Application of the Homotopy Analysis Method for Solving Equal-Width Wave and Modified Equal-Width Wave Equations Esmail Baboliana, Jamshid Saeidiana, and Mahmood Paripourb a Department of Mathematics and Computer Sciences, Tarbiat Moallem University, 599 Taleghani avenue, Tehran 1561836314, Iran b Hamedan University of Technology, Hamedan, 65156-579, Iran Reprint requests to J. S.; E-mail: [email protected] Z. Naturforsch. 64a, 685 – 690 (2009); received November 3, 2008 / revised March 1, 2009 Although the homotopy analysis method (HAM) is, by now, a well-known analytic method for handling functional equations, there is no general proof of its applicability to all kinds of equations. In this paper, by using this method to solve equal-width wave (EW) and modified equal-width wave (MEW) equations, we have made a new contribution to this field of research. Our goal is to emphasize on two points: one is the efficiency of HAM in handling these important family of equations and its superiority over other analytic methods like homotopy perturbation method (HPM), variational itera- tion method (VIM), and Adomian decomposition method (ADM). The other point is that although the considered two equations have different nonlinear terms, we have used the same auxiliary elements to solve them. Key words: Homotopy Analysis Method; Equal-Width Wave Equation; Modified Equal-Width Wave Equation; Nonlinear Equations. PACS numbers: 02.30.Jr; 02.60.Cb; 02.90.+p; 05.45.Yv 1. Introduction therein. In this method, Liao uses homotopy, a concept from topology, to continuously deform the nonlinear Nonlinear equations have been of considerable in- equation under study to a system of linear ones. terest among scientists because they model many sci- The reasons for the preference of HAM over other entific problems arising in different fields. Although, analytic techniques can be listed as follows: numerical methods have largely been applied to solve (i) HAM properly overcomes restrictions of pertur- these equations, some restrictions of numerical meth- bation techniques because it doesn’t need any small or ods have motivated scientists to search for analytic large parameters to be contained in the equation. solutions. So there are, also, analytic techniques for (ii) Liao, in his book [2], proves that this method solving nonlinear problems, among the classic ones is a generalization of some previously used techniques we can consider Lyapunov’s artificial small parameter such as δ-expansion method, artificial small parameter method, perturbation techniques, and the δ-expansion method, and ADM. Also, it is shown that HPM [14] is method. Recently the idea of giving analytic approx- just a special case of HAM [10, 13, 15]. imations to functional equations has largely devel- oped and some new techniques have been proposed. (iii) Unlike previous analytic techniques, HAM Among the newly developed ones are the Adomian provides us with a convenient way to adjust and con- trol the convergence region and rate of approximation decomposition method (ADM), the homotopy analy- h sis method (HAM), the variational iteration method series. This is done by using the so called ¯-curves, (VIM), the homotopy perturbation method (HPM), the see [2]. tanh method, the sine-cosine method, and the exp- The present work concerns with the application of function method. HAM in solving the equal-width wave (EW) equation Homotopy analysis method (HAM), first proposed and a modified form of this equation. We focus on two by Liao [1, 2], is an elegant method which has proved points: firstly, the applicability and efficiency of HAM its effectiveness and efficiency in solving many types in handling this family of wave-type equations and of functional equations, see [1 – 13] and the references its superiority in comparison with other techniques. 0932–0784 / 09 / 1100–0685 $ 06.00 c 2009 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 686 E. Babolian et al. · HAM for Equal-Width Wave and Modified Equal-Width Wave Equations Secondly, although the wave-type equations may have If the auxiliary linear operator, the initial guess, different nonlinear terms, when applying HAM we can the auxiliary parameter, and the auxiliary function are treat them by using a general choice of auxiliary ele- properly chosen, the series (2) converges at q = 1and ments. we have ∞ 2. Basic Idea of HAM u(r,t)=u0(r,t)+ ∑ um(r,t), (4) m=1 For a good understanding of HAM the reader is re- ferred to Liao’s book [2], also a general introduction which must be a solution to the original nonlinear and recent developments could be found in [16]. In this equation, this has been proved by Liao [2]. If we set h = − H(r,t)= section we briefly review the basic idea of HAM. Let ¯ 1and 1, equation (1) becomes us consider the following nonlinear equation in a gen- (1 − q)L[φ(r,t;q) − v (r,t)] + qN[φ(r,t;q)] = 0, eral form: 0 which is a special case and is used in the homotopy N[u(r,t)] = 0, perturbation method (HPM). where N is a nonlinear operator, u(r,t) is an unknown According to (3), the governing equations for function, and r and t denote spatial and temporal in- um(x,t) can be deduced from the zeroth-order defor- dependent variables, respectively. At this stage we ig- mation equation (1). Define the vector nore initial or boundary conditions, because they are u =[u (r,t),u (r,t),...,u (r,t)]. not used, directly, in construction of the so-called ‘ho- n 0 1 n motopy equation’. Actually we have to refer to these Differentiating equation (1) m times with respect to the conditions for solving the resulting linear system, for embedding parameter q, then setting q = 0 and finally more details see [2, 16, 17]. By means of generalizing dividing by m!, we have the so-called ‘mth-order de- the traditional homotopy method, Liao constructs the formation equation’, so-called ‘zeroth-order deformation equation’ L[um(r,t)−χmum−1(r,t)] = hH¯ (r,t)Rm(um−1,r,t), (5) (1−q)L[φ(r,t;q)−v0(r,t)] = qhH¯ (r,t)N[φ(r,t;q)], (1) where where q ∈ [0,1] is the embedding (or homotopy) pa- m− rameter,h ¯ = 0 is a non-zero auxiliary parameter – 1 ∂ 1N[φ(r,t;q)] Rm(um− ,r,t)= |q= , (6) which recently renamed to ‘convergence-control pa- 1 (m − 1)! ∂qm−1 0 rameter’, H(r,t) is an auxiliary function, L is an auxil- iary linear operator, v0(r,t) is an initial guess for u(r,t), and φ(r,t q) and ; is an unknown function. It is important , m ≤ χ = 0 1 that we have great freedom to choose auxiliary ele- m , m > . ments in HAM [2]. Obviously, when q = 0andq = 1 1 1 one has Directly substituting the series (2) into the zeroth- order deformation equation (1), equating coefficients φ(r,t;0)=v0(r,t), φ(r,t;1)=u(r,t). of like-power of the embedding parameter q, one can Thus as q increases from 0 to 1, the solution φ(r,t;q) get the same high-order deformation (5), as proved varies from the initial guess v0(r,t) to the solu- in [10, 15, 16]. Equation (5) is a linear one, so we tion u(r,t). Expanding φ(r,t;q) in Taylor series with have transformed a nonlinear equation to a set of lin- respect to q,wehave ear ones, which can be easily solved using an iterative u (r,t) ∞ procedure. After solving (5), we can substitute m m in (4) and obtain an approximation of arbitrary order. φ(r,t;q)=u0(r,t)+ ∑ um(r,t)q , (2) m=1 It worths noting that HAM, in many cases, leads to a series which gives the exact solution, see e. g. [17], and where using the auxiliary parameterh ¯ one can adjust and con- ∂mφ(r,t q) trol the convergence region and rate of approximation u (r,t)= 1 ; | . m m q=0 (3) h m! ∂q series. The discussion on how to use ¯ is made in [2]. E. Babolian et al. · HAM for Equal-Width Wave and Modified Equal-Width Wave Equations 687 When we are aiming to get approximate solutions, we ement scheme to the EW equation [20]. An spec- can use only the few first terms in (4), without having tral method, finite difference method, and cubic B- concern to get an exact solution series. spline collocation method also has been applied to In HAM what we are mainly concerned about is solve the aforementioned equation. Recently, Yusu- how to choose the initial guess (v0), the auxiliary lin- foglu and Bekir solved this equation using VIM and ear operator L, and the auxiliary function H(r,t).Liao, ADM [21]. in his book [2], has proposed three rules for choosing these components, actually here we get advantage of 4. Modified Equal-Width Wave Equation all informations about the equation such as its physi- cal background and boundary/initial conditions. In the The modified equal-width wave (MEW) equation is cases, like our considered equations, it’s common to formulated as follows: choose the auxiliary function to be H(r,t)=1, as 2 used by many authors [7, 9, 17]. When we don’t have ut + εu ux − µuxxt = 0. (9) enough information to decide on a set of base func- tions, this choice for H seems to be a good alterna- This equation has a solitary wave solution of the form tive. 1 u(x,t)=A (√ (x − ct − x )), It seems that when we don’t emphasize on a set of sech µ 0 (10) base function, for representing the solution, there is no difference between HAM and HPM.