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.
Highlights
Fractional calculus is one of the most novel types of calculus having a broad range of applications in many different scientific and engineering disciplines
2 Preliminaries 2.1 Fractional calculus firstly we present the definitions of the Riemann-Liouville and the Caputo of fractional derivative
The numerical solutions which are obtained by using the sinc-Galerkin method (SGM) for this problem are presented in Table and Table
Summary
Fractional calculus is one of the most novel types of calculus having a broad range of applications in many different scientific and engineering disciplines. In this paper we propose a new solution technique for approximate (or alternatively saying numerical) 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.
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