Mathematical Problems in Engineering

Volume 2015, Article ID 217348, 10 pages

http://dx.doi.org/10.1155/2015/217348

## Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems

^{1}Department of Management Information Systems, Bartin University, 74100 Bartin, Turkey^{2}Department of Mathematical Engineering, Yildiz Technical University, 34220 Istanbul, Turkey

Received 10 June 2014; Accepted 13 July 2014

Academic Editor: Abdon Atangana

Copyright © 2015 Sertan Alkan and Aydin Secer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We employ the sinc-Galerkin method to obtain approximate solutions of space-fractional order partial differential equations (FPDEs) with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions are compared with the exact solutions of the problems. These comparisons demonstrate that the sinc-Galerkin method is a very efficient tool in solving space-fractional partial differential equations.

#### 1. Introduction

Fractional calculus, which might be considered as an extension of classical calculus, is as old as the classical calculus and fractional differential equations have been often used to describe many scientific phenomena in biomedical engineering, image processing, earthquake engineering, signal processing, physics, statistics, electrochemistry, and control theory.

Because finding the exact or analytical solutions of fractional order differential equations is not an easy task, several different numerical solution techniques have been developed for the approximate solutions of these types of equations. Some of the well-known numerical techniques might be listed as generalized differential transform method [1, 2], finite difference method [3], Adomian decomposition method [4, 5], homotopy perturbation method [6–8], Haar wavelet method [9, 10], differential transform method [11–13], and Adams-Bashforth-Moulton scheme [14]. A detailed and informative study on fractional calculus can be found in [15]. Furthermore a relatively new analytical method was presented in [16] to solve time “The Time-Fractional Coupled-Korteweg-de-Vries Equations” via homotopy decomposition method by the same authors. The sinc methods were introduced in [17] and expanded in [18] by Frank Stenger. Sinc functions were firstly analyzed in [19, 20]. In [21], the sinc-Galerkin method is used to approximate solutions of nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. In [22], the sinc-Galerkin method is applied to nonlinear fourth-order differential equations with nonhomogeneous and homogeneous boundary conditions. In the paper at [23], the numerical solutions of Troesch’s problem are obtained by the sinc-Galerkin method and the results are compared with methods of Laplace, homotopy perturbation, splines, and perturbation. Reference [24] which contains short abstract version of current paper has been presented in an International Conference and Workshop on Mathematical Analysis 2014, Malaysia. In [25], the authors present a comparison between sinc-Galerkin method and sinc-collocation method to obtain approximate solutions of linear and nonlinear boundary value problems. Similarly, the wavelet-Galerkin method and the sinc-Galerkin method for solving nonhomogeneous heat equations are compared in [26]. The paper [27] offers an application of the sinc-Galerkin method for solving second-order singular Dirichlet-type boundary value problems. In [28], the sinc-Galerkin method is used to approximate solutions of fractional order ordinary differential equations in Caputo sense.

In this paper we propose a new solution technique for approximate solution of space-fractional order partial differential equations (FPDEs) with variable coefficients and boundary conditions by using the sinc-Galerkin method that has almost not been employed for the space-fractional order partial differential equations in the form with boundary conditions where is Caputo fractional derivative operator.

The paper is organized as follows. Section 2 presents basic theorems of fractional calculus and sinc-Galerkin method. In Section 3, we use the sinc-Galerkin method to obtain an approximate solution of a general space-fractional partial differential equation. In Section 4, we present three examples in order to illustrate the effectiveness and accuracy of the present method. The obtained results are compared with the exact results.

#### 2. Preliminaries

##### 2.1. Fractional Calculus

In this section, we present the definitions of the fractional Riemann-Liouville derivative and the Caputo of fractional derivatives. By using these definitions, we give the definition of the integration by parts of fractional order.

*Definition 1 (see [29]). *Let be a function; is a positive real number, and is the integer. , satisfy the inequality and the Euler gamma function. Then,(i)the left and right Riemann-Liouville fractional derivatives of order with respect to of function are given as
respectively;(ii)the left and right Caputo fractional derivatives of order with respect to of function are given as
respectively.

Now, we can write the definition of integration by parts of fractional order by using the relations given in (3)–(6).

*Definition 2. *If and is a function such that , one can write

##### 2.2. Properties of Sinc Basis Functions and Quadrature Interpolations

In this section, we recall notations and definitions of the sinc function state some known results and derive some useful formulas to be used in the next sections of the present paper.

###### 2.2.1. The Sinc Basis Functions

*Definition 3 (see [30]). *The function which defined all by
is called the* sinc function*.

*Definition 4 (see [30]). *Let be a function defined on and let . Define the series
where from (8) we have
If the series in (9) converges, it is called the* Whittaker cardinal function* of . They are based on the infinite strip in the complex plane
Generally, approximations can be constructed for infinite, semi-infinite, and finite intervals. Define the function
which is a conformal mapping from , the eye-shaped domain in the -plane, onto the infinite strip , where
This is shown in Figure 1. For the sinc-Galerkin method, the bases functions are derived from the composite translated sinc functions
where . The function is an inverse mapping of . We may define the range of on the real line as
evenly spaced nodes on the real line. The image which corresponds to these nodes is denoted by