Direct Solution ofnth-Order IVPs by Homotopy Analysis Method

A. Sami Bataineh,M. S. M. Noorani,I. Hashim

doi:10.1155/2009/842094

Abstract

Direct solution of a class ofnth-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78).

Highlights

Higher-order initial value problems IVPs arise in mathematical models for problems in physics and engineering
It seems more natural to provide direct numerical methods for solving the nthorder IVPs. It is the purpose of the present paper to present an alternative approach for the direct solution of nth-order IVPs based on the homotopy analysis method HAM
HAM yields a very rapid convergence of the solution series and in most cases, usually only a few Differential Equations and Nonlinear Mechanics iterations leading to very accurate solutions

Summary

Introduction

Higher-order initial value problems IVPs arise in mathematical models for problems in physics and engineering. The analytic homotopy analysis method HAM , initially proposed by Professor Liao in his Ph.D. thesis 1 , is a powerful method for solving both linear and nonlinear problems. The interested reader can refer to the much-cited book 2 for a systematic and clear exposition on this method In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering 3–17. This method has been successfully employed to solve many types of nonlinear problems in science and engineering 3–17 All of these successful applications verified the validity, effectiveness and flexibility of the HAM. Chowdhury and Hashim demonstrated the applicability of the analytic homotopy-perturbation method for solving nth-order IVPs. The aim of this paper is to apply HAM and NHAM for the first time to obtain approximate solutions of nth-order IVPs directly. Numerical comparison will be made against the seven- and eight-order Runge-Kutta method RK78

Basic Ideas of HAM

Example 1

Example 3

Conclusions