Application of homotopy perturbation method for solving eighth-order boundary value problems

Abstract

Homotopy perturbation method is applied to the numerical solution for solving eighth-order boundary value problems. Comparison of the result obtained by the present method with that obtained by modified decomposition method [M. Mesrovic, The modified decomposition method for eighth-order boundary value problems, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.015] reveals that the present method is very effective and convenient.

Introduction

A new perturbation method called homotopy perturbation method (HPM) was proposed by He in 1997 and systematical description in 2000 which is, in fact, a coupling of the traditional perturbation method and homotopy in topology [1]. This new method was further developed and improved by He and applied to nonlinear oscillators with discontinuities [2], nonlinear wave equations [3], asymptotology [4], boundary value problem [5], Limit cycle and bifurcation of nonlinear problems [6] and many other subjects. Thus He’s method is a universal one which can solve various kinds of nonlinear equations. For example, it was applied to the quadratic Ricatti differential equation by Abbasbandy [7]; to the axisymmetric flow over a stretching sheet by Ariel et al. [8]; to the nonlinear systems of reaction–diffusion equations by Ganji and Sadighi [9]; to the Helmholtz equation and fifth-order KdV equation by Rafei and Ganji [10]; for the thin film flow of a fourth grade fluid down a vertical cylinder by Siddiqui et al. [11]; to the nonlinear Voltra–Fredholm integral equations by Ghasemi et al. [12]. Recently various powerful mathematical methods such as variational iteration method [13], [14], [15], [16], [17], [18], [19], [20], [21], Exp-function method [22], [23], F-expansion method [24], Adomian decomposition method [25] and others [26], [27] have been proposed to obtain exact and approximate analytic solutions for linear and nonlinear problems.

In this paper, we consider the general eighth-order boundary value problems of the type:

$$y^{\text{(viii)}}(x)=f(x\text{,}y\text{,}{y}^{\prime }\text{,}{y}^{″}\text{,}{y}^{‴}\text{,}{y}^{\text{(iv)}}\text{,}{y}^{\text{(v)}}\text{,}{y}^{\text{(vi)}}\text{,}{y}^{\text{(vii)}})\text{,}a<x<b\text{,}$$

with boundary conditions

$$\begin{array}{cc}\hfill & y(a)={\alpha }_0\text{,}{y}^{\prime }(a)={\alpha }_1\text{,}{y}^{″}(a)={\alpha }_2\text{,}{y}^{‴}(a)={\alpha }_3\text{,}\hfill \\ \hfill & y^{\text{(iv)}}(a)={\alpha }_4\text{,}{y}^{\text{(v)}}(a)={\alpha }_5\text{,}y(b)={\beta }_0\text{,}{y}^{\prime }(b)={\beta }_1\text{,}\hfill \end{array}$$

where f is continuous function on [a,b] and the parameters ${\alpha }_i\text{,}i=0\text{,}1\text{,}\dots \text{,}6$ and ${\beta }_i\text{,}i=0\text{,}1$ are real constants.

Such type of boundary value problems arise in the mathematical modeling of the viscoelastic flows and other branches of mathematical, physical and engineering sciences, see [28], [29], [30], [31] and references therein. Several numerical methods including spectral Galerkin and collocation [29], [30], sixth B-spline method [28], decomposition method [32], spline collocation approximation [33], Chow–Yorke algorithm [34] and others [35], [36] have been developed for solving the problem of type (1). This paper, applies the homotopy perturbation method [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] to the discussed problem.