Two-dimensional Bernoulli wavelets with satisﬁer function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz–Galerkin method are applied for the numerical solution of the time fractional diffusion-wave equation. In this way, a satisﬁer function which satisﬁes all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz–Galerkin method with satisﬁer function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.


Introduction
Many phenomena in various field of the science, can be modeled very successfully by time-fractional differential equations. In this paper we focus on the following fractional diffusion-wave equation ( where L [ 0; T [ 0; 1\q 2 is the order of the fractional derivative in the Caputo sense, f 0 ; f 1 ; g 0 and g 1 are known and sufficiently smooth functions, while the function u is to be determined. In the case q ¼ 2, this equation is named telegraph equation. Recently, considerable amount of papers have been proposed methods for solving the FDWE [2][3][4][5][6][7][8][9][10][11][12][13]. Chen et al. [1] obtained the analytical solution by the method of separation of variables and proposed the numerical solution with finite difference method. In [2], Bhrawy et al. applied a spectral tau method based on the Jacobi operational matrix to solve the problem. Liu et al. [3] proposed the fractional predictor-corrector method to solve this problem. In [4], Mainardi derived the fundamental solutions for the FDWE. The combination of the compact difference method and alternating direction implicit method are used for solving two-dimensional fractional Cattaneo equation in [5]. A fully discrete difference scheme is recommended for a diffusion-wave system by Wess [6]. Heydari et al. [7] applied fractional operational matrix (FOM) of integration for the Legendre wavelets (LWs) to solve the problem. In [8] a compact finite-difference scheme is used for the fourth-order fractional diffusion-wave system. In [9] finite difference schemes of second-order are proposed for the time-fractional diffusion-wave equation. In [10], Hu and Zhang used finitedifference methods for fourth-order fractional diffusionwave. Sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation applied by Darzi et al. [11]. Hosseini et al. [12] employed the meshless local radial point interpolation method which is based on the Galerkin weak form and radial point interpolation approximation for solving FDWE.
The Ritz-Galerkin method is the method to transform a continuous problem to a discrete problem. Several partial differential equations are numerically solved by Ritz-Galerkin method, but using of the appropriate satisfier function in the Ritz-Galerkin method is taken into consideration recently, see for instance [14][15][16][17][18][19][20]. The satisfier function fulfills all the problem conditions. In conclusion, employing of it in Ritz-Galerkin method provides the facility to satisfy the problem conditions, also leads to a system of algebraic equations of smaller size and hence reduces the computation time.
In mathematical research, wavelet theory is a relatively new and growing area. It has been used in a wide range of engineering; for instance, wavelets are very successfully applied in signal analysis for waveform representations and segmentations [21]. Wavelets allow the accurate representation of many types of functions and operators [22,23]. Furthermore, wavelets make a connection with fast numerical algorithms [24].
In this paper, we define two-dimensional BWs for the first time. Moreover, this is the first time the Ritz-Galerkin method in the two-dimensional BWs basis and with utilizing the satisfier function is employed to give an approximate solution of FDWE. We also compare our results with those results obtained by [3] and [7]. Comparison for the numerical examples shows the more accuracy and less computations of our scheme in comparison to other published methods.
This paper is separated in to the following sections: In Sect. 2, we introduce basic formulation of wavelets and the Bernoulli wavelets. In Sect. 3, we construct two different satisfier functions and apply the Ritz-Galerkin method in the two-dimensional BWs basis for numerically discretize the problem. Section 4 presents and discusses the numerical results for two test examples, whilst Sect. 5 includes the conclusions of this paper.
Returning now to the Ritz-Galerkin approximation

Numerical results and comparisons
In this section, we apply the numerical scheme in the previous section for finding the approximation solutions of two examples of FDWE. We compare our results with obtained results in [3] and [7]. In all examples the package of Mathematica ver. 10.4 has been used. The approximate norm-2 of absolute error is given as with the homogenous initial and boundary conditions, and let hðx; tÞ ¼ 2xð1 À xÞt 2Àq Cð3 À qÞ þ 2txð1 À xÞ þ 2t 2 : Now, we apply the numerical method presented in this paper for which is the exact solution. In [7], Heydari et al. used fractional operational matrix of integration for the LWs to solve this problem for q ¼ 1:1; 1:3; 1:5; 1:7; 1:9. The best absolute error of the approximate solutions at some different points, in [7], with

Example 2
Consider the following FDWE [3,7]: o q uðx; tÞ ot q þ ouðx; tÞ ot ¼ o 2 uðx; tÞ ox 2 þ hðx; tÞ; ðx; tÞ 2 ½0; 1 Â ½0; 1; 1\q 2; ð4:2Þ where hðx; tÞ ¼ 6t 3Àq e x Cð4 À qÞ þ 3t 2 e x À t 3 e x : The initial and boundary conditions are determined correspondingly to the exact solution uðx; tÞ ¼ e x t 3 : From (3.18), we obtain wðx; tÞ ¼ t 3 ð1 À x þ exÞ, then apply the numerical method presented in this paper for Tables 1 and 2 present, respectively, the absolute error and the L 2 norm error for uðx; tÞ À e uðx; tÞ with different values of q. In Table 3, the absolute error of u(x, t) and CPU times for q ¼ 1:5 with k 1 ¼ k 2 ¼ 1 and different values of M 1 ¼ M 2 are given. It is seen from Table 3 that, with increase in the number of the two-dimensional Bernoulli wavelets basis, the approximate values of u(x, t) converge to the exact solutions. In Figs. 1 and 2, the exact and approximate solutions, and also the absolute difference between exact and approximate solutions of u(x, t) with q ¼ 1:1 and q ¼ 1:9 are plotted, respectively. Also, the graphs of the absolute errors for q ¼ 1:5 at t ¼ 0:5 and x ¼ 0:5 are shown in Fig. 3.
Liu et al. [3] employed the fractional predictor-corrector method and solved this problem with q ¼ 1:85 and different values of time and space step sizes. They obtained 1:6341 Â 10 À3 for the best maximum absolute error. Moreover, in Table 1 we compare our results with obtained results in [7]. These comparisons show the more accuracy and less computations of our technique in comparison to other published methods.     Fig. 1 The graphs of the Exact (blue) and approximate (red) solutions (left side) and the absolute error (right side) of u(x, t) in q ¼ 1:1, for Example 2 Fig. 2 The graphs of the Exact (blue) and approximate (red) solutions (left side) and the absolute error (right side) of u(x, t) in q ¼ 1:9, for Example 2

Conclusion
In this paper, the two-dimensional BWs was defined. Then the satisfier function in Ritz-Galerkin method with the two-dimensional BWs basis was successfully applied to solve the second-order time FDWE. Using of satisfier function in the Ritz-Galerkin method is an efficient tool to put on the initial and boundary conditions. Furthermore, a small number of basis elements were sufficient to derive accurate numerical solutions. Also, our results were compared with obtained results in [3] and [7]. Comparison for the numerical examples shows the more accuracy and less computations of the proposed method in comparison to other published methods.
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