Variational Homotopy Perturbation Method for Solving Higher Dimensional Initial Boundary Value Problems

Muhammad Aslam Noor,Syed Tauseef Mohyud-Din

doi:10.1155/2008/696734

Abstract

We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method. This method is called the variational homotopy perturbation method (VHPM). We use this method for solving higher dimensional initial boundary value problems with variable coefficients. The developed algorithm is quite efficient and is practically well suited for use in these problems. The proposed scheme finds the solution without any discritization, transformation, or restrictive assumptions and avoids the round-off errors. Several examples are given to check the reliability and efficiency of the proposed technique.

Highlights

The numerical and analytical solutions of higher dimensional initial boundary value problems of variable coefficients, linear and nonlinear, are of considerable significance for applied sciences
We develop the variational homotopy perturbation method Variational homotopy perturbation method (VHPM) for solving nonlinear problems
The proposed method is successfully implemented by using the initial conditions only

Summary

Introduction

The numerical and analytical solutions of higher dimensional initial boundary value problems of variable coefficients, linear and nonlinear, are of considerable significance for applied sciences. For implementation of the Adomian decomposition method, one has to find the so-called the Adomian polynomial, which is itself a difficult problem To overcome these difficulties and drawbacks, He 8–18 developed variational iteration method for solving linear and nonlinear problems, which arise in various branches of pure and applied sciences. The fact that the proposed technique solves nonlinear problems without using the so-called Adomian’s polynomials is a clear advantage of this algorithm over the decomposition method. In this algorithm, the correct functional is developed 8, 15–19, 21–25 and the Lagrange multipliers are calculated optimally via variational theory.

Variational iteration method

Homotopy perturbation method

Numerical applications

Conclusions