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-kindChebyshev wavelets expansion of twodimensions 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 problemunder 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.

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 [1], Sumudu transform method in [2], and decomposition method in [3] were used to construct analytical approximate solutions of fractional diffusion-wave equations, respectively.Finite difference schemes in [4][5][6][7] were widely used to solve the numerical solutions of the fractional diffusion-wave equations.The authors of [8] 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 [9].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][13][14]).In [15], 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 [16].The author of [17] proposed a numerical method based on the Legendre wavelets with their operational matrix of fractional integral to solve the time-fractional diffusionwave 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][19][20][21][22].Among them, the second-kind Chebyshev wavelets have gained much attention due to their useful properties ( [23][24][25][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 [27] 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 diffusionwave 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.

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 (),  > 0, is said to be in the space   ,  ∈ R, if there is a real number  with  >  such that () =    0 (), where  0 () ∈ [0, ∞), and Definition 2 (see [28]).The Riemann-Liouville fractional integral operator   of order  ( ≥ 0) for a function () ∈   ( ≥ −1) is defined as Definition 3 (see [28]).The Caputo fractional derivative operator   of order  ( ≥ 0) for a function () ∈   1 is defined as Some important properties of the operators   and   are needed in this paper; we only mention the following properties: (1) (2)     () = (),     () =  − (),  > .
The following theorems give the convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion.

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 timefractional diffusion-wave equations.Proof.The analytical form of the shifted Chebyshev polynomials [30] of the second-kind  *  of degree  is given by

Numerical Examples
In this section, we give some numerical examples to demonstrate the efficiency and reliability of the proposed method.In all the examples, the package of Matlab 2016a has been used.with the initial conditions  (, 0) = 0, and the boundary conditions where The problem is solved by the proposed method for  = 2,  = 3 and the CPU time is 3.818 seconds.Figure 1 shows the approximate solution and the absolute error of this problem for  = 1.1.Table 1 gives the absolute errors for different values of  at different points.To make a comparison, in Table 2, we list the absolute error obtained by the Legendre wavelets method in [17] based on fractional operational matrix of integral with  = 3,  = 3.From Figure 1 and Table 1, it can be seen that the presented method is very efficient and accurate in solving this problem.
with the initial conditions and the boundary conditions where (, ) = ((6/Γ(4 − )) 3− + 3 2 −  3 )  .The exact solution of this problem is (, ) =  3   .The problem is solved by the proposed method for  = 2,  = 6 and the CPU time is 28.79 seconds.Figure 2 shows the approximate solution and the absolute error of this problem for  = 1.3.Table 3 gives the absolute error for different values of  at different points with  = 2,  = 6.In order to compare our results with obtained results in [17], we list the absolute error obtained by the Legendre wavelets method in [17] with  = 2,  = 5 in Table 4. Table 5 lists the maximum absolute errors obtained by the proposed method for different choices of  and  at the points (  ,   ), where   = /40,   = /40, ,  = 0, 1, 2, . . ., 40.It is obvious that the results provided by the proposed method are more accurate than those given in [17].with the initial conditions  (, 0) = 0, and the boundary conditions where (, ) = (Γ(4)/Γ(4 − )) 3− sin 2  + 3 2 sin 2  − 2 3 cos(2).The exact solution of this problem is (, ) =  3 sin 2 .The problem is solved by the proposed method for  = 2,  = 6 and the CPU time is 28.42 seconds.Figure 3 shows the approximate solution and the absolute error of this problem in the case of  = 1.7,  = 2, and  = 6.Table 6 gives the absolute error of the approximate solutions for different values of  at different points with  = 2,  = 6.Table 7 lists the maximum absolute error obtained by the proposed method for different choices of  and  at the points (  ,   ), where   = /40,   = /40, ,  = 0, 1, 2, . . ., 40.
with the initial conditions and the boundary conditions where (, ) = ((Γ(4+)/Γ(4)) 3 +(+3) +2 +1− +3 −)  .The exact solution of this problem is (, ) = ( +3 + )  .The space-time graph of the approximate solution and the absolute error for  = 1.9,  = 2, and  = 6 is presented in Figure 4.The absolute error for different values of  at different points with  = 2 and  = 6 where (, ) = ((√) − 4 2 − 1)  2 + .The exact solution of this problem is (, ) =   2 + .This problem is solved by the proposed method with  = 2,  = 3 to 8. The absolute error of the approximate solutions for some different points are shown in Table 10. Figure 5 shows the approximate solution and the absolute error of this problem in the case of  = 2 and  = 6.The CPU time for this problem is 28.82 seconds.
From Figures 1-5 and Tables 1-10, it can be seen that the proposed method is very efficient and accurate in solving this problem and the obtained approximate solutions are very close to the exact ones for all chosen  with 1 <  ≤ 2. It is worth noting that more accurate result can be obtained by increasing basis functions.Remark 6.It should be noted that the presented method can also be applied to the calculation of long time .We only need to extend the second-kind Chebyshev wavelet basis functions on the interval [0, 1] to the general interval [, ].We choose Example 2 to illustrate the feasibility of this method.For example, Figure 6 shows the behavior of exact and approximation solution and corresponding error at  = 10.

Conclusion
In this paper, the collocation method based on the secondkind Chebyshev wavelets has been applied to the timefractional diffusion-wave equations.We derived the fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense via the shifted Chebyshev polynomials of the second kind.Convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two-dimension were given.The second-kind Chebyshev wavelets and their properties have been used to convert the problems under consideration into some corresponding

Table 1 :
The absolute error for Example 1 for some different values 1 <  ≤ 2 at some different points.

Table 3 :
The absolute errors for Example 2 for some different values 1 <  ≤ 2 at some different points.

Table 5 :
Maximum absolute error for Example 2 with various choices of  and .

Table 6 :
The absolute error for Example 3 for some different values 1 <  ≤ 2 at some different points.

Table 7 :
Maximum absolute error for Example 3 with various choices of  and .

Table 8 :
The absolute errors for Example 4 for some different values 1 <  ≤ 2 at some different points.

Table 9 :
Maximum absolute error for Example 4 with various choices of  and .

Table 10 :
The absolute errors for Example 5 at some different points with  = 2 and  = 3 to 8.