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Finite difference approach for variable order reaction–subdiffusion equations
- M. Adel^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1862-x
© The Author(s) 2018
- Received: 4 April 2018
- Accepted: 23 October 2018
- Published: 5 November 2018
Abstract
The fractional reaction–subdiffusion equation is one of the most famous subdiffusion equations. These equations are widely used in recent years to simulate many physical phenomena. In this paper, we consider a new version of such equations, namely the variable order linear and nonlinear reaction–subdiffusion equation. A numerical study is introduced using the weighted average methods for the variable order linear and nonlinear reaction–subdiffusion equations. A stability analysis of the proposed method is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. The paper is ended with the results of numerical examples that support the theoretical analysis.
Keywords
- Weighted average finite difference approximations
- Variable order linear and nonlinear reaction–subdiffusion equation
- Stability analysis
- Numerical example
MSC
- 65N12
- 65M60
- 41A30
1 Introduction
Fractional calculus is a very hot area of research due to its ability to study many applications in physics and engineering, which cannot be studied by the ordinary calculus. There are many applications of this important type of calculus [1–4]. The approximate and numerical techniques must be used [5–9] because most fractional differential equations (FDEs) do not have exact solutions. Recently, several numerical methods to solve fractional differential equations have been proposed, such as variational iteration method [10], homotopy perturbation method [11], Adomian decomposition method [12], homotopy analysis method [13], finite difference method (FDM) [14, 15], and spectral methods [5, 6].
Many physical processes appear to exhibit fractional order behavior that may vary with time or space. This fact enables us to consider the order of the fractional integrals and derivatives to be a function of time or some other variables. The objective of this work is to identify the most appropriate definition of a variable-order operator for modeling dynamic systems and to assign the order of the derivative to give a physical meaning that will facilitate the understanding of its use in problems of vibration and control. However, until now, only few researchers have considered the numerical analysis of variable-order differential equations; see, for example, [16–20].
2 Fractional reaction–subdiffusion equation
3 Finite difference scheme for a variable order fractional reaction–subdiffusion equation
In this section, we will use the weighted average FDM to obtain a discretization finite difference formula of the variable order linear and nonlinear reaction–subdiffusion equation (1). For some positive constants M and N, we use Δt and Δx to denote the time-step length and space-step length, respectively. The coordinates of the mesh points are \(x_{j}=j\Delta x\) \((j=0,1,2,\dots,N)\), and \(t_{m}=m \Delta t\), \((m=0,1,2,\dots,M)\) and the values of the solution \(y(x, t)\) on these grid points are \(y(x_{j}, t_{m})\equiv y_{j}^{m}\simeq Y_{j}^{m}\), where \(\Delta x =\frac{L}{N}\), and \(\Delta t=\frac{T}{M}\).
Now, we are going to obtain a finite difference scheme of the linear and nonlinear variable order reaction–subdiffusion equation (1). In our study we take \(k_{\alpha}=\varepsilon=1\).
Remark 1
It is worthy to report here that the number of arithmetic operations required to solve the system of equations (20) is approximately \(\frac{2}{3}(m+1)^{3} \), see [31].
Theorem 1
The difference equations (20) are uniquely solvable.
Proof
Because \(\phi> 0\), the coefficient matrix of the difference equations (20) is a strictly diagonally dominant matrix. Therefore, A is a nonsingular matrix, which proves the theorem. □
Lemma 1
- (1)
\(\rho_{0}^{1-\alpha}=1\); \(\rho_{1}^{1-\alpha}=\alpha-1\); \(\rho _{k}^{1-\alpha}<0\), \(k=2,3,\dots\);
- (2)
\(\sum_{k=0}^{\infty}\rho_{k}^{1-\alpha}=1\); \(\forall n \in N^{+}\), \(-\sum_{k=1}^{n}\rho_{k}^{1-\alpha}<1\).
Proof
See [32]. □
4 Stability analysis
In this section, we use the John von Neumann method in the stability analysis of the weighted average scheme (17). In our study we neglected the source term (i.e., \(g(x,t)=0\)).
Proposition 1
Proposition 2
Proposition 3
Theorem 2
Proof
5 Numerical results
Absolute error between the exact and numerical solutions of the variable order nonlinear reaction–subdiffusion equation (30) for different values of \(\alpha(x,t) \), Δt and Δx
α(x,t) | \(\Delta x=\frac{1}{5}\),\(\Delta t=\frac {1}{25}\) | \(\Delta x=\frac{1}{10}\) ,\(\Delta t=\frac{1}{100}\) | \(\Delta x=\frac{1}{20}\),\(\Delta t=\frac{1}{400}\) |
---|---|---|---|
\(e^{xt-5}\) | 5.449 × 10^{−3} | 2.604 × 10^{−3} | 8.085 × 10^{−4} |
[33] | 3.9838 × 10^{−3} | 1.0056 × 10^{−3} | 4.5969 × 10^{−4} |
\(\frac{10-(xt)^{2}}{300}\) | 5.585 × 10^{−3} | 2.776 × 10^{−3} | 8.532 × 10^{−4} |
[33] | 4.0042 × 10^{−3} | 1.0109 × 10^{−3} | 4.3241 × 10^{−4} |
\(\frac{15-x^{2}+t^{4}}{400}\) | 5.594 × 10^{−3} | 2.791 × 10^{−3} | 8.566 × 10^{−4} |
[33] | 4.0025 × 10^{−3} | 1.0093 × 10^{−3} | 4.2750 × 10^{−4} |
\(\frac{15+\cos(xt)}{300}\) | 5.672 × 10^{−3} | 2.912 × 10^{−3} | 8.881 × 10^{−4} |
[33] | 4.0133 × 10^{−3} | 1.0128 × 10^{−3} | 4.1130 × 10^{−4} |
\(\frac{5^{xt}-\cos(xt)}{40}\) | 5.341 × 10^{−3} | 2.598 × 10^{−3} | 8.018 × 10^{−4} |
[33] | 3.9431 × 10^{−3} | 9.9547 × 10^{−3} | 4.3483 × 10^{−4} |
\(\frac{e^{xt}+\sin(xt)}{50}\) | 5.451 × 10^{−3} | 2.728 × 10^{−3} | 8.356 × 10^{−4} |
[33] | 3.9715 × 10^{−3} | 1.0019 × 10^{−3} | 4.2391 × 10^{−4} |
\(\frac{10-\sin^{3}(xt)}{300}\) | 5.583 × 10^{−3} | 2.776 × 10^{−3} | 8.532 × 10^{−4} |
[33] | 4.0037 × 10^{−3} | 1.0103 × 10^{−3} | 4.3143 × 10^{−4} |
\(\frac{25-x^{4}+\sin ^{3}(t)}{500}\) | 5.653 × 10^{−3} | 2.881 × 10^{−3} | 8.794 × 10^{−4} |
[33] | 4.0097 × 10^{−3} | 1.0118 × 10^{−3} | 4.1491 × 10^{−4} |
\(\frac{20-\sin^{3}(x)+\cos^{5}(t)}{400}\) | 5.668 × 10^{−3} | 2.898 × 10^{−3} | 8.843 × 10^{−4} |
[33] | 4.0148 × 10^{−3} | 1.0132 × 10^{−3} | 4.1486 × 10^{−4} |
Example 5.1
Dependency of maximum absolute error on Δx, Δt with \(\lambda=0\), \(\alpha(x,t)=\frac{12-\sin^{3}(xt)}{12}\), where the final time is \(T=0.1\)
Δx | Δt | Maximum absolute error |
---|---|---|
\(\frac{1}{10}\) | \(\frac{1}{20}\) | 3.208 × 10^{−3} |
\(\frac{1}{10}\) | \(\frac{1}{30}\) | 2.756 × 10^{−3} |
\(\frac{1}{20}\) | \(\frac{1}{40}\) | 2.607 × 10^{−3} |
\(\frac{1}{30}\) | \(\frac{1}{50}\) | 2.589 × 10^{−3} |
\(\frac{1}{40}\) | \(\frac{1}{50}\) | 2.562 × 10^{−3} |
According to Remark 1, the number of arithmetic operations required to solve the system in this case is approximately \(\frac{2}{3}(91+1)^{3} \simeq519{,}125\).
Example 5.2
6 Conclusion and remarks
This paper presents a class of numerical methods for solving the variable order linear and nonlinear reaction–subdiffusion equation. The contribution in this paper is a generalization of the work done by Sweilam et al. [3]. This class of methods is very close to the weighted average FDM. Special attention is given to the stability of the fractional finite weighted average FDM. For this we have resorted to a kind of fractional John von Neumann stability analysis. From the theoretical study we can conclude that this procedure is suitable for the fractional finite weighted average FDM and leads to very good predictions for the stability bounds. The stability of the fractional finite weighted average FDM presented strongly depends on the value of the weighting parameter λ. 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. By comparing the results in this paper with the results in [33], we found that the same order of maximum error was obtained under the same values of \(\alpha(x,t)\), Δx and Δt.
All computations in this paper were performed using MATLAB software.
Declarations
Acknowledgements
The author is very grateful for the editor’s and the referee’s careful reading and comments on this paper, which greatly improved the presentation of the paper.
Availability of data and materials
All data and material are available for everyone.
Funding
This work is supported by Faculty of Science, Cairo University.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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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