Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

Zhou, Fengying; Xu, Xiaoyong

doi:https://doi.org/10.1155/2017/2610804

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusion Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 2610804 | https://doi.org/10.1155/2017/2610804

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

Fengying Zhou¹and Xiaoyong Xu¹

Academic Editor: Ming Mei

Received11 May 2017

Accepted06 Aug 2017

Published07 Sept 2017

Abstract

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.

1. Introduction

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 ([1]) 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 [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 [4–7] 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 ([12–14]). 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 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 [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 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 [28]). The Riemann-Liouville fractional integral operator of order for a function is defined as

Definition 3 (see [28]). 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 [28].

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 [29]). 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 ;
(ii)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 ;
(ii) where

Proof. (i) where . Similar to the proof of Theorem 1 in [29], 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 [30] 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 .

To solve this problem, we assume where is an unknown matrix which should be determined and is as in (13). By integrating two times with respect to on both sides of (52) and together with (2), we have Also by integrating (53) two times with respect to , we obtainPutting in (55) and considering the boundary conditions (51), we getThus, we haveWriteSoApplying fractional differentiation of order and order one on both sides of (59) with respect to , we get whereNow by substituting (53), (59), and (60) into (49), replacing by , and taking collocation points , , , we obtain the following linear system of algebraic equations: . By solving this system to determine the unknown matrix , we can achieve an approximate solution for the problem by substituting into (59).

6. 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.

Example 1. Consider the following time-fractional diffusion-wave equation ([17]) of order with the initial conditions and the boundary conditions where . The exact solution of this problem is .
The problem is solved by the proposed method for and the CPU time is 3.818 seconds. Figure 1 shows the approximate solution and the absolute error of this problem for . 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 . From Figure 1 and Table 1, it can be seen that the presented method is very efficient and accurate in solving this problem.

(a)

(b)

Example 2. Consider the following time-fractional diffusion-wave equation ([17]) of order with the initial conditions and the boundary conditions where . The exact solution of this problem is .
The problem is solved by the proposed method for and the CPU time is 28.79 seconds. Figure 2 shows the approximate solution and the absolute error of this problem for . Table 3 gives the absolute error for different values of at different points with . 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 in Table 4. Table 5 lists the maximum absolute errors obtained by the proposed method for different choices of and at the points , where . It is obvious that the results provided by the proposed method are more accurate than those given in [17].

(a)

(b)

Example 3. Consider the following time-fractional diffusion-wave equation of order : with the initial conditions and the boundary conditions where . The exact solution of this problem is .
The problem is solved by the proposed method for , 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 , and . Table 6 gives the absolute error of the approximate solutions for different values of at different points with . Table 7 lists the maximum absolute error obtained by the proposed method for different choices of and at the points , where .

(a)

(b)

Example 4. Consider the following time-fractional diffusion-wave equation of order with the initial conditions and the boundary conditions where . The exact solution of this problem is .
The space-time graph of the approximate solution and the absolute error for , and is presented in Figure 4. The absolute error for different values of at different points with and is tabulated in Table 8. Table 9 lists the maximum absolute error obtained by the proposed method for different choices of and at the points , where . The CPU time for this problem is 28.32 seconds.

(a)

(b)

Example 5. Consider the following time-fractional diffusion-wave equation: with the initial conditions and the boundary conditions where . The exact solution of this problem is .
This problem is solved by the proposed method with , to . 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 and . The CPU time for this problem is 28.82 seconds.

(a)

(b)

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 . 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 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 .

(a)

(b)

7. Conclusion

In this paper, the collocation method based on the second-kind Chebyshev wavelets has been applied to the time-fractional 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 linear systems of algebraic equations. The proposed method is very convenient for solving time-fractional diffusion-wave equations, since the boundary conditions are taken into account automatically during the process of establishing the expression of the approximate solution. Applicability and accuracy have been tested on some numerical examples. We also compared our results with the results obtained by Legendre wavelet, which showed that the proposed approach was more accurate. Furthermore, the proposed method can be expected to solve other types of fractional partial differential equations numerically.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the manuscript.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11601076) and the Youth Science Foundation of Jiangxi Province (Grant nos. 20151BAB211004 and 20151BAB211012).

References

J. Chen, F. Liu, V. Anh, S. Shen, Q. Liu, and C. Liao, “The analytical solution and numerical solution of the fractional diffusion-wave equation with damping,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1737–1748, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
R. Darzi, B. Mohammadzade, S. Mousavi, and R. Beheshti, “Sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation,” Journal of Mathematics and Computer Science, vol. 2013, no. 6, pp. 79–84, 2013.
View at: Google Scholar
S. S. Ray, “Exact solutions for time-fractional diffusion-wave equations by decomposition method,” Physica Scripta, vol. 75, no. 1, pp. 53–61, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
J. Huang, Y. Tang, L. Vázquez, and J. Yang, “Two finite difference schemes for time fractional diffusion-wave equation,” Numerical Algorithms, vol. 64, no. 4, pp. 707–720, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
F. Zeng, “Second-order stable finite difference schemes for the time-fractional diffusion-wave equation,” Journal of Scientific Computing, vol. 65, no. 1, pp. 411–430, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-Z. Sun and X. Wu, “A fully discrete difference scheme for a diffusion-wave system,” Applied Numerical Mathematics, vol. 56, no. 2, pp. 193–209, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
X. Hu and L. Zhang, “A compact finite difference scheme for the fourth-order fractional diffusion-wave system,” Computer Physics Communications, vol. 182, no. 8, pp. 1645–1650, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
V. R. Hosseini, E. Shivanian, and W. Chen, “Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping,” Journal of Computational Physics, vol. 312, pp. 307–332, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
A. Esen, O. Tasbozan, Y. Ucar, and N. M. Yagmurlu, “A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations,” Tbilisi Mathematical Journal, vol. 8, no. 2, pp. 181–193, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Mao, A. Xiao, Z. Yu, and L. Shi, “Finite difference and sinc-collocation approximations to a class of fractional diffusion-wave equations,” Journal of Applied Mathematics, vol. 2014, Article ID 536030, 11 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Mao, A. Xiao, Z. Yu, and L. Shi, “Sinc-chebyshev collocation method for a class of fractional diffusion-wave equations,” The Scientific World Journal, vol. 2014, Article ID 143983, 7 pages, 2014.
View at: Publisher Site | Google Scholar
S. Nemati and S. Sedaghat, “Matrix method based on the second kind Chebyshev polynomials for solving time fractional diffusion-wave equations,” Journal of Applied Mathematics and Computing, vol. 51, no. 1-2, pp. 189–207, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931–4943, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
A. H. Bhrawy, E. H. Doha, D. Baleanu, and S. S. Ezz-Eldien, “A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations,” Journal of Computational Physics, vol. 293, pp. 142–156, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
M. R. Hooshmandasl, M. H. Heydari, and C. Cattani, “Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions,” European Physical Journal Plus, vol. 131, no. 8, article 268, 2016.
View at: Publisher Site | Google Scholar
Z. Barikbin, “Two-dimensional Bernoulli wavelets with satisfier function in the Ritz—Galerkin method for the time fractional diffusion-wave equation with damping,” Mathematical Sciences, vol. 11, no. 3, pp. 195–202, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
M. H. Heydari, M. R. Hooshmandasl, F. M. M. Maalek Ghaini, and C. Cattani, “Wavelets method for the time fractional diffusion-wave equation,” Physics Letters A, vol. 379, no. 3, pp. 71–76, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Li, N. Sun, B. Zheng, Q. Wang, and Y. Zhang, “Wavelet operational matrix method for solving the Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 3, pp. 483–493, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
S. Balaji, “Legendre wavelet operational matrix method for solution of fractional order Riccati differential equation,” Journal of the Egyptian Mathematical Society, vol. 23, no. 2, pp. 263–270, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
M. H. Heydari, M. R. Hooshmandasl, and F. Mohammadi, “Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions,” Applied Mathematics and Computation, vol. 234, pp. 267–276, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Li, “Solving a nonlinear fractional differential equation using Chebyshev wavelets,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2284–2292, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Wang and L. Zhu, “Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method,” Advances in Difference Equations, vol. 2017, article 27, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
L. Zhu and Q. Fan, “Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2333–2341, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, “New wavelets collocation method for solving second-order multipoint boundary value problems using chebyshev polynomials of third and fourth kinds,” Abstract and Applied Analysis, vol. 2013, Article ID 542839, 9 pages, 2013.
View at: Publisher Site | Google Scholar
Y. Chen, L. Sun, X. Li, and X. Fu, “Numerical solution of nonlinear fractional integral differential equations by using the second kind Chebyshev wavelets,” Computer Modeling in Engineering and Sciences, vol. 90, no. 5, pp. 359–378, 2013.
View at: Google Scholar | MathSciNet
A. K. Gupta and S. Saha Ray, “Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method,” Applied Mathematical Modelling, vol. 39, no. 17, pp. 5121–5130, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
İ. Çelik, “Chebyshev wavelet collocation method for solving generalized Burgers-Huxley equation,” Mathematical Methods in the Applied Sciences, vol. 39, no. 3, pp. 366–377, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
View at: MathSciNet
F. Zhou and X. Xu, “Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets,” Applied Mathematics and Computation, vol. 247, pp. 353–367, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
N. H. Sweilam, A. M. Nagy, and A. A. El-Sayed, “Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation,” Chaos, Solitons and Fractals, vol. 73, pp. 141–147, 2015.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2017 Fengying Zhou and Xiaoyong Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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