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

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.Themethod 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.


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][7][8], Haar wavelet method [9,10], differential transform method [11][12][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 Dirichlettype 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 2 Mathematical Problems in Engineering 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  0   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.

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  : [, ] × [, ] → R be a function;  is a positive real number, and  is the integer.,  satisfy the inequality  − 1 ≤  <  and Γ the Euler gamma function.Then, (i) the left and right Riemann-Liouville fractional derivatives of order  with respect to  of (, ) function are given as (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)

The Sinc Basis Functions
Definition 3 (see [30]).The function which defined all  ∈ C by is called the sinc function.
Definition 4 (see [30]).Let  be a function defined on R and let ℎ > 0. 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  =  −1 () =   /(1 +   ) is an inverse mapping of  = ().We may define the range of  −1 on the real line as evenly spaced nodes {ℎ} ∞ =−∞ on the real line.The image which corresponds to these nodes is denoted by

Sinc Function Interpolation and Quadrature
Definition 5 (see [21]).Let   be a simply connected domain in the complex plane  and let   denote the boundary of   .Let ,  be points on   and let  be a conformal map and   = (ℎ),  = ±1, ±2, . ... Definition 6 (see [21]).Let (  ) be the class of functions  that are analytic in   and satisfy in which and those on the boundary of   satisfy Theorem 7 (see [21]).Let Γ be (0, 1),  ∈ (  ); then, for ℎ > 0 sufficiently small, where For the sinc-Galerkin method, the infinite quadrature rule must be truncated to a finite sum.The following theorem indicates the conditions under which an exponential convergence results.
Theorem 8 (see [21]).If there exist positive constants , , and  such that then the error bound for the quadrature rule The infinite sum in (21) is truncated with the use of (23) to arrive at inequality (24).Making the selections where ⟦ ⋅ ⟧ is an integer part of the statement and  is the integer value which specifies the grid size, then We used these theorems to approximate the integrals that arise in the formulation of the discrete systems corresponding to a second-order boundary value problem.

The Sinc-Galerkin Method
Consider fractional boundary value problem with boundary conditions where  0   is Caputo fractional derivative operator.An approximate solution for (, ) is represented by the formula where   =   +   + 1 and   =   +   + 1.The basis functions {  (, )} are given by where The unknown coefficients   in ( 29) are determined by orthogonalizing the residual with respect to the functions {  (, )}, −  ≤  ≤   , −  ≤  ≤   .This yields the discrete Galerkin system where inner product is defined by where () is weight function and it is convenient to take for the problem ( 27)- (28).
Lemma 9 (see [23]).Let  be the conformal one-to-one mapping of the simply connected domain   onto   , given by (12).Then The following theorems which can easily be proven by using Lemma 9 and definitions are used to solve (27).
Theorem 11.For 0 <  < 1, the following relations hold: where () = ()  ()().By the definition of the Riemann-Liouville fractional derivative, we have We will use the sinc quadrature rule given with ( 26) to compute it because the integral given in last equality is divergent on the interval [, 1].For this purpose, a conformal map and its inverse image that denotes the sinc grid points are given by respectively.Then, according to equality (26) we can write where ℎ  = / √ .As a result, it can be written in the following way: Now, applying the sinc quadrature rule with respect to  and  in last integral, we obtain Consequently, using   ()| =  =  (0) ln =  ln we obtain This completes the proof.
Replacing each term of ( 32) with the approximation defined in Theorems 10 and 11, replacing (  ,   ) by   , and dividing by ℎ  ℎ  we obtain the following theorem.27)-( 28) is (29), then the discrete sinc-Galerkin system for the determination of the unknown coefficients {  , −  ≤  ≤   , −  ≤  ≤   } is given by

Examples
In this section, the present method will be tested on three different problems.

Example 1. Consider fractional boundary value problem
+  (, ) ,  (0, ) =  (1, ) = 0,  (, 0) =  (, 1) = 0 (55) which has the following exact solution: for The numerical solutions which are obtained by using the sinc-Galerkin method (SGM) for this problem are presented in Tables 1 and 2 for different values.Also, the graphs of exact and approximate solutions for different values are presented in Figures 2 and 3.
The numerical solutions which are obtained by using the sinc-Galerkin method (SGM) for this problem are presented in Tables 3 and 4. In addition, in Figures 4 and 5, the graphs of exact and approximate solutions for different values are presented.Example 3 (see [32]).Consider the fractional convection-diffusion equation    The numerical solutions which are obtained by using the sinc-Galerkin method (SGM) for this problem are presented in Tables 5 and 6.In addition, in Figures 6 and 7, the graphs of exact and approximate solutions for different values are presented.

Conclusion
In this study, we use the sinc-Galerkin method to obtain approximate solutions of boundary value problems for spacefractional partial differential equations with variable coefficients.In order to illustrate the efficiency and accuracy of the present method, the method is applied to three examples in the literature and the obtained results are compared with exact solutions.As a result, it is shown that sinc-Galerkin   method is very effective and accurate for obtaining approximate solutions of space-fractional differential equations with variable coefficients.In the future, we plan to extend the present numerical solution technique to nonlinear spacefractional partial differential equations.

30 Figure 2 :Figure 3 :
Figure 2: Graphs of approximate solutions for different values.

30 Figure 4 :Figure 5 :
Figure 4: Graphs of approximate solutions for different values.

30 Figure 6 :
Figure 6: Graphs of approximate solutions for different values.