Convergence of the Homotopy Perturbation Method

Int. J. Nonlinear Sci. Numer. Simul., Vol. 12 (2011), pp. 9–14

Copyright © 2011 De Gruyter. DOI 10.1515/IJNSNS.2011.020

Convergence of the Homotopy Perturbation Method

M. Turkyilmazoglu1;

Painleve equation [5], the Burger’s equation [6] and the quadratic Riccati differential equation [7]. One may refer 1 to the recent publications [8] and [9] for application of the Mathematics Department, University of Hacettepe, homotopy method in biological fields. Although a great Beytepe, Ankara, Turkey deal of equations were solved using the homotopy perturAbstract. The present paper is devoted to the analysis of bation method, the question of convergence of the method the homotopy perturbation method. The question of conver- still remains unanswered. Very recently, a routine convergence of the method is resolved. It is theoretically proven gence theorem was outlined in [10]. However, this theorem that under a special constraint the homotopy perturbation requires knowledge of the exact solution in prior, as demonmethod converges to the exact solution of sought solution strated in [10] via some examples. of nonlinear ordinary or partial differential equation. An In the present paper, we investigate the homotopy pererror estimate is also provided. Examples clearly demon- turbation technique from a mathematical point of view. The strate, why and on what interval, the corresponding homo- main purpose is to theoretically analyze the method and to topy series generated by the homotopy perturbation method show eventually that under certain circumstances the hoconverges to the exact solution. motopy perturbation method converges to the exact desired solution, without a priori knowledge of the exact solution. Keywords. Nonlinear equations, approximate solution, Our another objective is to address the error estimate of the homotopy perturbation method, convergence. approximate solution. A theorem is given that is justified through basic examples from ordinary and partial differenPACS® (2010). 02.70.-c, 02.60.Lj, 02.30.Jr, 45.10.Hj. tial equations well studied in the literature. The presented theory in this paper provides information about not only the convergence, but also the interval of convergence of the homotopy series. In the rest of the paper, Section 2 lays down the basis for 1 Introduction the homotopy perturbation method. A theory is outlined in The search for a better and easy to use tool for the solution Section 3 for convergence and the error estimate. Illustraof nonlinear equations that illuminate the nonlinear phetive examples in Section 4 are followed by the conclusions nomena of real life problems of science and engineering in Section 5. has recently received a continuing interest. Various methods, therefore, were proposed to find approximate solutions of nonlinear equations. One of the most recent popular techniques is the homotopy perturbation method, which is a combination of the classical per- 2 The Homotopy Perturbation Method turbation technique and the homotopy concept as used in The homotopy perturbation method was first proposed by topology. In the homotopy perturbation method, which re- the Chinese mathematician He [1]. The essential idea of quires neither a small parameter nor a linear term in a dif- this method is to introduce a homotopy parameter, say p, ferential equation, a homotopy with an embedding param- which varies from 0 to 1. At p D 0, the system of equaeter p 2 Œ0; 1 is constructed. In [1] a basic idea of ho- tions is usually reduced to a simplified form which normally motopy perturbation method for solving nonlinear differ- admits a rather simple solution. As p gradually increases ential equations was presented. In this method, the solu- continuously toward 1, the system goes through a sequence tion is considered as the sum of an infinite series which of deformations, and the solution at each stage is close to converges rapidly to the accurate solution. Numerous non- that at the previous stage of the deformation. Eventually at linear problems were recently treated by the method, see p D 1, the system takes the original form of the equation amongst them, the nonlinear Schrödinger equations [2], the and the final stage of the deformation gives the desired sononlinear equations arising in heat transfer [3] and [4], the lution. To illustrate the basic ideas of this method, consider * Corresponding author: M. Turkyilmazoglu, Mathematics Department, University of Hacettepe, 06532–Beytepe, Ankara, Turkey; E-mail: [email protected]. Received: June 22, 2011. Accepted: September 20, 2011.

the nonlinear boundary value problem N.u.r// D 0I r 2 ;

du D 0I r 2 ; (1) B u.r/; dn

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M. Turkyilmazoglu

where u.r/, that is defined over the region , is the function to be solved under the boundary constraints in B, that are defined over the boundary of . He’s homotopy perturbation technique [1, 11] defines a homotopy u.r; p/ W R Œ0; 1 ! R and then constructs

is lacking in [10], which is essential for the study of error analysis of the resulting homotopy series approximations. Unlike the theorem stated in [10], the subsequent proposed theorems here do not have such drawbacks, including further an error estimate.

(2) Theorem 1. Suppose that A R be a Banach space donated with a suitable norm kk (depending on the physical where L is a suitable auxiliary linear operator and u0 is an problem considered), over which the sequence uk .t / of (4) initial approximation of equation (1) satisfying exactly the is defined. Assume also that the initial approximation u0 .t / remains inside the ball of the solution u.t /. Taking r 2 R boundary conditions. It is obvious from equation (2) that to be a constant, the following statements hold true: (3) (i) If kvkC1 .t /k rkvk .t /k for all k, given some 0 < H.u; 0/ D L.u/ L.u0 /; H.u; 1/ D N.u/: P k r < 1, then the series solution u.t; p/ D 1 kD0 uk .t /p , As p moves from 0 to 1, u.r; p/ moves from u0 .r/ to u.r/. defined in (4) converges absolutely at p D 1 to (5) over the In topology, this is called a deformation and L.u/ L.u0 / domain of definition of t . and N.u/ are said to be homotopic. Our basic assumption k, given some r > 1, is that the solution of equation (2) when equated to zero can (ii) If kvkC1 .t /k rkvk .t /k for all P1 uk .t /p k , defined then the series solution u.t; p/ D kD0 be expressed as a power series in p in (4) diverges at p D 1 over the domain of definition of t . H.u; p/ D .1 p/ŒL.u/ L.u0 / C pN.u/;

u.r; p/ D u0 .r/ C pu1 .r/ C p 2 u2 .r/ C

Proof. (i) According to the ratio test for the power series in p, the proof is clear. However, in order to give an estimate D uk .r/p k : (4) to the truncation error of homotopy perturbation method, kD0 we shortly give the whole proof here. If Sn .t / denote the sequence of partial sum of the series (5), we need to show The approximate solution of equation (1), therefore, can be that S .t / is a Cauchy sequence in A. For this purpose, n readily obtained as consider 1 X

u.r/ D lim u.r; p/ D p!1

1 X kD0

uk .r/:

(5)

kSnC1 .t / Sn .t /k D kunC1 .t /k rkun .t /k r 2 kun1 .t /k r nC1 ku0 .t /k:

(6)

Whenever the series (4) is known to be convergent, then (5) It should be remarked that owing to (6), all the approximarepresents the exact solution of (1); see [11] for a proof. tions produced by the homotopy method (2) will lie within the ball of u.t /. For every m; n 2 N , n m, making use of (6) and the triangle inequality successively, we have 3 A Convergence Theorem and Error Estimate kSn .t / Sm .t /k D k.Sn .t / Sn1 .t // C .Sn1 .t / Using the methodology underlined above, the number of Sn2 .t // C C .SmC1 .t / Sm .t //k problems treated by the homotopy perturbation method approach a couple of hundred now. However, the very basic 1 r nm mC1 r ku0 .t /k: (7) question of why the series obtained by setting p D 1 in (4) 1r should be convergent remains unanswered till today. Despite the fact that [10] gave a sufficient condition for the Since 0 < r < 1, we get from (7) convergence based on the Banach contraction mapping thelim kSn .t / Sm .t /k D 0: (8) orem, it is vulnerable to several deficiencies. First of all, aln;m!1 though the theorem in [10] relies upon the contraction mapping, it is not always practical to show this property of the Therefore, Sn .t / is a Cauchy sequence in the Banach space operator. Second, it appears that the theorem requires aux- A, and this implies that the series solution (5) is convergent. dm This completes the proof of (i). iliary parameter L to be only in the form L D dr m (which is believed to correspond to the traditional Taylor series exThe proof of (ii) follows from the fact that under the hypansion of the solution). Third, as clearly noticed from the pothesis supplied in (ii), there exists a number l, l > r > 1, presented examples in [10], to show the convergence, ex- so that the interval of convergence of the power series (4) act solution is demanded, which is impossible for most of is jpj < 1= l < 1, which obviously excludes the case of the engineering applications. And finally, an error estimate p D 1.


11


Remark 1. Since the finite number of terms does not affect the convergence, theorem 1 is equally valid if the inequalities stated in (i) and (ii) are true for sufficiently large k. kv .t/k needs to be pursued in Therefore, the ratio rat D kvkC1 k .t/k practice. Remark 2. By enforcing the ratio in (i) to hold true in the infinite limit, the validity region of t for the homotopy series solution can also be constructed. Theorem 2. If the series solution defined in (5) is convergent, then it converges to the exact solution of the nonlinear problem (1). Proof. By the hypothesis, limn!1 un .t / D 0, and further producing the homotopy series coefficients uk .t / of (4) from (2)–3, it can be shown that N

1 X

uk .t / D 0:

kD0

This completes the proof.

that governs the steady free convection flow over a vertical semi-infinite flat plate which is embedded in a fluid saturated porous medium of ambient temperature [13] and also the steady-state boundary-layer flows over a permeable stretching sheet [14]. In accordance with Theorem 1, choosd , the homotopy series solution ing u0 .t / D 0 and L D dt via the homotopy perturbation (2) can be calculated as u.t / D t

2 t3 C t5 C : 3 15

(13)

If we employ the absolute value norm in R, then Theorem 1 ensures the convergency of (13) provided that lim

n!1

junC1 .t /j < 1; jun .t /j

that yields the convergence region 2 < t < 2 for the obtained homotopy series (13). (9) To get a larger region of convergence in t , let us change the linear operator and the initial guess, respectively as, d C 2 and u0 .t / D 1 e 2t . Subsequently, it can be L D dt shown that the terms in the homotopy series (5) satisfy

P Theorem 3. Assume that the series solution 1 nD0 un .t /, 1 unC1 .t / D .1 e 2t /: (14) defined in (5)Pis convergent to the solution u.t /. If the trunun .t / 2 u .t / is used as an approximation to the cated series M nD0 n solution u.t / of problem (1), then an upper bound for the Hence, regarding (14), Theorem 1 ensures the convergence error EM .t /, is estimated as of the corresponding homotopy series for all values of t satisfying t > ln23 . r M C1 ku0 .t /k: (10) EM .t / 1r Example 2. Consider now the second-order nonlinear difProof. Making use of inequality (7) of Theorem 1, we im- ferential equation mediately obtain 2u00 C u u2 D 0; u.0/ D 0; u.1/ D 1; (15) nM 1r (11) that governs the steady mixed convection flow past a plane ku.t / SM .t /k r M C1 ku0 .t /k; 1r of arbitrary shape under the boundary layer and Darcy– and taking into account .1 r nM / < 1, (11) directly leads Boussinesq approximations [15]. With the choices of L D d2 to the desired formula (10). This completes the proof. 1 and u0 .t / D 1 e t , the homotopy (2) generates dt 2 a homotopy series (5) whose ratios of successive terms are tabulated in table 1 taking into consideration the L2 norm in R. As observed from the table, the convergence of the homotopy series to the exact solution takes place, but at a 4 Illustrative Examples considerably slow rate. Since the limit of ratio is very close To illustrate validity of the outlined theorems, we take into to 1, it apparently requires more terms for the convergence account the following examples taken from the homotopy of the homotopy series (5). perturbation studies in the literature. 4.1

Example 3. Consider now the nonlinear partial differential Burger’s equation

Convergence cases

Example 1. Consider the first-order nonlinear differential equation [12] u0 C u2 D 1;

u.0/ D 0;

(12)

u t C uux D uxx ;

u.x; 0/ D 2x;

(16)

that has been found to describe various kinds of phenomena, such as a mathematical model of turbulence and the


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M. Turkyilmazoglu

M D2 0.749659

M D 20 0.941188

M D 40 0.969703

M D 60 0.980241

M D 80 0.98512

M D 100 0.98529

Table 1. The ratio of successive terms in the homotopy series corresponding to equation (15). M denotes the number of

terms in the homotopy series (5). approximate theory of the flow through a shock wave trav- (2) then generates the subsequent homotopy series eling in a viscous fluid [16]. Equation (16) admits an exact 1 solution given by X .x C y C z cos x cos y cos z/ u.x; t / D nD0 2x 2n u.x; t / D : (17) t t 2nC1 1 C 2t (22) .2n/Š .2n C 1/Š To approximate the exact solution (17), we choose the auxiliary parameters as u0 .x; t / D 2x and L D @t@ . Then, which leads to limn!1 junC1 j D 0. So, the homotopy sejun j the homotopy (2) turns out to be ries (22) converges to the exact solution of (20) for all t . u t .x; t; p/Cp.u.x; t; p/ux .x; p; t /uxx .x; t; p// D 0; u.x; 0; p/ D 2x:

(18)

Equation (18) produces the below homotopy series for the solution of (16)

Example 5. As a final example, we consider the classical Blasius flat-plate flow problem of fluid mechanics governed by the nonlinear initial value of third-order y 000 C

yy 00 D 0; 2

y.0/ D y 0 .0/ D y 0 .1/ 1 D 0; 2 Œ0; 1/:

2

3

(23)

4

u.x; t / D 2x 4xt C 8xt 16xt C 32xt C C .1/n 2nC1 xt n C ;

With the transformations (19) yD

u ;

t D ;

whose convergence to the exact solution takes place for ju j < 1. Thus, in accordance with theorem 1, where is a scaling parameter taken here as 4, system (23) limn!1 junC1 nj is converted into this holds exactly for the values of 1=2 < t < 1=2. Example 4. Consider now the following fourth-order parabolic partial differential equation arising in the study of the transverse vibrations of a uniform flexible beam [17] ut t C

yCz zCx 1 uxxxx C 1 uyyyy 2 cos x 2 cos y xCy C 1 uzzzz D 0; (20) 2 cos z

u.x; y; z; 0/ D u t .x; y; z; 0/ D x C y C z .cos x C cos y C cos z/; whose exact solution is given by u.x; t / D .x C y C z cos x cos y cos z/e t : (21)

u000 C

uu00 D 0; 22

u.0/ D u0 .0/ D u0 .1/ 1 D 0; t 2 Œ0; 1/:

We can define the residual error based on L2 norm Z 1 yy 00 2 1=2 Res D Œy 000 C ; d 2 0 and the residual error based on L1 norm Z 1 yy 00 jd; Res D jy 000 C 2 0

(24)

(25)

(26)

and also the residual error based on maximum norm ˇ yy 00 ˇˇ ˇ Res D Maxx2RC ˇy 000 C ˇ: 2 3

2

(27)

d d t To approximate the exact solution (21), if we choose the Taking L D dt , the 3 C dt 2 with u0 .t / D 1 C t C e auxiliary parameters respectively, u0 .x; t / D .x C y C z values of residual errors from the homotopy (2) are shown 2 cos x cos y cos z/.1 t / and L D @t@ 2 , the homotopy in Table 2. Figure 1 demonstrates the ratio in Theorem 1


13


M D1 0.488667 0.706018 0.984375

Res (25) (26) (27)

M D 10 2:760 102 5:926 102 1:403 102

M D 20 3:893 103 5:306 103 1:832 103

M D 30 6:736 104 2:682 104 3:112 104

Table 2. The residual errors Res based on, respectively, (25), (26) and (26) for the homotopy series solutions corresponding to equation (24). M denotes the number of terms in the homotopy series (5).

whose exact separable solution is given by

1.0

u.x; t / D e xt :

0.8

(30)

To approximate the exact solution (30), if we choose the auxiliary parameters u0 .x; t / D e x and L D @t@ , then the homotopy (2) turns out to be

rat

0.6

0.4

u t .x; t; p/ C p.ux .x; p; t / uxxt .x; t; p// D 0;

0.2

0.0

u.x; 0; p/ D e x : 0

5

10

15

20

25

k

Figure 1. A list plot of the ratio rat versus k from the theo-

rem to reveal the convergence of the HPM solution for the Blasius equation (24).

(31)

The homotopy series solution of (29) from (31) can be found as u.x; t / D

e x .720 C 45 360t C 46 440t 2 C 13 320t 3 720 C 1470t 4 C 66t 5 C /; (32)

whose radius of convergence is zero, so that the homotopy which was computed from the homotopy series (5). The series (32) is convergent only at the point t D 0. convergence is assured further by the value of ratio less than Two features are worth mentioning about the prounity, suggesting that limit tends to 0.83. posed convergence judgment. The first is that, in many cases it is hard to judge whether the value of the ratio kvkC1 .t /k=kvk .t /k is less than or greater than unity, es4.2 Non-convergence Cases pecially for partial differential equations. The proposed method then requires numerical evaluation of the ratio with Example 6. Let us now consider the following heat transa carefully chosen norm as implemented in Examples 2 and fer problem governed by the nonlinear ordinary differential 5, or the remainder of the homotopy series in (5) should be equation [18] forced to vanish using some estimates. The second important issue is that, particular choices shown in this study were (28) already made in the literature by several authors and our .1 C 2u/u0 C u D 0; u.0/ D 1; t 2 RC : intention is to prove by means of our theorems why their When the auxiliary parameters are taken as u0 .t / D 1 choices lead to non-convergent series or convergent series d , the homotopy (2) generates a homotopy with a restricted interval of convergence. As also demonas well as L D dt series of the form (5). In line with theorem 1, enforcing strated through Example 1, for different auxiliary linear opku .t/k < 1 under the maximum norm defined erator L and initial approximation u0 , the homotopy series limn!1 kunC1 n .t/k over R, it can be easily deduced that the region of conver- could be constructed so as to change the convergence regency occurs for t 2 R . However, this region is not com- gion or even the convergence property itself, which can be plied with the physical region corresponding to (28), so the ensured by the theorems provided. homotopy perturbation method diverges for this example. Example 7. As a final example, let us consider the linear 5 Concluding Remarks partial differential equation In this paper, the homotopy perturbation method has been analyzed with an aim to investigate the conditions which re(29) sult in the convergence of the generated homotopy solutions u t C ux 2uxxt D 0; u.x; 0/ D e x ;


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M. Turkyilmazoglu

Burgers equation in fluid, Num. Meth. Part. Dif. Eqns. 26 of the nonlinear ordinary and partial differential equations. (2010), 917–930. The theorems outlined in the paper have proved that if specific values are assigned to the auxiliary parameters in the [7] S. Abbasbandy, Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s dehomotopy perturbation method, then the approximate hocomposition method, Appl. Math. Comput. 173 (2006), 493– motopy results successfully converge to the exact solution. 500. Examples have been provided to verify the theory. Via the [8] S. Das, P. K. Gupta and Rajeev, A Fractional predator-prey theorems provided here, not only is the question of converModel and its Solution, Int. J. Nonlinear Sci. Numer. Simulation 10 (2009), 873–876. gence of the homotopy series answered, but also the region of validity of the space variable ensuring the convergence is [9] S. Das and P. K. Gupta, A mathematical model on fractional Lotka-Volterra equations, Journal of Theoretical Bidetermined. The physical problems involving stronger nonology 277 (2011), 1–6. linearities can also be dealt with the homotopy method un[10] J. Biazar and H. Ghazvini, Convergence of the homotopy der the assurance of convergence provided by the presented perturbation method for partial differential equations, Nontheorems. linear Anal. Real World Appl. 366 (2009), 2633–2640.

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