Abstract
This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed.
1. Introduction
Fractional models have been increasingly shown by many scientists to describe adequately the problems with memory and nonlocal properties in fluid mechanics, viscoelasticity, physics, biology, chemistry, finance, and other areas of applications [1–6]. In particular, the fractional diffusion-wave equation has been used to model many important physical phenomena ranging from amorphous, colloid, glassy, and porous materials through fractals, percolation clusters, and random and disordered media to comb structures, dielectrics and semiconductors, polymers, and biological systems [7–10]. It is a generalization of the classical diffusion-wave equation by replacing the integer-order time derivative with a fractional derivative of order (). This equation can be derived from the anomalous superdiffusion in continuous time random walk which is generally non-Markovian processes [11].
Although the considerable work on the numerical solution of fractional diffusion equations has been done [12–15], there are very limited numerical methods for solving the fractional diffusion-wave equations [16–18]. However, all the above mentioned papers dealt with the fractional diffusion-wave equations by finite difference methods. It is well known that any algorithm based on the finite difference discretization of a fractional derivative has to take into account its memory or nonlocal structure; thus this means a high storage requirement [19].
In the present paper, we consider the following differential equation with the Caputo fractional derivative and a variable coefficient: with the initial conditions, and the boundary conditions, where and are space and time variables, respectively, is a continuous function, and denotes the field variable. For , the fractional equation (1) is known as the fractional diffusion-wave equation which fills the gaps between the diffusion equation and wave equation [16, 20].
We develop a sinc-Chebyshev collocation method to solve numerically problem (1) with (2) and (3). Since a fractional derivative is a nonlocal operator, it is natural to consider a global scheme such as the collocation method for its numerical solution [19, 21]. The required approximate solution is expanded as a series with the elements of shifted Chebyshev polynomials in time and sinc functions in space with unknown coefficients. By utilizing the collocation technique and some properties of the shifted Chebyshev polynomials and sinc functions, the problem is reduced to the solution to a system of linear algebraic equations. And a matrix representation of the system is obtained to calculate the solution. The presented method is effective and convenient.
The remainder of this paper is organized as follows: in the next section, we introduce some necessary definitions and relevant results for developing this method. Section 3 is devoted to constructing and analyzing the numerical algorithm. As a result, a system of linear algebraic equations is formed and the solution of the considered problem is obtained. In Section 4, the numerical example is given to demonstrate the effectiveness and convergence of the proposed method. A brief conclusion is given in the final section.
2. Notations and Some Preliminary Results
In this section, we introduce some basic definitions and derive several preliminary results for developing the presented method.
2.1. The Caputo Fractional Derivative
Definition 1 (see [22]). Let . The operator defined on by for is called the Riemann-Liouville fractional integral operator of order . For, , we set , that is, the identity operator.
Definition 2 (see [22]). Let and . The Caputo fractional differential operator for is defined as
2.2. The Composite Translated Sinc Functions
The sinc functions and their properties are discussed in [23, 24]. For any , the translated sinc functions with equidistant space nodes are given as where the sinc functions are defined on the whole real line by If is defined on , then for any the series is called the Whittaker cardinal expansion of whenever this series converges. can be approximated by truncating (8).
To construct our needed approximations on the interval , we choose which maps the finite interval onto . The basic functions on are taken to be the composite translated sinc functions: Thus we may define the inverse image of the equidistant space node as The class of functions such that the known exponential convergence rate exists for the sinc interpolation is denoted by and defined in the following text.
Definition 3 (see [21]). Let be the class of functions which are analytic in and satisfy where , and on the boundary of (denoted ).
Theorem 4 (see [21, 23]). If , then, for all , Further, one assumes that there are positive constants and so that . And if one selects , then, for all .
The above expressions show that the sinc interpolation on converges exponentially. We also require the following derivatives of the composite translated sinc functions evaluated at the nodes. Consider
2.3. The Shifted Chebyshev Polynomials
The Chebyshev polynomials are a well-known family of orthogonal polynomials defined on the interval and can be determined with the aid of the recurrence formulae [25, 26]:
In order to use these polynomials on the interval , it is necessary to define the so-called shifted Chebyshev polynomials by the variable substitution: . Let the shifted Chebyshev polynomials be denoted by . The analytic form of the shifted Chebyshev polynomials is given by Specially, and .
Caputo’s fractional derivative of order for the shifted Chebyshev polynomials is given by where
3. The Derivation of the Sinc-Chebyshev Collocation Method
In order to solve problem (1) with (2) and (3), first of all, we approximate by the composite translated sinc functions and shifted Chebyshev polynomials as It is noted that the approximate solution satisfies the boundary conditions in (3) since , tend to zeros when tends to and . For discretizing (1) with (2), the lemma is given as follows.
Lemma 5. Let and be spatial collocation points given in (11). Then the following relations hold: where and .
Proof. By (16), (20), and (22), it follows that Taking into account (17), we obtain Using (16) and (19), one has The proof is completed.
We are now ready to solve problem (1) with (2) and (3). A collocation scheme is constructed by substituting (22) for into (1) and evaluating the result at the points in (11) and . For suitable temporal collocation points, we use the roots of the shifted Chebyshev polynomials . Therefore, using Lemma 5, we have Also by applying (22) to the initial conditions (2) and collocating in points , we obtain
To obtain a matrix representation of the above equations, we let where So we get a system of linear equations with unknown parameters . And this system can be expressed in a matrix form Equation (31) can be solved easily for the unknown coefficients . Consequently given in (22) can be calculated.
4. Numerical Examples
To validate the effectiveness of the proposed method for problem (1) with (2) and (3), we consider the example given in [16]. The exact solution of the above problem is [27] where is the one-parameter Mittag-Leffler function.
To solve the above problem with by using the method described in Section 3, we choose and , and this leads to . We will report the accuracy and efficiency of the method based on the -errors and -errors. Figure 1 gives the 3D diagrams of the numerical and exact solutions on the whole computational domain with . A good agreement of the numerical solution with the exact one is achieved. In Table 1, we list the numerical and exact solutions at some points for different numbers of collocation points with . Furthermore, Figure 2 shows the absolute error function obtained by the presented method with and . In Figure 3, we plot the curves of the absolute errors at for different numbers of collocation points. From Figures 2 and 3, we see that the proposed method can provide accurate results only using a small number of collocation points.
To explore the dependence of errors on the parameters , we represent the -error and -error in semi-log scale. Firstly, the computational investigation is concerned with the spatial error. To this end, we fix the polynomial degree , a value large enough such that the error stemming from the temporal approximation is negligible. In Figure 4, we plot the error as functions of , where a logarithmic scale is used for the spatial-error-axis. As expected, the error shows an exponential decay, since in this semi-log representation one observes that the error variations are approximately linear versus [19].
Now we check the temporal error, which is more interesting because of the fractional derivative in time. For a similar reason mentioned above, we fix a large enough value to avoid contamination of the spatial error. We present the error as a function of the shifted Chebyshev polynomial degree in Figure 5, where a logarithmic scale is now used for the temporal-error-axis. From Figure 5, it is clearly observed that the temporal error depends on the discretization parameters .
5. Conclusion
In this paper, we develop and analyze the efficient numerical methods for the fractional diffusion-wave equation. Based on the collocation technique, the sinc functions and shifted Chebyshev polynomials are used to reduce the problem to the solution of a system of linear algebraic equations. And a matrix representation of the above equations is obtained. In the numerical example, the solution obtained by this method is in excellent agreement with the exact one. The effectiveness and convergence of the presented method are confirmed through the numerical experimentation. One issue of future work is to develop the theory analysis of the method for the proposed fractional differential equation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Grant nos. 11271311 and 11371016), the Chinese Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (Grant no. IRT1179), and Hunan Province Innovation Foundation for Postgraduate (Grant no. CX2013B252).