Abstract
The two-dimensional nonlinear wave equations are considered. Solution to the problem is approximated by using optimal homotopy asymptotic method (OHAM). The residual and convergence of the proposed method to nonlinear wave equation are presented through graphs. The resultant analytic series solution of the two-dimensional nonlinear wave equation shows the effectiveness of the proposed method. The comparison of results has been made with the existing results available in the literature.
1. Introduction
The wave equations play a vital role in diverse areas of engineering, physics, and scientific applications. An enormous amount of research work is already available in the study of wave equations [1, 2]. This paper deals with the two-dimensional nonlinear wave equation of the form
The differential equations (DEs) can be solved analytically by a number of perturbation techniques [3, 4]. These techniques are fairly simple in calculating the solutions, but their limitations are based on the assumption of small parameters. Therefore, the researchers are on the go for some new techniques to overcome these limitations.
The idea of homotopy was pooled with perturbation. Liao [5] proposed homotopy analysis method (HAM) in his Ph.D. dissertation and applied it to various nonlinear engineering problems [6–8]. The homotopy perturbation method (HPM) was initially introduced by He [9–13]. HPM has been extensively used by several researchers successfully for physical models [14–16]. Some useful comparisons between HAM and HPM were done by Domairry and Liang [17, 18].
Recently Marinca and Herişanu [19–21] introduced OHAM for the solution of nonlinear problems which made the perturbation methods independent of the assumption of small parameters, and Ullah et al. [22–26] have extended and applied OHAM successfully for numerous nonlinear phenomena.
The motive of this paper is to apply OHAM for the solution of two-dimensional nonlinear wave equations. In [19–21] OHAM has been proved to be useful for obtaining an approximate solution of nonlinear differential equations. Here, we have proved that OHAM is more useful and reliable for the solution of two-dimensional nonlinear wave equations, hence, showing its validity and greater potential for the solution of transient physical phenomenon in science and engineering.
Section 2 has the basic idea of OHAM formulated for the solution of partial differential equations. In Section 3, the effectiveness of OHAM for two-dimensional nonlinear wave equation has been studied.
2. Basic Formulation of OHAM
Consider the partial differential equation of the following form: where is a differential operator, is an unknown function, and denote spatial and temporal independent variables, respectively, is the boundary of , and is a known analytic function. can be divided into two parts: and such that where is the simpler part of the partial differential equation which is easier to solve and contains the remaining part of .
According to OHAM, one can construct an optimal homotopy which satisfies Here the auxiliary function is nonzero for and . Equation (4) is called optimal homotopy equation. Clearly, we have Obviously, when and we obtain and , respectively. Thus, as varies from to , the solution approaches from to , where is obtained from (4) for : Next, we choose auxiliary function in the form To get an approximate solution, we expand by Taylor’s series about in the following manner: Substituting (8) into (4) and equating the coefficient of like powers of , we obtain zeroth-order problem, given by (6), the first- and second-order problems are given by (9) and (10), respectively, and the general governing equations for are given by (11) as follows: where are the coefficient of in the expansion of about the embedding parameter . One has It should be underscored that the for is governed by the linear equations with linear boundary conditions that come from the original problem, which can be easily solved.
It has been observed that the convergence of the series equation (8) depends on the auxiliary constants . If it is convergent at , one has Substituting (13) into (1), it results in the following expression for residual: In actual computation . If then is the Exact solution of the problem. Generally it does not happen, especially in nonlinear problems.
For determining auxiliary constants, , , there are a number of methods like Galerkin’s method, Ritz method, least squares method, and collocation method. The method of least squares can be applied as follows: The th-order approximate solution can be obtained by these optimal constants. The more general auxiliary function is useful for convergence, which depends on constants , can be optimally identified by (16), and is useful in error minimization.
3. Application of OHAM to Two-Dimensional Nonlinear Wave Equations
To demonstrate the effectiveness of the formulation of OHAM, we consider two-dimensional nonlinear wave equations of the form (1) with initial conditions Applying the method formulated in Section 2 leads to
Zeroth-Order Problem. Consider Its solution is
First-Order Problem. Consider Its solution is
Second-Order Problem. Consider Its solution is
Third-Order Problem. Consider Its solution is Adding (20), (22), (24), and (26), we obtain The residual can be calculated by using (14). For calculations of the constants , , and , using (27) in (17) and applying the procedure mentioned in (13) and (14), we get The Exact solution is [2] and HPM solution is [2]
4. Results and Discussions
The formulation presented in Section 2 provides accurate solutions for the problems demonstrated in Section 3. We have used Mathematica 7 for most of our computational work. In Table 1 and Figures 1, 2, 3, we have compared the OHAM results with the results obtained by HPM and Exact for various values of and at spatial domain for Table 1 and at different values of and fixed value of for Figures 1–3. In Table 2, we have presented absolute errors at different values of and . Figure 4 presents the residual at a spatial domain at . The convergence of OHAM is presented in Figure 5 at .
5. Conclusion
In this paper, we studied the two-dimensional nonlinear wave equation. We have used the OHAM to approximate the solution of the titled problem. It is clear from the present work that OHAM can successfully be applied to nonlinear phenomena like the one we had. The obtained results are accurate which shows the effectiveness and validity of the proposed method. It is observed that OHAM is simpler in applicability and more convenient to control the convergence and involve less computational work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.