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
2.2.2. 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 onto such that and . If the inverse map of is denoted by , define and , .
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 be , ; then, for 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 (21) is 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.
3. The Sinc-Galerkin Method
Consider fractional boundary value problem with boundary conditions where is Caputo fractional derivative operator. An approximate solution for is represented by the formula where and . 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 10 (see [31]). The following relations hold: where
Proof. See [31].
Theorem 11. For , the following relations hold: where , , and .
Proof. The inner product with sinc basis elements of is given by
Using Definition 2, we can write
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 . 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 . 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 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 we obtain the following theorem.
Theorem 12. If the assumed approximate solution of the boundary-value problem (27)-(28) is (29), then the discrete sinc-Galerkin system for the determination of the unknown coefficients is given by
We introduce the following notations in order the write the system above in a matrix-vector form. Let , be the matrices , with th entry as given by Lemma 9. Further, is an diagonal matrix whose diagonal entries are
The matrices , , and are similarly defined though of size . Introducing this notation in (47) leads to the matrix form
where
Firstly multiplying this term with and secondly multiplying it with yield the equivalent system
where
for
Furthermore,
, , , and have dimensions , , , and , respectively. At last, the matrices and have th entries given by and , respectively.
To obtain the approximate solution equation (29), we need to solve the system for which requires solving (51) for . Solution of (51) for is shown in [26].
4. Examples
In this section, the present method will be tested on three different problems.
Example 1. Consider fractional boundary value problem 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.
Example 2. Consider fractional boundary value problem 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 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
In particular, if , , , , .
The problem 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 5 and 6. In addition, in Figures 6 and 7, the graphs of exact and approximate solutions for different values are presented.
5. Conclusion
In this study, we use the sinc-Galerkin method to obtain approximate solutions of boundary value problems for space-fractional 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 space-fractional partial differential equations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.