Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems

M A Hajji,Q M Al-Mdallal

doi:10.1155/2013/875930

Abstract

This paper presents a modified Galerkin method based on sinc basis functions to numerically solve nonlinear boundary value problems. The modifications allow for the accurate approximation of the solution with accurate derivatives at the endpoints. The algorithm is applied to well-known problems: Bratu and Thomas-Fermi problems. Numerical results demonstrate the clear advantage of the suggested modifications in obtaining accurate numerical solutions as well as accurate derivatives at the endpoints.

Highlights

In this paper, we consider the following class of nonlinear boundary value problems (NBVPs) of the following form: yy′′(xx) + λλλλ 󶀡󶀡xxx xx (xx)󶀱󶀱 = gg (xx), xx x, (1)subject to yy = αα0, yy = αα1, (2)where bbb b. is class of NBVPs covers many interesting and important problems
To the authors best knowledge, neither Bratu nor omas-Fermi problem has been investigated using a Galerkin method based on sinc basis functions
E Sinc-Galerkin method was developed back in 1993, see [25]. It has been used by many authors in the numerical treatment of several types of differential equations, see, for example, [26,27,28]. e success of the Sinc-Galerkin method led us to consider it for the numerical treatment of two important well-known problems, namely, Bratu and omasFermi problems

Summary

Introduction

To the authors best knowledge, neither Bratu nor omas-Fermi problem has been investigated using a Galerkin method based on sinc basis functions. E Sinc-Galerkin method was developed back in 1993, see [25] It has been used by many authors in the numerical treatment of several types of differential equations, see, for example, [26,27,28]. E success of the Sinc-Galerkin method led us to consider it for the numerical treatment of two important well-known problems, namely, Bratu and omasFermi problems. Modi cations for the treatment of nonhomogeneous boundary condition to better approximate the derivative of the solution at the endpoints are presented at the end of Section 3.

Sinc Basis Functions Properties and Quadrature Interpolations

The Sinc-Galerkin Approach

Numerical Examples

Conclusion