Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems

Yanqin Liu

doi:10.1155/2012/727031

Abstract

A variational homotopy perturbation method (VHPM) which is based on variational iteration method and homotopy perturbation method is applied to solve the approximate solution of the fractional initial boundary value problems. The nonlinear terms can be easily handled by the use of He′s polynomials. It is observed that the variational iteration method is very efficient and easier to implements; illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

Highlights

It has turned out that many phenomena in engineering, physics, chemistry, and other sciences 1–3 can be described very successfully by models using mathematical tools for fractional calculus
Comparisons of like powers of p give solutions of various orders. This method does not resort to linearization or assumptions of weak nonlinearity, the solutions generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems
A variational homotopy perturbation method which is based on homotopy perturbation method and variational iteration method is used to solve fractional partial equations

Summary

Introduction

It has turned out that many phenomena in engineering, physics, chemistry, and other sciences 1–3 can be described very successfully by models using mathematical tools for fractional calculus. The importance of obtaining the exact and approximate solutions of fractional nonlinear equations in physics and mathematics is still a significant problem that needs new methods to discover exact and approximate solutions. These nonlinear fractional differential equations are difficult to get their exact solutions 4–7. Some semianalytical techniques have largely been used to solve these equations, such as, Adomian decomposition method 8, 9 , variational iteration method 10–12 , differential transform method 13, 14 , Laplace decomposition method 15, 16 , and homotopy perturbation method 17–23. This paper considers the effectiveness of the variational homotopy perturbation method in solving fractional partial equations

Description of the VHPM

Approximate Solutions of Fractional Equations

Conclusion