A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains

Behzad Ghanbari,Jafar Biazar

doi:10.4236/am.2011.28136

Abstract

Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the following type of nonlinear boundary value problems in finite domain. In framework of HAM a convenient way to adjust and control the convergence region and rate of convergence of the obtained series solutions, by defining the so-called control parameter h , is provided. This paper aims to propose an efficient way of finding the proper values of h.Such values of parameter can be determined at the any order of approximations of HAM series solutions by solving of a nonlinear polynomial equation. Some examples of nonlinear initial value problems in finite domain are used to illustrate the validity of the proposed approach. Numerical results confirm that obtained series solutions agree very well with the exact solutions.

Highlights

Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the following type of nonlinear boundary value problems in finite domain
This paper aims to propose an efficient way of finding the proper values of. Such values of parameter can be determined at the any order of approximations of HAM series solutions by solving of a nonlinear polynomial equation
Different from all perturbation and traditional non-perturbation methods, the HAM provides us a simple way to ensure the convergence of solution series, and the HAM is valid even for strongly nonlinear problems

Summary

Introduction

We consider the following type of nonlinear boundary value problems in a finite domain (as considered in [1]): u(n) x f x,u, u , ,u(n 1) , a x b (1). [10], homotopy perturbation method (HPM) [11], analysis method (HAM) [12], have been studied for obtaining approximate solutions to boundary value problems. Different from all perturbation and traditional non-perturbation methods, the HAM provides us a simple way to ensure the convergence of solution series, and the HAM is valid even for strongly nonlinear problems. Different from all perturbation and previous non-perturbation methods, the HAM provides us with great freedom to choose proper base functions to approximate a nonlinear problem. These advantages make the method to be a powerful and flexible tool in mathematics and engineering, which can be readily distinguished from existing numerically and analytically methods.

The Implement of HAM to BVPs

Proposed Approach

Numerical Examples and the nonlinear operator N is given by

Conclusions