Research Article | Open Access
Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method
The second-kind Chebyshev wavelets collocation method is applied for solving a class of time-fractional diffusion-wave equation. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the second kind. Moreover, convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two dimensions are given. During the process of establishing the expression of the solution, all the initial and boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Based on the collocation technique, the second-kind Chebyshev wavelets are used to reduce the problem to the solution of a system of linear algebraic equations. Several examples are provided to confirm the reliability and effectiveness of the proposed method.
Many phenomena in various fields of the science and engineering can be modeled by fractional differential equations, in which time-fractional diffusion-wave equation is a mathematical model of a wide class of important physical phenomena. It is obtained from the diffusion-wave equation by replacing the second-order time derivative term by a fractional derivative order . In this paper, our study focuses on the following time-fractional diffusion-wave () with the Caputo fractional derivative with the initial condition and the boundary conditions where are given functions with second-order continuous derivatives and is a constant and is a given known function.
It is noted that most fractional diffusion-wave equations do not have closed form solutions. Many researchers have proposed various methods to solve the time-fractional diffusion-wave equations from the perspective of analytical solution and numerical solution. The method of separation of variables in , Sumudu transform method in , and decomposition method in  were used to construct analytical approximate solutions of fractional diffusion-wave equations, respectively. Finite difference schemes in [4–7] were widely used to solve the numerical solutions of the fractional diffusion-wave equations. The authors of  employed radial point interpolation method for solving the fractional diffusion-wave equations. B-spline collocation method was proposed to solve the fractional diffusion-wave equations in . In [10, 11], Sinc-finite difference method and Sinc-Chebyshev method were employed for solving the fractional diffusion-wave equations respectively. Recently methods based on operational matrix of Jacobi and Chebyshev polynomials were proposed to deal with the fractional diffusion-wave equations ([12–14]). In , the authors applied fractional order Legendre functions method depending on the choices of two parameters to solve the fractional diffusion-wave equations. Two-dimensional Bernoulli wavelets with satisfier function in the Ritz-Galerkin method were proposed for the time-fractional diffusion-wave equation in . The author of  proposed a numerical method based on the Legendre wavelets with their operational matrix of fractional integral to solve the time-fractional diffusion-wave equations.
Since fractional derivative is a nonlocal operator, it is natural to consider a global scheme such as the collocation method for its numerical solution. Spectral methods are widely used in seeking numerical solutions of fractional order differential equations, due to their excellent error properties and exponential rates of convergence for smooth problems. Collocation methods, one of the three most common spectral schemes, have been applied successfully to numerical simulations of many problems in science and engineering.
Wavelets, as another basis set and very well-localized functions, are considerably useful for solving differential and integral equations. Particularly, orthogonal wavelets are widely used in approximating numerical solutions of various types of fractional order differential equations in the relevant literatures; see [18–22]. Among them, the second-kind Chebyshev wavelets have gained much attention due to their useful properties ([23–26]) and can handle different types of differential problems. It is observed that most papers using these wavelets methods to approximate numerical solutions of fractional order differential equations are based on the operational matrix of fractional integral or fractional derivatives. It is inevitable to produce approximation error during the process of constructing the operational matrix. Regarding this point, analysis in  shows some disadvantages of using the operational matrix of Legendre and Chebyshev wavelets.
Inspired and motivated by the work mentioned above, our main purpose of this paper is to extend the second-kind Chebyshev wavelets for solving time-fractional diffusion-wave equations (1)–(3) and to show that it is not necessary to establish the operational matrix of fractional integrals and fractional derivatives when applying wavelets to solve various types of fractional partial differential equations. To reduce the approximation error at most during the calculation process, fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of the shifted Chebyshev polynomials of the second kind. By utilizing the collocation method and some properties of the second-kind Chebyshev wavelets, the problem under consideration is reduced to the solution to a system of linear algebraic equations. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically.
The rest of the paper is organized as follows. Section 2 describes some necessary definitions and preliminaries of calculus. Section 3 gives the convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two-dimension. Section 4 is devoted to deriving the fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense by means of shifted Chebyshev polynomials of the second kind. The proposed method is described for solving time-fractional diffusion-wave equations in Section 5. In Section 6, numerical results of some test problems are presented. Finally, a brief conclusion is given in Section 7.
2. Definitions and Preliminaries
In this section, we present some necessary definitions and preliminaries of the fractional calculus theory which will be used later. The widely used definitions of fractional integral and fractional derivative are the Riemann-Liouville definition and the Caputo definition, respectively.
Definition 1. A real function , , is said to be in the space , , if there is a real number with such that , where , and if , .
Definition 2 (see ). The Riemann-Liouville fractional integral operator of order for a function is defined as
Definition 3 (see ). The Caputo fractional derivative operator of order for a function is defined as
Some important properties of the operators and are needed in this paper; we only mention the following properties:(1) for .(2), , .(3), , where the ceiling function denotes the integer smaller than or equal to and . One can see more details about fractional calculus in .
3. The Second-Kind Chebyshev Wavelets and Their Properties
The second-kind Chebyshev wavelets have four arguments: can assume any positive integer, , , is the degree of the second-kind Chebyshev polynomials and is the normalized time. They are defined on the interval as where. Here are the second-kind Chebyshev polynomials of degree which are orthogonal with respect to the weight function on the interval and satisfy the following recursive formula: Note that when dealing with the second-kind Chebyshev wavelets the weight function has to be dilated and translated as
A function defined over may be expanded by the second-kind Chebyshev wavelets as where in which denotes the inner product in . If the infinite series in (10) is truncated, then it can be written as where and are matrices given by
Observe that is an orthonormal set over . A function defined over may be expanded as where If the infinite series in (14) is truncated, then (14) can be written aswhere and are matrices given by Write and , ; then can be written as where is a matrix given by .
The following theorems give the convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion.
Theorem 4 (see ). Let be a second-order derivative square-integrable function defined on with bounded second-order derivative; say for some constant ; then
(i) can be expanded as an infinite sum of the second-kind Chebyshev wavelets and the series converges to uniformly; that is, where ;
Theorem 5. Supposing that is a continuous function defined on , , , and are bounded with some positive constant . Then, for any positive integer ,
(i) the series converges to uniformly in ; that is, where ;
Proof. (i) where . Similar to the proof of Theorem 1 in , we get, for , where . So, for and , Note that, for , Hence, for and , since and .
For and , where we use the variable substitution and the definition of the second-kind Chebyshev polynomials. Therefore, for and , we obtain by and . Similarly, for and , we also have Relations (29), (31), and (32) show that the series converges to uniformly in .
(ii) According to the orthonormality of , we have By relations (29), (31), and (32), it gives The proof is completed.
4. The Fractional Integral of a Single Second-Kind Chebyshev Wavelet
In this section, fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of the shifted second-kind Chebyshev polynomials , which plays an important role in dealing with the time-fractional diffusion-wave equations.
Theorem 6. The fractional integral of a Chebyshev wavelet defined on the interval with compact support is given by where .
Proof. The analytical form of the shifted Chebyshev polynomials  of the second-kind of degree is given byAccording to the relation between and , it gives thatBy interchanging the summation and substituting with , can be written asLet ; then Therefore So, when , let ; then Similarly, when , we have Applying the Riemann-Liouville fractional integral of order with respect to on , we obtain Thus, we have The proof is completed.
For example, in the case of , we obtain where
5. Description of the Proposed Method
Consider the time-fractional diffusion-wave equation with the following form: with initial condition and boundary conditions where are given functions with second-order continuous derivatives in and is a given function in .