Asian Journal of Applied Sciences

Year: 2012 | Volume: 5 | Issue: 2 | Page No.: 85-95
DOI: 10.3923/ajaps.2012.85.95

Analysis of Homotopy Perturbation Method for Solution of Hyperbolic Equations with an Integral Condition

P.R. Sharma and Giriraj Methi

Abstract: Aim of the study was to obtain exact solutions of a hyperbolic equation with nonlocal boundary conditions using HPM, Pade approximant and Laplace transform. Some examples are discussed to illustrate the method which shows the effectiveness, accuracy and fast convergence of the method. Also relative error is discussed with the exact solution and HPM and Pade approximant solution.

Keywords: non-local boundary condition, homotopy perturbation method and Partial differential equations

INTRODUCTION

Partial differential equations with nonlocal boundary specifications have received much attention in last couple of years. However, most of the research papers were related to parabolic partial differential equations (Batten, 1963; Yurchuk, 1986; Cahlon et al., 1995; Bougoffa, 2005). Integral nonlocal boundary conditions can be used when it is impossible to directly determine the values of the sought quantity on the boundary and we know its total amount or integral average on space domain. There are some problems in modern physics and technology that can be modelled by hyperbolic boundary value problems with nonlocal boundary conditions (Shi, 1993; Belin, 2001; Pulkina, 1999).

The objective of this study was to solve a boundary value problem with non-local integral condition for a class of hyperbolic partial differential equations using Homotopy Perturbation Method (HPM).

THE HOMOGENOUS EQUATION

Consider the homogeneous hyperbolic equation:

in the rectangular domain Ω = (0, 1)x(0, T).

With initial conditions:

Dirichlet boundary condition:

and the non-local boundary condition:

Assuming that

φ(x), Ψ(x) ε I₂(0.1) are known function and satisfy the compatibility conditions φ(1)-φ(0) = 0, Ψ(1)-Ψ(0) = 0 and .

Firstly, as in Belin (2001) the Eq. 1-5 are reduced to an equivalent problem given below:

Lemma 1: Problem given by Eq. 1-5 is equivalent to the following problem:

Proof: Let w(x, t) be a solution of the Eq. 1-5. Integrating Eq. 1 with respect to x over (0, 1) and using Eq. 5, we obtain w_x(1.t)-w_x(0, t) = 0.

Let w(x, t) be a solution of the Eq. 6, we will show that:

For this, integrating with respect to x, we obtain:

by virtue of compatibility conditions:

we get:

Hence, the presence of integral condition complicates the application of standard methods therefore using Lemma-1 to eliminate the non-local condition and reduces it to the classical boundary condition.

METHOD OF SOLUTION

Consider the following nonlinear differential equation:

with the boundary conditions:

where, A is a general differential operator, B is a boundary operator, f(r) is known analytic function and γ is the boundary of the domain D.

The operator A can generally be divided into two parts L and N, where, L is linear and N is nonlinear, therefore, Eq. 7 can be written as:

By using Homotopy technique, we construct a homotopy u(r, p):Dx[0, 1]→R which satisfies:

where, pε[0, 1] is an embedding parameter and w₀ is the initial approximation of Eq. 7 which satisfies the boundary conditions.

Obviously, we get:

The changing process of p from zero to unity is just that of u(r, p) changing from w₀(r) to w(r). This is called deformation and also L(u)-L(w₀) and A(u)-f(r) are called homotopic in topology. If, the embedding parameter p (0≤x≤1) is considered as a small parameter, applying the classical perturbation technique, we can assume that the solution of Eq. 10 can be given as power series in p that is:

and setting p = 1, results in approximate solution of Eq. 7 as:

The series of Eq. 13 is convergent for most of the cases (He, 2000, 2003). However, the convergent rate depends on the nonlinear operator N and the following suggestions have already been made by He (1999):

APPLICATIONS

Feasibility and efficiency of the HPM is demonstrated by following examples:

Example 1: Consider the problem Eq. 1-5 with the initial conditions:

and boundary conditions:

Where:

By Lemma 1, this non-local boundary problem can be transformed into the following form:

Where:

To find the solution of Eq. 19 by HPM, the Homotopy is constructed in the following form

Substituting the Eq. 12 into Eq. 20 and equating the coefficients of like powers of p, we get:

Let us choose w(0, x) = (1-t) cos(2πr), as an initial approximation, then from the Eq. 21 the following terms are calculated successively:

and so on.

Now, taking only three term approximation, we get:

Taking the Laplace transform of Eq. 23, we have:

For the sake of the simplicity, let s = 1/t, then:

replace we obtain in terms of s as follows:

Taking the inverse Laplace transform of Eq. 26, we obtain:

which is the exact solution of Eq. 19.

Variation of w(x, t) for different values of x and t is shown through in Fig. 1.

Example 2: Consider the Eq. 1-5 with the initial conditions:

and boundary conditions:

Again by Lemma 1, this problem is equivalent to the following problem:

Where:

To find the solution of Eq. 27 by HPM, the Homotopy is constructed in the following form:

Substituting the Eq. 12 into the Eq. 29 and equating the coefficients of like powers of p, we get:

and so on.

Let us choose w(x, 0) = t sin (2πx), as an initial approximation, then from the Eq. 30 the following terms are calculated successively:

and so on.

Now, taking only three terms approximation, we get:

Taking the Laplace transform of Eq. 36, we have:

For the sake of the simplicity, let s = 1/t, then:

replace we obtain in terms of s as follows:

Taking the inverse Laplace transform of Eq. 39, we obtain:

which is the exact solution of Eq. 31.

Variation of w(x, t) for different values of x and t is shown through in Fig. 2.

THE INHOMOGENEOUS HYPERBOLIC EQUATION

Consider the inhomogeneous hyperbolic equation:

in the rectangular domain Ω = (0, 1)x(0, T). Initial conditions given by:

Dirichlet boundary condition:

and the non-local boundary condition:

where, φ(x), η(x) ε L₂(0, 1) are known functions and satisfy the compatibility conditions:

Similarly, reducing the Eq. 33-37 to an equivalent problem given below.

Lemma 2: Problem given by Eq. 33-37 is equivalent to the following problem:

Introducing the new unknown function:

Where:

Then, Eq. 39 is transformed into:

Where:

and:

Now, to illustrate the technique the following example is presented.

Example 3: Consider the following problem:

Hence, using Lemma 2 and transformation w (x, t) = W (x, t)+v (x, t) where:

the Eq. 44-47 will be transformed to the following form:

To find the solution of the Eq. 48 by HPM, the Homotopy is constructed in the following form:

Substituting the Eq. 12 into Eq. 49 and equating the coefficients of like powers of p, we get:

and so on.

Let us choose as an initial approximation, then from the Eq. 50 the following terms are calculated successively:

and so on.

Now, taking only three terms approximation, we get:

Taking the Laplace transform of Eq. 60, we have:

For the sake of the simplicity, let then:

replace , we obtain in terms of s as follows:

Taking the inverse Laplace transform of Eq. 63, we obtain:

Which is the exact solution of equation. Variation of w (x, t) for different values of x and t is shown through in Fig. 3.

CONCLUSION

Homotopy perturbation method combined with Pade approximant and Laplace transform is applied hyperbolic equation with integral boundary condition. Exact solution is obtained without discretisation, perturbation and linearization. It is observed that HPM along with Pade and Laplace transform enhances reliability, consistency and fast convergences.

ACKNOWLEDGMENTS

We are thankful to the referee for his valuable suggestions.

REFERENCES