- Research
- Open Access
A space–time spectral collocation algorithm for the variable order fractional wave equation
- A. H. Bhrawy^{1}Email author,
- E. H. Doha^{2},
- J. F. Alzaidy^{3} and
- M. A. Abdelkawy^{1, 4}
- Received: 5 May 2016
- Accepted: 22 July 2016
- Published: 2 August 2016
Abstract
The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space–time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi–Gauss–Lobatto collocation scheme for the spatial discretization and the shifted Jacobi–Gauss–Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.
Keywords
- Variable-order fractional derivative
- Collocation method
- Jacobi polynomials
- Gauss quadrature
- Fractional wave equation
Background
The subject of fractional calculus is one of the branches of applied mathematics which deals with derivatives and integrals of any arbitrary order (Hilfer 2000; Kilbas and Trujillo 2002; Kilbas et al. 2006). Fractional partial differential equations are describing the phenomena in many various areas such as fluid mechanics, physics, engineering, biology (Miller and Ross 1993; Giona and Roman 1992; Rossikhin and Shitikova 1997; Podlubny 1999; West 2007). The concept of variable-order fractional allows the power of the fractional operator to be a function of the independent variable (Coimbra 2003; Chechkin et al. 2005; Evans and Jacob 2007; Sun et al. 2009; Coimbra et al. 2005; Coimbra and Ramirez 2007). Few numerical methods have been introduced and discussed to solve the variable-order fractional problems (Sun et al. 2012; Ma et al. 2012; Zeng et al. 2015; Fu et al. 2015; Abdelkawy et al. 2015). Bhrawy and Zaky (2015a) proposed a new algorithm for solving one-and two-dimensional variable-order cable equations based on Jacobi spectral collocation approximation together with the Jacobi operational matrix for variable-order fractional derivative. Chen et al. (2014) proposed an implicit alternating direct method for the two-dimensional variable-order fractional percolation equation also discussed the stability and convergence of the implicit alternating direct method.
Spectral methods (Canuto et al. 2006; Saadatmandi and Dehghan 2011; Doha and Bhrawy 2012; Bhrawy and Zaky 2015b, c; Bhrawy et al. 2016a) have been widely used in many fields in the last four decades. In the early times, the spectral method based on Fourier expansion has been used in few fields such as a simple geometric field and periodic boundary conditions. Recently, they have been developed theoretically and used as powerful techniques to solve various kinds of problems. Based on the accuracy and exponential rates of convergence, spectral methods have an excellent reputation when compared with others numerical methods. The expression of the problem solution as a finite series of polynomials/functions is the major step of all types of spectral methods. Then, the coefficients of this expansion will be chosen such that the absolute error is diminished as well as possible.
The spectral collocation method (Canuto et al. 2006; Bhrawy and Alofi 2013; Gu and Chen 2014; Bhrawy and Abdelkawy 2015; Bhrawy 2016a) is a specific type of spectral methods, that is more applicable and widely used to solve almost types of differential (Bhrawy et al. 2016b; Tatari and Haghighi 2014), integral (Bhrawy et al. 2016c; Rahmoune 2013), integro-differential (Jiang and Ma 2013; Ma and Huang 2014) and delay differential (Bhrawy et al. 2015a; Reutskiy 2015) equations. While, the numerical solution will be enforced to almost satisfy the partial differential equations (PDEs) in spectral collocation method. In other words, the residuals may be permitting to be zero at chosen points. Wei and Chen (2012) proposed Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equations. Bhrawy and Alofi (2012) introduced the spectral shifted Jacobi–Gauss collocation method for solving the Lane–Emden type equation. Bhrawy et al. (2015b) proposed the spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. Bhrawy (2016b) proposed Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations.
The aim of this paper is to find the numerical solution of the space–time variable order fractional wave equation subject to initial-boundary conditions. The wave equation is an important second-order partial differential equation for the description of waves as they occur in physics such as sound waves, light waves and water waves. Variable order wave equation appears in areas such as acoustics, electromagnetics, and fluid dynamics. This paper extends the SJ–GL-C and SJ–GR-C schemes in order to solve the space-time variable order fractional wave equation. The proposed collocation scheme is investigated for both temporal and spatial discretizations. The SJ–GL-C and SJ–GR-C are proposed, with a suitable modification for treating the boundary and initial conditions, for spatial and temporal discretizations. This treatment, for the conditions, improves the accuracy of the scheme greatly. Therefore, the space–time variable order fractional wave equation with its conditions is reduced to system of algebraic equations which is far easier to be solved. Finally, numerical examples with comparisons lighting the high accuracy and effectiveness of the proposed algorithm are presented.
The present paper is presented as follows. The definitions of the fractional calculus and some properties of Jacobi polynomials are introduced in “Preliminaries” section. The spectral collocation methods for the space–time variable order fractional wave problem subject to initial-boundary conditions are presented in “Jacobi collocation method” section and then illustrated with two examples in “Numerical examples” section. The “Conclusion” is included in the last section.
Preliminaries
We first recall some definitions and preliminaries of the variable-order fractional differential and integral operators and some knowledge of orthogonal shifted Jacobi polynomials that are most relevant to spectral approximations.
Definition 1
Definition 2
Definition 3
Jacobi collocation method
In this section, we introduce a numerical algorithm extends the SJ–GL-C and SJ–GR-C schemes in order to solve the space-time variable order fractional wave equation. The collocation points are selected at the SJ–GR and SJ–GL interpolation nodes for temporal and spatial variables, respectively. The core of the proposed method consists of discretizing the space–time variable order fractional wave equation to create a system of algebraic equations of the unknown coefficients. This system can be then easily solved with a standard numerical scheme.
Numerical examples
This section reports two numerical examples to demonstrate the high accuracy and applicability of the proposed method. We also compare the results given from our scheme and those reported in the literature. The comparisons reveal that our method is very effective and convenient.
Example 1
x | t | Our method | NSFD method (Sweilam and Assiri 2015) | ||
---|---|---|---|---|---|
\(\theta _{1}=\theta _{2}=0\) | \(\theta _{1}=\theta _{2}=\frac{1}{2}\) | \(\theta _{1}=1,\theta _{2}=0\) | \(N=1000\) | ||
\(\vartheta _{1}=\vartheta _{2}=0\) | \(\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\) | \(\vartheta _{1}=0,\vartheta _{2}=1\) | \(M=125\) | ||
0 | 1 | 2.15204 × 10^{−14} | 3.55271 × 10^{−14} | 0 | 0 |
0.8 | 2.1985 × 10^{−15} | 9.9476 × 10^{−14} | 3.55271 × 10^{−14} | 2.70905 × 10^{−3} | |
1.6 | 1.22734 × 10^{−14} | 1.84741 × 10^{−13} | 1.42109 × 10^{−14} | 2.47795 × 10^{−3} | |
2.4 | 1.60228 × 10^{−14} | 5.68434 × 10^{−14} | 7.10543 × 10^{−14} | 2.12119 × 10^{−3} | |
3.2 | 3.54371 × 10^{−15} | 1.42109 × 10^{−14} | 1.42109 × 10^{−14} | 1.62219 × 10^{−3} | |
4 | 1.69769 × 10^{−14} | 2.84217 × 10^{−14} | 2.84217 × 10^{−14} | 9.1329 × 10^{−4} | |
4.8 | 1.93105 × 10^{−14} | 8.52651 × 10^{−14} | 5.68434 × 10^{−14} | 9.988 × 10^{−5} | |
5.6 | 5.34145 × 10^{−14} | 0 | 0 | 1.52948 × 10^{−3} | |
6.4 | 2.16945 × 10^{−14} | 1.42109 × 10^{−14} | 2.13163 × 10^{−14} | 3.50169 × 10^{−3} | |
7.2 | 5.49134 × 10^{−14} | 3.55271 × 10-15 | 0 | 6.15396 × 10^{−3} | |
8 | 1.37362 × 10^{−15} | 1.42109 × 10^{−14} | 0 | 0 |
T | Our method | NSFD method (Sweilam and Assiri 2015) | ||
---|---|---|---|---|
\(\theta _{1}=\theta _{2}=0\) | \(\theta _{1}=\theta _{2}=\frac{1}{2}\) | \(\theta _{1}=1,\theta _{2}=0\) | \(N=1000\) | |
\(\vartheta _{1}=\vartheta _{2}=0\) | \(\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\) | \(\vartheta _{1}=0,\vartheta _{2}=1\) | \(M=125\) | |
1 | \(3.88506\times 10^{-14}\) | \(3.55271\times 10^{-14}\) | \(5.68434\times 10^{-14}\) | \(6.1539\times 10^{-3}\) |
4 | \(4.86067\times 10^{-13}\) | \(6.82121\times 10^{-13}\) | \(7.95808\times 10^{-13}\) | \(3.4818\times 10^{-3}\) |
8 | \(2.00448\times 10^{-11}\) | \(5.00222\times 10^{-12}\) | \(6.36646\times 10^{-12}\) | \(9.0641\times 10^{-5}\) |
Figure 1 displays the space-graph of the numerical solution of problem (1) with \(N=8,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\). While, Fig. 2 compares graphically the curves of numerical and exact solutions of problem (1) for the different values of t at \(N=8,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\). Moreover, we represent in Figs. 3 and 4 the absolute error curves obtained by the present method at \(t=0.5\) and \(x=5\) with \(N=8,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\), respectively. This demonstrates that the proposed method leads to an accurate approximation and yields exponential convergence rates.
Example 2
The MAEs of problem 2
N | Our method with several choices of \(N,\,M\) | ||
---|---|---|---|
(0, 0, 0, 0) | \(\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) | \(\left(-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) | |
4 | \(7.3277\times 10^{-2}\) | \(4.46797\times 10^{-2}\) | 0.107228 |
8 | \(1.024\times 10^{-4}\) | \(1.09006\times 10^{-4}\) | \(1.14665\times 10^{-4}\) |
12 | \(1.55737\times 10^{-8}\) | \(2.31041\times 10^{-8}\) | \(3.18938\times 10^{-8}\) |
16 | \(1.75149\times 10^{-12}\) | \(3.27383\times 10^{-12}\) | \(2.14628\times 10^{-12}\) |
20 | \(3.70814\times 10^{-14}\) | \(5.32907\times 10^{-14}\) | \(6.51701\times 10^{-14}\) |
- 1.
Case 1, \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\).
- 2.
Case 2, \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\).
- 3.
Case 3, \(\theta _{1}=\theta _{2}=-\frac{1}{2}, \,\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\).
Conclusions
We presented a collocation method to achieve an accurate numerical solution for variable-order fractional wave problem subject to initial-boundary conditions. One of the most advantages of the present technique is that a fully spectral method was implemented for the time and space variables by using SJ–GR-C and SJ–G-C approximations respectively. The problem with its conditions was then reduced to an algebraic system. The greatest feature of the present scheme is, adding few terms of the SJ–G and SJ–GR collocation points, a full agreement between the approximate and exact solutions was achieved. Through the numerical examples and specially the comparison between the obtained approximate solution and those obtained by other approximations, we demonstrate the validity and high accuracy of the present method.
Declarations
Authors’ contributions
The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.
Acknowledgements
This article was funded by the Deanship of Scientific Research DSR, King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are very grateful to the reviewers for their comments and suggestions which have improved the paper.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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