1 2 depends only on V
0 V1 V
119894
In order to obtain119867119899 we give 119899-order derivative of both
sides of (18) with respect to 119901 and let 119901 = 0 that is
120597119899
119873(V (119901))
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
suminfin
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
(19)
According toTheorem 2
120597119899
119873(suminfin
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
120597119899
suminfin
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
=
120597119899
sum119899
119894=0119867119894119901119894
120597119901119899
100381610038161003816100381610038161003816100381610038161003816119901=0
= 119899119867119899
(20)
We know that 119867119894
just depends on V0 V1 V
119894so
(120597119899
sum119899
119894=0119867119894119901119894
)120597119901119899
|119901=0
= 119899119867119899 Substituting (20) in (19) leads
us to find119867119894in the following form
119867119899=
1
119899
120597119899
119873(sum119899
119894=0V119894119901119894
)
120597119901119899
10038161003816100381610038161003816100381610038161003816100381610038161003816119901=0
(21)
which is called for the first time by Ghorbani as the Hepolynomials [3]
Theorem 4 The homotopy perturbation method for solvingnonlinear functional equations is the Adomian decompositionmethod with the homotopyH(V 119901) given by
H (V 119901) = V minus 119891 (119909) minus 119901119873 (V) (22)
Proof Substituting (11) and (18) into (10) and equating theterms with the identical powers of 119901 we have
H (V 119901) =infin
sum
119894=0
V119894119901119894
minus 119891 (119909) minus 119901
infin
sum
119894=0
119867119894119901119894
= 0
H (V 119901) = V0minus 119891 (119909) +
infin
sum
119894=0
(V119894+1
minus 119867119894) 119901119894+1
= 0
(23)
1199010 V0minus 119891 (119909) = 0
119901119899+1 V119899+1
minus 119867119899= 0 119899 = 0 1 2
(24)
From (24) we have
V0= 119891 (119909)
V119899+1
= 119867119899 119899 = 0 1 2
(25)
According to Theorem 3 we have119867119899= 119860119899 Substituting (25)
in (11) leads us to
V = V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119891 (119909) + 1198600119901 + 119860
11199012
+ sdot sdot sdot
(26)
so
lim119901rarr1
V = 119891 (119909) + 1198601+ 1198602+ sdot sdot sdot
= 119891 (119909) +
infin
sum
119894=0
119860119894=
infin
sum
119894=0
119906119894= 119906
(27)
Therefore by letting
H (V 119901) = V minus 119891 (119909) minus 119901119873 (V) (28)
we observe that the power series V0+ V1119901 + V
21199012
+ sdot sdot sdot
corresponds to the solution of the equation H(V 119901) = V minus119891(119909) minus 119901119873(V) = 0 and becomes the approximate solution of(1) if 119901 rarr 1 This shows that the homotopy perturbationmethod is the Adomian decomposition method with thehomotopy H(V 119901) given by (28) The proof of Theorem 4 iscompleted
These two approaches give the same equations for high-order approximations This is mainly because the Taylorseries of a given function is unique which is a basic theoryin calculus Thus nothing is new in Ghorbanirsquos definitionexcept the new name ldquoHersquos polynomialrdquo He just employed theearly ideas of ADM
4 Mathematical Problems in Engineering
Example 5 (see [20]) Consider the following nonlinear Vol-terra integral equation
119910 (119909) = 119909 + int
119909
0
1199102
(119905) 119889119905 (29)
with the exact solution 119910(119909) = tan119909
We apply standard ADM and HPM For applying stan-dard HPM we construct following homotopy
H (119906 119901) = 119906 (119909) minus 119909 minus 119901int
119909
0
[119906 (119905)]2
119889119905 = 0 (30)
In view of (13) we have
1199010 V0(119909) minus 119909 = 0
119901119899 V119899+1
(119909) minus int
119909
0
119867(V0 V1 V
119899) 119889119905 = 0 119899 ge 0
(31)
Now if we apply ADM for solving (29) substituting (3) and(4) in (29) leads to
infin
sum
119894=0
119906119894(119909) = 119909 + int
119909
0
infin
sum
119894=0
119860119894119889119905 (32)
In view of (7) we have following recursive formula
1199060(119909) = 119909
119906119899+1
(119909) = int
119909
0
119860119899119889119905 119899 ge 0
(33)
According to Theorem 3 we have 119860119899= 119867(V
0 V1 V
119899) By
solving (31) and (33) we have
119906 (119909) =
infin
sum
119894=0
119906119894(119909) = lim
119901rarr1
V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119909 +
1199093
3
+
21199095
15
+
171199097
315
+
621199099
2835
+ sdot sdot sdot = tan119909
(34)
5 Conclusion
It has been shown that the standard HPM provides exactlythe same solutions as the standard Adomian decompositionmethod for solving functional equations It has been provedthat Hersquos polynomials are only Adomianrsquos polynomials withdifferent name
References
[1] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008
[2] J-L Li ldquoAdomianrsquos decomposition method and homotopyperturbationmethod in solving nonlinear equationsrdquo Journal ofComputational and Applied Mathematics vol 228 no 1 pp168ndash173 2009
[3] A Ghorbani ldquoBeyond Adomian polynomials he polynomialsrdquoChaos Solitons and Fractals vol 39 no 3 pp 1486ndash1492 2009
[4] N Bellomo and R A Monaco ldquoComparison between Ado-mianrsquos decompositionmethods and perturbation techniques foronlinear random differential equationsrdquo Journal of Mathemati-cal Analysis and Applications vol 110 pp 495ndash502 1985
[5] R Rach ldquoOn the Adomian (decomposition) method andcomparisons with Picardrsquos methodrdquo Journal of MathematicalAnalysis and Applications vol 128 no 2 pp 480ndash483 1987
[6] H Sadat ldquoEquivalence between the Adomian decompositionmethod and a perturbationmethodrdquoPhysica Scripta vol 82 no4 Article ID 045004 2010
[7] S Liang and D J Jeffrey ldquoComparison of homotopy analysismethod and homotopy perturbation method through an evo-lution equationrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 12 pp 4057ndash4064 2009
[8] M Sajid and T Hayat ldquoComparison of HAM and HPM meth-ods in nonlinear heat conduction and convection equationsrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2296ndash2301 2008
[9] GAdomian YCherruault andKAbbaoui ldquoAnonperturbativeanalytical solution of immune response with time-delays andpossible generalizationrdquo Mathematical and Computer Mod-elling vol 24 no 10 pp 89ndash96 1996
[10] K Abbaoui and Y Cherruault ldquoConvergence of Adomianrsquosmethod applied to differential equationsrdquo Computers amp Mathe-matics with Applications vol 28 no 5 pp 103ndash109 1994
[11] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[12] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[13] J-H He ldquoNew interpretation of homotopy perturbationmethodrdquo International Journal of Modern Physics B vol 20 no18 pp 2561ndash2568 2006
[14] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995
[15] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981
[16] H Jafari and S Momani ldquoSolving fractional diffusion and waveequations bymodified homotopy perturbationmethodrdquoPhysicsLetters A vol 370 no 5-6 pp 388ndash396 2007
[17] H Jafari M Alipour and H Tajadodi ldquoConvergence of homo-topy perturbation method for solving integral equationsrdquo ThaiJournal of Mathematics vol 8 no 3 pp 511ndash520 2010
[18] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] Y Zhu Q Chang and S Wu ldquoA new algorithm for calculatingAdomian polynomialsrdquoAppliedMathematics and Computationvol 169 no 1 pp 402ndash416 2005
[20] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer 1st edition 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Example 5 (see [20]) Consider the following nonlinear Vol-terra integral equation
119910 (119909) = 119909 + int
119909
0
1199102
(119905) 119889119905 (29)
with the exact solution 119910(119909) = tan119909
We apply standard ADM and HPM For applying stan-dard HPM we construct following homotopy
H (119906 119901) = 119906 (119909) minus 119909 minus 119901int
119909
0
[119906 (119905)]2
119889119905 = 0 (30)
In view of (13) we have
1199010 V0(119909) minus 119909 = 0
119901119899 V119899+1
(119909) minus int
119909
0
119867(V0 V1 V
119899) 119889119905 = 0 119899 ge 0
(31)
Now if we apply ADM for solving (29) substituting (3) and(4) in (29) leads to
infin
sum
119894=0
119906119894(119909) = 119909 + int
119909
0
infin
sum
119894=0
119860119894119889119905 (32)
In view of (7) we have following recursive formula
1199060(119909) = 119909
119906119899+1
(119909) = int
119909
0
119860119899119889119905 119899 ge 0
(33)
According to Theorem 3 we have 119860119899= 119867(V
0 V1 V
119899) By
solving (31) and (33) we have
119906 (119909) =
infin
sum
119894=0
119906119894(119909) = lim
119901rarr1
V0+ V1119901 + V21199012
+ sdot sdot sdot
= 119909 +
1199093
3
+
21199095
15
+
171199097
315
+
621199099
2835
+ sdot sdot sdot = tan119909
(34)
5 Conclusion
It has been shown that the standard HPM provides exactlythe same solutions as the standard Adomian decompositionmethod for solving functional equations It has been provedthat Hersquos polynomials are only Adomianrsquos polynomials withdifferent name
References
[1] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008
[2] J-L Li ldquoAdomianrsquos decomposition method and homotopyperturbationmethod in solving nonlinear equationsrdquo Journal ofComputational and Applied Mathematics vol 228 no 1 pp168ndash173 2009
[3] A Ghorbani ldquoBeyond Adomian polynomials he polynomialsrdquoChaos Solitons and Fractals vol 39 no 3 pp 1486ndash1492 2009
[4] N Bellomo and R A Monaco ldquoComparison between Ado-mianrsquos decompositionmethods and perturbation techniques foronlinear random differential equationsrdquo Journal of Mathemati-cal Analysis and Applications vol 110 pp 495ndash502 1985
[5] R Rach ldquoOn the Adomian (decomposition) method andcomparisons with Picardrsquos methodrdquo Journal of MathematicalAnalysis and Applications vol 128 no 2 pp 480ndash483 1987
[6] H Sadat ldquoEquivalence between the Adomian decompositionmethod and a perturbationmethodrdquoPhysica Scripta vol 82 no4 Article ID 045004 2010
[7] S Liang and D J Jeffrey ldquoComparison of homotopy analysismethod and homotopy perturbation method through an evo-lution equationrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 12 pp 4057ndash4064 2009
[8] M Sajid and T Hayat ldquoComparison of HAM and HPM meth-ods in nonlinear heat conduction and convection equationsrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2296ndash2301 2008
[9] GAdomian YCherruault andKAbbaoui ldquoAnonperturbativeanalytical solution of immune response with time-delays andpossible generalizationrdquo Mathematical and Computer Mod-elling vol 24 no 10 pp 89ndash96 1996
[10] K Abbaoui and Y Cherruault ldquoConvergence of Adomianrsquosmethod applied to differential equationsrdquo Computers amp Mathe-matics with Applications vol 28 no 5 pp 103ndash109 1994
[11] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[12] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[13] J-H He ldquoNew interpretation of homotopy perturbationmethodrdquo International Journal of Modern Physics B vol 20 no18 pp 2561ndash2568 2006
[14] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995
[15] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons New York NY USA 1981
[16] H Jafari and S Momani ldquoSolving fractional diffusion and waveequations bymodified homotopy perturbationmethodrdquoPhysicsLetters A vol 370 no 5-6 pp 388ndash396 2007
[17] H Jafari M Alipour and H Tajadodi ldquoConvergence of homo-topy perturbation method for solving integral equationsrdquo ThaiJournal of Mathematics vol 8 no 3 pp 511ndash520 2010
[18] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] Y Zhu Q Chang and S Wu ldquoA new algorithm for calculatingAdomian polynomialsrdquoAppliedMathematics and Computationvol 169 no 1 pp 402ndash416 2005
[20] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer 1st edition 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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![Adomian Decomposition Method with Modified Bernstein ...JournalofAppliedMathematics collocationtechniquetosolvesomedierentialandintegral equations[]. Dfinition (Bernstein basis polynomials).](https://static.fdocuments.in/doc/165x107/6113765049d5e97b5a692ce3/adomian-decomposition-method-with-modified-bernstein-journalofappliedmathematics.jpg)