Open Access Research

Sinc-Galerkin method for approximate solutions of fractional order boundary value problems

Aydin Secer1*, Sertan Alkan2, Mehmet Ali Akinlar1 and Mustafa Bayram1

Author Affiliations

1 Department of Mathematical Engineering, Yildiz Technical University, Istanbul, 34220, Turkey

2 Department of Management Information Systems, Bartin University, Bartin, 74100, Turkey

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Boundary Value Problems 2013, 2013:281  doi:10.1186/1687-2770-2013-281

Published: 30 December 2013

Abstract

In this paper we present an approximate solution of a fractional order two-point boundary value problem (FBVP). We use the sinc-Galerkin method that has almost not been employed for the fractional order differential equations. We expand the solution function in a finite series in terms of composite translated sinc functions and some unknown coefficients. These coefficients are determined by writing the original FBVP as a bilinear form with respect to some base functions. The bilinear forms are expressed by some appropriate integrals. These integrals are approximately solved by sinc quadrature rule where a conformal map and its inverse are evaluated at sinc grid points. Obtained results are presented as two new theorems. In order to illustrate the applicability and accuracy of the present method, the method is applied to some specific examples, and simulations of the approximate solutions are provided. The results are compared with the ones obtained by the Cubic splines. Because there are only a few studies regarding the application of sinc-type methods to fractional order differential equations, this study is going to be a totally new contribution and highly useful for the researchers in fractional calculus area of scientific research.

Keywords:
fractional order two point boundary value problem; sinc-Galerkin method; fractional derivatives; quadrature rule; Riemann-Liouville derivative; Caputo derivative; Mathematica