On the convergence of Homotopy perturbation method

Journal of the Egyptian Mathematical Society (2014) xxx, xxx–xxx

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems

ORIGINAL ARTICLE

On the convergence of Homotopy perturbation method Zainab Ayati

a,*

, Jafar Biazar

b

a Department of Engineering Sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, PC 44891-63157 Rudsar-Vajargah, Iran b Faculty of Mathematical Science, University of Guilan, Department of Applied Mathematics, P.O. Box 41635-19141, PC 4193-833697 Rasht, Iran

Received 24 September 2013; revised 5 June 2014; accepted 18 June 2014

KEYWORDS Homotopy perturbation method; Functional equation; Convergence

Abstract In many papers, Homotopy perturbation method has been presented as a method for solving non-linear equations of various kinds. Using Homotopy perturbation method, it is possible to find the exact solution or a closed approximate to the solution of the problem. But, only a few works have been considered the problem of convergence of the method. In this paper, convergence of Homotopy perturbation method has been elaborated briefly. 2000 MATHEMATICS SUBJECT CLASSIFICATION:

65k10; 65N12; 65Q05

ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction Mathematical modeling of many physical systems leads to functional equations, in various fields of physics and engineering. There are some methods to obtain approximate solutions of this kind of equations. One of them is Homotopy perturbation method. The method introduced by He in 1998 [1] has been developed and improved by himself. He applied it to boundary value problems, non-linear wave equations, and * Corresponding author. Tel.: +98 1313233509. E-mail address: [email protected] (Z. Ayati). Peer review under responsibility of Egyptian Mathematical Society.

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many other subjects [2–6]. Homotopy perturbation method can be considered as a universal method capable of solving various kinds of non-linear functional equations. For example, it was also applied to non-linear Schrodinger equations [7], to systems of Volterra integral equations of the second kind [8], to generalized Hirota–Satsuma coupled KdV equation [11], and to many other equations [2–12]. In this method the solution is considered as the summation of an infinite series which usually converge rapidly to the exact solution. This method continuously deforms, the difficult problems under study into a simple problem, easy to solve. Almost all perturbation methods are based on the assumption of the existence of a small parameter in the equation. But most non-linear problems have not such a small parameter. This method has been proposed to eliminate the small parameters. In this paper, a proof of convergence of the HPM is presented.

1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2014.06.015 Please cite this article in press as: Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.06.015

2

Z. Ayati, J. Biazar Then

2. Basic idea of homotopy perturbation method To illustrate the basic concept of homotopy perturbation method, consider the following non-linear functional equation AðuÞ  fðrÞ ¼ 0 ; r 2 X;

ð1Þ

with boundary conditions; Bðu; @u=@nÞ ¼ 0 ; r 2 C;

ð2Þ

where A is a general functional operator, B is a boundary operator, f(r) is a known analytic function, and C is the boundary of the domain X. Generally speaking, the operator A can be divided into two parts L and N, where L is a linear, while N is a non-linear operator. Eq. (1), therefore, can be rewritten as follows; LðuÞ þ NðuÞ  fðrÞ ¼ 0:

ð3Þ

We construct a homotopy v(r, p):X · [0, 1] fi R which satisfies Hðv; pÞ ¼ ð1  pÞ½LðvÞ  Lðu0 Þ þ p½AðvÞ  fðrÞ ¼ 0;

ð4Þ

or Hðv; pÞ ¼ LðvÞ  Lðu0 Þ þ pLðu0 Þ þ p½NðvÞ  fðrÞ ¼ 0;

ð5Þ

where p 2 [0, 1] is an embedding parameter, and u0 is an initial approximation for the solution of Eq. (1), which satisfies the boundary conditions. According to HPM, we can first use the embedding parameter p as a small parameter, and assume that the solution of Eq. (5) can be written as a power series in p: v ¼ v0 þ v1 p þ v2 p2 þ    ¼

1 X vi pi :

ð6Þ

i¼0

Considering p = 1, the approximate solution of Eq. (2) will be obtained as follows; u ¼ limv ¼ v0 þ v1 þ v2 þ   

ð7Þ

p!1

3. Convergence of the method Let’s rewrite the Eq. (5) as the following; LðvÞ  Lðu0 Þ ¼ p½fðrÞ  Lðu0 Þ  NðvÞ:

ð8Þ

Substituting (7) into (8) leads to; ! " !# 1 1 X X i i L vi p  Lðu0 Þ ¼ p fðrÞ  Lðu0 Þ  N vi p :

ð9Þ

i¼0

i¼0

1 X N vi pi

!

i¼0

1 n X X 1 @n N vi pi ¼ n! @pn n¼0 i¼0

!! pi :

ð13Þ

;

n ¼ 0; 1; 2; . . . ;

p¼0

We set Hn ðv0 ; v1 ; . . . ; vn Þ ¼

n X 1 @n N vi pi n! @pn i¼0

!! p¼0

ð14Þ where Hn s are the so-called He’s polynomials [13]. Then; ! 1 1 X X i N vi p ¼ Hi pi : ð15Þ i¼0

n¼0

Substituting (15) into (10), we drive; " # 1 1 X X i Lðvi Þ  Lðu0 Þ ¼ p fðrÞ  Lðu0 Þ  Hi p : i¼0

ð16Þ

n¼0

By equating the terms with the identical powers in p: 8 0 p : Lðv0 Þ  Lðu0 Þ ¼ 0; > > > 1 > > : Lðv1 Þ ¼ fðrÞ  Lðu0 Þ  H0 ; p > > > > p2 : Lðv2 Þ ¼ H1 ; > < .. > . > > > > > p3 : Lðvnþ1 Þ ¼ Hn ; > > > > : .. . So we derive 8 v0 ¼ u0 ; > > > > v1 ¼ L1 ½fðrÞ  u0  L1 ðH0 Þ; > > > > 1 > > < v2 ¼ L ðH1 Þ; . > .. > > > > > vnþ1 ¼ L1 ðHn Þ; > > > >. : ..

ð17Þ

ð18Þ

Theorem 3.1. Homotopy perturbation method used the solution of Eq. (1) is equivalent to determining the following sequence; sn ¼ v1 þ . . . þ vn ; s0 ¼ 0;

ð19Þ

by using the iterative scheme:

So "

1 1 X X Lðvi Þ  Lðu0 Þ ¼ p fðrÞ  Lðu0 Þ  N vi pi i¼0

!# :

snþ1 ¼ L1 Nn ðsn þ v0 Þ  u0 þ L1 ðfðrÞÞ; ð10Þ

i¼0

where

P1

According to Maclaurin expansion of N to p, we have; ! !! 1 1 1 X X X 1 @n i i N vi p ¼ N vi p n! @pn i¼0 n¼0 i¼0

 i i¼0 vi p with respect

Nn

n X vi i¼0

Pi :

ð11Þ

ð20Þ

! ¼

n X Hi ;

n ¼ 0; 1; 2; . . . :

ð21Þ

n¼0

Proof. For n = 0, from (20), we have;

p¼0

From [13], we get !! 1 X @n i N v p i @pn i¼0

p¼0

¼

n X @n N vi pi n @p i¼0

s1 ¼ L1 N0 ðs0 þ v0 Þ  u0 þ L1 ðfðrÞÞ; ¼ L1 ðH0 Þ  u0 þ L1 ðfðrÞÞ:

!! : p¼0

ð12Þ

ð22Þ

Then v1 ¼ L1 ðH0 Þ  u0 þ L1 ðfðrÞÞ:

ð23Þ

Please cite this article in press as: Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.06.015

On the convergence of Homotopy perturbation method

3

For n = 1:

lim snþ1 ¼ L1 lim Nn ðsn þ v0 Þ  u0 þ L1 ðfðrÞÞ x!1 ! n X 1 vi  u0 þ L1 ðfðrÞÞ: ¼ L lim Nn

x!1

s2 ¼ L1 N1 ðs1 þ v0 Þ  u0 þ L1 ðfðrÞÞ ¼ L1 ðH0 þ H1 Þ  u0 þ L1 ðfðrÞÞ;

ð24Þ

x!1

1

¼ L ðH1 Þ þ v1 : 1

s ¼ L

According to s2 = v1 + v2 , we get; 1

v2 ¼ L ðH1 Þ:

snþ1 ¼ L1 Nn ðsn þ v0 Þ  u0 þ L1 ðfðrÞÞ ! n X ¼ L1 Hi  u0 þ L1 ðfðrÞÞ;

x!1

i¼0

1

i¼0

But by Eqs. (21) and (15) for p = 1, we drive; ! 1 1 X X Hi ¼ N vi : i¼0

n¼0

ð34Þ

1 X ¼ L Hi  u0 þ L1 ðfðrÞÞ:

ð25Þ

This theorem will be proved by strong induction. Let’s assume that vk+1 = L1(Hk), for k = 1, 2, . . . , n  1, so

i¼0

n X lim Hi  u0 þ L1 ðfðrÞÞ

ð35Þ

i¼0

So

n X ¼  L1 ðHi Þ  u0 þ L1 ðfðrÞÞ ¼ v1 þ v2 þ . . . þ vn

1 X s ¼ L N vi

!

1

n¼0 1

 L ðHn Þ:

 u0 þ L1 ðfðrÞÞ;

i¼0

ð26Þ

1



1

s ¼ L Nðs þ v0 Þ  u0 þ L ðfðrÞÞ:

Then, from (19) , it can be driven; vnþ1 ¼ L1 ðHn Þ:

ð27Þ

Which is the same as the result of (18) from HPM, and the theorem is proved. h

Lemma 1. Eq. (29) is equivalent to; LðuÞ þ NðuÞ  fðrÞ ¼ 0:

ð36Þ

Proof. We rewrite Eq. (29) as fallows; Theorem 3.2. Let B be a Banach space. (a)

P1

i¼0 vi

s þ u0 ¼ L1 Nðs þ v0 Þ þ L1 ðfðrÞÞ:

obtained by (18), convergence to s 2 B, if

By applying the operator L to Eq. (36) we derive;

9ð0 6 k < 1Þ; s:t ð8n 2 N ) kvn k 6 kkvn1 kÞ: P (b) s ¼ 1 n¼1 vn , satisfies in

ð28Þ

s ¼ L1 Nðs þ v0 Þ  u0 þ L1 ðfðrÞÞ:

ð29Þ

(a) we have ksnþ1  sn k ¼ kvnþ1 k 6 kkvn k 6 k2 kvn1 k 6    6k

kv0 k:

ki ¼

kmþ1 kv0 k: 1k

9s 2 B; s:t lim sn ¼ n!1

vn ¼ s:

n¼1

(b) From Eq. (20), we have;

kvi k–0;

kvi k ¼ 0: P1 In Theorem 3.2, n¼0 vi converges to exact solution, when 0 6 ki < 1. If vi and v0i are obtained by two different homotopy, and ki < k0i for each N, the rate of convergence of P P1 i 2 1 0 v is higher than v . i n¼0 n¼0 i 0

Example 1. Consider the Lane–Emden equation in the following form

ð32Þ

2 u00 þ u0 þ u ¼ x5 þ 30x3 ; uð0Þ ¼ 0; u0 ð0Þ ¼ 0: x

Then {sn}, is Cauchy sequence in Banach space, and it is convergent, i.e., 1 X

kviþ1 k ; kvi k

ð31Þ

So n;m!1

(

ð30Þ

ksn  sm k ¼ kðsn  sn1 Þ þ ðsn1  sn2 Þ þ  þ ðsmþ1  sm Þk; 6 ksn  sn1 k þ ksn1  sn2 k þ  þ ksmþ1  sm k;   6 kn kv0 k þ kn1 kv0 k þ  þ kmþ1 kv0 k 6 kn þ kn1 þ  þ kmþ1 kv0 k;  mþ1  þ  þ kn þ  kv0 k 6 kmþ1 ð1 þ k þ  þ kn þ Þkv0 k; 6 k

lim ksn  sm k ¼ 0:

Lðs þ v0 Þ þ Nðs þ v0 Þ ¼ ðfðrÞÞ: ð38Þ P1 By considering u ¼ s þ v0 ¼ n¼0 vi , Eq. (36), has been derived which is the original equation. Then solution of Eq. (29) is the same as solution of A(u)  f(r) = 0. h Definition 1. for every i 2 N, we define;

For any n, m 2 N, n P m, we drive;

6

Lðs þ u0 Þ ¼ Nðs þ v0 Þ þ ðfðrÞÞ: But u0 = v0 , then;

Proof.

nþ1

ð37Þ

ð33Þ

ð39Þ

With the exact solution u(x) = x5. To solve Eq. (39) by homotopy perturbation method, we construct the following Homotopy; 2 u00  U000 ¼ pðx5 þ 30x3  u0  u  u000 Þ: x

ð40Þ

Let’s consider the solution u as the summation of a series;

Please cite this article in press as: Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.06.015

4 u¼

Z. Ayati, J. Biazar 1 X ui :

ð41Þ

i¼0

Substituting (41) into (40) leads to; 1 1 1 X X 2X u00i  U000 ¼ pðx5 þ 30x3  u0i  ui  u000 Þ: x i¼0 i¼0 i¼0

v1 v2 v3 v4 v5

¼ 561 x7 þ x5 ; 1 ¼  5040 x9  561 x7 ; 1 1 ¼ 665280 x11 þ 5040 x9 ; 1 13 1 ¼  121080960 x  665280 x11 ; 1 15 1 ¼ 29059430400 x þ 121080960 x13 ;

Then;

Beginning with u0 = 0, we get ;

k1 ¼ 0:0177387914; k2 ¼ 0:0110722610; k3 ¼ 0:00756010906; k4 ¼ 0:00548724954;

u1 ¼ 421 x7 þ 32 x5 ; 11 7 1 u2 ¼  252 x  34 x5  4024 x9 ;

k5 ¼ 0:00416293765;

25 7 1 x9 þ 216 x7 þ 38 x5 þ 332640 x11 ; u3 ¼ 36288 137 271 179 7 1 u4 ¼  19958400 x11  435456 x9  9072 x  163 x5  51891840 x13 ; 7 13 8419 11 2245 9 601 7 u5 ¼ 148262400 x þ 1197504000 x þ 5225472 x þ 54432 x þ 323 x5 1 þ 10897286400 x15 ;

k6 ¼ 0:00326590091:

By comparison between the obtained results in Example 3.2.1, it can be concluded that the rate of convergence of homotopy (42) is higher than homotopy (41) (see Fig. 1). 4. Conclusion

By considered kfðxÞk ¼ max jfðxÞj, we have; 06x61

k1 ¼ 0:5210503471; k3 ¼ 0:5093374003;

k2 ¼ 0:5139910140; k4 ¼ 0:5062439696;

k5 ¼ 0:5041785188;

k6 ¼ 0:5027965117:

The Homotopy perturbation method is a simple and powerful tool for obtaining the solution of functional equations. In this article, we have proved convergence for Homotopy perturbation method. As a result from this paper, under assumption of Theorem 3.2, homotopy perturbation method is convergence to exact solution of the problem. The problem of study of convergence conditions for differential equations, integral equations, integro-differential equations and theirs systems is also under study in our research group.

If the linear part of equation is consider as follows;   2 0 00 2 d 2 du x : Lu ¼ u þ u ¼ x x dx dx Then we construct the following homotopy     2 0 2 0 00 00 v þ v  u0 þ u0 ¼ pðx5 þ 30x3  v  u000 x x 2  u00 Þ; x

Acknowledgement ð42Þ

where L1 ðuÞ ¼

Z

x

x2 0

Z

x

x2 uðxÞdx dx:

0

Suppose u = v0 + pv1 + p2v2 +   , and v0 = u0 = 0. So;

Fig. 1

Plots of solution of HPM and exact solution for Ex. 3.2.1.

I appreciate the referee for his close attention and the time that spend to read my article in details. The reviewer’s precise comments was insightful for making this paper more comprehensive. References [1] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (1999) 257–262. [2] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. NonLinear Mech. 35 (1) (2000) 37–43. [3] J.H. He, A modified perturbation technique depending upon an artificial parameter, Mechanical 35 (2000) 299–311. [4] J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (2006) 87–88. [5] J.H. He, New interpretation of homotopy perturbation method, Int. J. Modern Phys. B 20 (2006) 2561–2568. [6] J. H He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fract. 26 (2005) 695–700. [7] J. Biazar, H. Ghazvini, Exact solutions for non-linear Schro¨dinger equations by He’s homotopy perturbation method, Phys. Lett. A 366 (2007) 79–84. [8] J. Biazar, H. Ghazvini, He’s homotopy perturbation method for solving system of Volterra integral equations of the second kind, Chaos Solitons Fract. 39 (2) (2009) 770–777. [9] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A 352 (2006) 404–410. [10] Qi. Wang, Homotopy perturbation method for fractional KdV-Burgers equation, Chaos Solitons Fract. 35 (2008) 843–850.

Please cite this article in press as: Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.06.015

On the convergence of Homotopy perturbation method [11] D.D. Ganji, M. Rafei, Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation by homotopy perturbation method, Phys. Lett. A 356 (2006) 131–137. [12] S. Abbasbandy, Application of He’s homotopy perturbation method to functional integral equations, Chaos Solitons Fract. 31 (5) (2007) 1243–1247.

5 [13] A. Ghorbani, Beyond Adomian’s polynomials: He polynomials, Chaos Solitons Fract. 39 (3) (2009) 1486–1492.

Please cite this article in press as: Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.06.015