Homotopy Perturbation Method for Solving Wave-Like Nonlinear Equations with Initial-Boundary Conditions

Afgan Aslanov Mathematics and Computing Department, Beykent University, 34396 Istanbul, Turkey Correspondence should be addressed to Afgan Aslanov, afganaslanov@beykent.edu.tr Received 4 May 2011; Revised 27 June 2011; Accepted 20 July 2011 Academic Editor: Pham Huu Anh Ngoc Copyright q 2011 Afgan Aslanov. This 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. The homotopy perturbation method is employed to obtain approximate analytical solutions of the wave-like nonlinear equations with initial-boundary conditions. An efficient way of choosing the auxiliary operator is presented. The results demonstrate reliability and efficiency of the method.


Introduction
In this paper, we consider the equation where f, g, ϕ, ψ, and h are known functions.Problems like 1.1 -1.2 -1.3 model many problems in classical and quantum mechanics, solitons, and matter physics 1, 2 .If f is a function of u only, we obtain a Klein-Gordon or sine-Gordon-type equations.
In the last decade, some various approximate methods have been developed, such as the homotopy perturbation method HPM 3-13 and Adomian's decomposition method ADM 14-20 to solve linear and nonlinear differential equations.
Unlike the various approximation techniques for solving nonlinear wave type problems, which are usually valid for initial value problems without boundary conditions or some special type of problems homogenous, etc. , our technique is applicable for all initial-boundary problems of type 1.1 -1.2 -1.3 .Chowdhury and Hashim 9 applied the HPM for solving Klein-Gordon and sine-Gordon equations, with initial conditions 1.2 .El-Sayed 19 and Wazwaz and Gorguis 20 used ADM for solving wave-like and heatlike problems.Their approaches cannot be applied for all wave-like equations with initialboundary conditions since the operator L u tt cannot control the boundary condition 1.3see Example 2.1 below.
The central idea here is that the problem has a unique solution see, e.g., 21 and therefore there exists an inverse of the operator The main idea of HPM is to introduce a homotopy parameter, say p, which takes values from 0 to 1.When p 0 the equation usually reduces to a sufficiently simplified form linear or very easy nonlinear .As p increases to 1, the equation goes through a sequence of "deformations" homotopics and at p 1 takes the original form of the equation.
We rewrite 1.1 as Lu f x, t, u, u x , u y , u xy g x, t , 1.5 and construct the following homotopy: Usually we take v 0 as a solution of the problem 1.1 -1.2 -1.3 with f 0 or simply v 0 0. Assume that the solution of 1.6 is in the form and substituting 1.7 into 1.6 and equating terms of the same powers of p we obtain a system of equations for u 0 , u 1 , u 2 , . ... Solving these system of equations we obtain a solution in the form

Applications
The HPM and ADM offer excellent choices for obtaining the closed-form analytical solutions of wave-like equations.Chowdhury and Hashim 9 , El-Sayed 19 , and Wazwaz and Gorguis 20 recently showed how the HPM and ADM can be applied to find an analytic approximate solution of wave-like equations with initial conditions 1.2 .They mainly used the operator L u tt for solving the wave-like problems.But in case of inhomogeneous nonlinear or even linear equations with initial-boundary conditions, these approaches have some difficulties.If we construct the standard homotopy with L u tt , for solving wave-like equations with initial-boundary conditions, usually in the second or even in the first stage of HPM, we obtain an "overdetermined" or very difficult problem.To explain these difficulties we consider the following example.
Example 2.1.Consider the linear problem

2.1
The exact solution is

2.2
Using our HPM technique we can easily find this solution.Indeed, let us take L u tt − u xx , v 0 0 and construct the homotopy Now substituting 1.7 into u and equating the coefficients of like powers of p, we get a system of linear equations
Now let us show that this problem can not be solved when L is taken as L u tt .Indeed if we choose v 0 0 or v 0 −x 3 which seems most natural and appropriate and L u tt and construct the homotopy we obtain an overdetermined problem for u 0 in the form which has no solution.
If in the first stage we consider the boundary conditions u 0 x, 0 −x 3 , u 0t x, 0 0, u 0 0, t 0, we obtain u 0 −x 3 and in the second stage we need to solve the problem which is overdetermined again and has no solution.

2.9
The exact solution is u e t x.We take again L u tt − u xx , v 0 0, and construct the homotopy x − e t x 0.

2.10
Substituting 1.7 into 2.10 we obtain x − e t x 0,

2.11
and equating terms of the coefficients of like powers of p gives x, 0 0, u 3 0, t 0, . . . .

2.12
Solving these equations we obtain see   for x < t.In a similar manner we have and therefore The absolute errors between the exact and three term approximation of the series solution for some values of x, t ∈ 0, 1 × 0, 1 are shown in Table 1.
Example 2.3.Now we consider the problem

2.19
The exact solution is u x t 2 x − t 2 x 2 .We take v 0 0, and construct the homotopy Now substituting 1.7 into 2.20 and equating terms of the coefficients of like powers of p, we obtain

2.29
Continuing we obtain

Conclusion
Homotopy perturbation method has been successful for solving many linear and nonlinear wave-type problems.However, it has difficulties in dealing with initial boundary problems, namely, in including all initial and boundary conditions together into the process of homotopy perturbation and computation.Our main goal is to construct the homotopy perturbation scheme containing all initial and boundary conditions together.The goal is achieved by involving an auxiliary operator which includes both variables x and t.

Table 1 :
Maximum errors for Example 2.2.
tt − u xx .Substituting 1.7 into u and equating the coefficients of like powers of p, we get a system of linear equations