Open Access Research

An efficient computer application of the sinc-Galerkin approximation for nonlinear boundary value problems

Aydin Secer1*, Muhammet Kurulay2, Mustafa Bayram1 and Mehmet Ali Akinlar3

Author affiliations

1 Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, İstanbul, 34210, Turkey

2 Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Davutpasa, İstanbul, 34210, Turkey

3 Department of Mathematics, Bilecik University, Bilecik, 11210, Turkey

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Citation and License

Boundary Value Problems 2012, 2012:117  doi:10.1186/1687-2770-2012-117

Published: 24 October 2012

Abstract

A powerful technique based on the sinc-Galerkin method is presented for obtaining numerical solutions of second-order nonlinear Dirichlet-type boundary value problems (BVPs). The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. Without any numerical integration, the differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products; therefore, the evaluation is based on solving a matrix system. The solution is obtained by constructing the nonlinear (or linear) matrix system using Maple and the accuracy is compared with the Newton method. The main aspect of the technique presented here is that the obtained solution is valid for various boundary conditions in both linear and nonlinear equations and it is not affected by any singularities that can occur in variable coefficients or a nonlinear part of the equation. This is a powerful side of the method when being compared to other models.

Keywords:
Maple; sinc-Galerkin approximation; sinc basis function; nonlinear matrix system; Newton method