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Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula
- Mohamed M Khader^{1, 2}Email author and
- Mohamed H Adel^{3}
https://doi.org/10.1186/s13662-015-0731-0
© Khader and Adel 2016
- Received: 21 September 2015
- Accepted: 20 December 2015
- Published: 2 February 2016
Abstract
In this article, a numerical study is introduced for solving the fractional wave equations by using an efficient class of finite difference methods. The proposed scheme is based on the Hermite formula. The stability and the convergence analysis of the proposed methods are given by a recently proposed procedure similar to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Finally, a numerical example is presented to confirm the theoretical results.
Keywords
- finite difference methods
- Hermite formula
- fractional wave equation
- stability and convergence analysis
1 Introduction
In recent years, it has turned out that many phenomena in engineering, physics, chemistry, and other sciences can be described very successfully by models using mathematical tools from fractional calculus, i.e. the theory of derivatives and integrals of fractional (non-integer) order. There are many applications of the fractional differential equations (FDEs), see [1–10], and the studied models have received a great deal of attention like in the fields of viscoelastic materials [1], solutes in natural porous or fractured media [3], and anomalous diffusion. Most FDEs do not have exact solutions, so approximate and numerical techniques must be used [11–21]. Recently, several numerical methods to solve FDEs have been given such as the variational iteration method [16], the Adomian’s decomposition method [21], the collocation method [22–24], and the finite difference method [15, 17, 25]. In this section, we introduce the Riemann-Liouville definitions of the fractional derivative operator \(D^{\alpha}\) [26, 27].
Definition 1
In the last few years there have appeared many papers studying the proposed model (2)-(4) [15, 17]. In this paper, we study the time fractional case and use an efficient class of finite difference methods based on the Hermite formula to solve this model.
The plan of the paper is as follows. In Section 2, we give some approximate formulas of the fractional derivatives and numerical finite difference scheme. In Section 3, we study the stability and the accuracy of the presented scheme. In Section 4, we introduce numerical solutions of fractional wave equation. The paper ends with some conclusions in Section 5.
2 Finite difference scheme of the fractional wave equation
In this section, we introduce an efficient class of FDM and use it to obtain the discretization finite difference formula of the time fractional wave equation (2). For some positive integer numbers N and M, we use the notations Δx and Δt for the space-step length and the time-step length, respectively. The coordinates of the mesh points are \(x_{j}=j\Delta x\) (\(j=0,1,\ldots,N\)), and \(t_{m}=m \Delta t\) (\(m=0,1,\ldots,M\)) and the values of the solution \(u(x, t)\) on these grid points are \(u(x_{j}, t_{m})\equiv u_{j}^{m}\simeq U_{j}^{m}\), where \(\Delta x =h=\frac{L}{N}\), \(\Delta t=k=\frac{T}{M}\).
For more details as regards discretization in fractional calculus see [9, 25, 28].
- (1)
\(\omega^{(\alpha)}_{r}> 0\), \(r=1,2,\ldots\) .
- (2)
\(\omega^{(\alpha)}_{r}>\omega^{(\alpha +1)}_{r}\), \(r=1,2,\ldots\) .
Proposition 1
Proof
3 Stability analysis
In this section, we use the von Neumann method to study the stability analysis of the finite difference scheme (14) for the force free case (i.e., \(f(x,t)=0\)).
Proposition 2
Proof
Proposition 3
Assume that \(\xi_{m}\) (\(m=1,2,\ldots,M\)) is the solution of (17), with the condition \(k^{\alpha} < \frac{h^{2}}{6\Gamma(3-\alpha)}\), then \(|\xi_{m}|\leq|\xi_{0}|\) (\(m=1,2,\ldots,M\)).
Proof
It is easy to prove it by the mathematical induction together with Proposition 2. □
Theorem 1
Proof
From Proposition 3 and (30), \(\| T^{m}\|_{2} \leq \| T^{0}\|_{2}\), \(m=1,2,\ldots,M\), which means that the difference scheme is stable. □
By the Lax equivalence theorem [28] we can show that the numerical solution converges to the exact solution as \(h, k\rightarrow0\).
4 Numerical results
The exact solution of equation (2) in this case is \(u(x,t)=\sin (\pi x)(t^{2}-t)\).
The maximum error with different values of Δ x and Δ t for \(\pmb{\alpha=1.5}\) and \(\pmb{T=0.2}\)
Δx | \(\frac{1}{5}\) | \(\frac{1}{10}\) | \(\frac{1}{20}\) | \(\frac{1}{30}\) | \(\frac{1}{30}\) | \(\frac{1}{40}\) | \(\frac{1}{40}\) | \(\frac{1}{45}\) |
Δt | \(\frac{1}{50}\) | \(\frac{1}{100}\) | \(\frac {1}{150}\) | \(\frac{1}{150}\) | \(\frac{1}{200}\) | \(\frac{1}{200}\) | \(\frac{1}{210}\) | \(\frac{1}{220}\) |
max. error | 0.01149 | 0.00361 | 0.00120 | 0.00115 | 0.00021 | 0.00019 | 0.00006 | 0.00004 |
The maximum error with different values of Δ x , Δ t for \(\pmb{\alpha=1.7}\) and \(\pmb{T=0.4}\)
Δx | \(\frac{1}{10}\) | \(\frac{1}{20}\) | \(\frac{1}{30}\) | \(\frac{1}{50}\) | \(\frac{1}{50}\) | \(\frac{1}{60}\) | \(\frac{1}{60}\) | \(\frac{1}{70}\) |
Δt | \(\frac{1}{50}\) | \(\frac{1}{100}\) | \(\frac {1}{200}\) | \(\frac{1}{250}\) | \(\frac{1}{300}\) | \(\frac{1}{400}\) | \(\frac{1}{450}\) | \(\frac{1}{480}\) |
max. error | 0.01396 | 0.01064 | 0.00736 | 0.00653 | 0.00586 | 0.00494 | 0.00460 | 0.00443 |
5 Conclusion and remarks
This paper presents a class of numerical methods for solving the fractional wave equations. This class of methods depends on the finite difference method based on the Hermite formula. Special attention is given to the study of the stability and the convergence of the fractional finite difference scheme. To execute this aim we have resorted to a kind of fractional von Neumann stability analysis. From the theoretical study we can conclude that this procedure is suitable for the proposed fractional finite difference scheme and leads to very good predictions for the stability bounds. Numerical solutions and exact solutions of the proposed problem are compared and the derived stability condition is checked numerically. From this comparison, we can conclude that the numerical solutions are in excellent agreement with the exact solutions. All computations in this paper were run with the Matlab programming package.
Declarations
Acknowledgements
The authors are very grateful for the editor’s and the referee’s careful reading and comments on this paper.
Open Access This 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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