/ / Article
Special Issue

## Theory, Methods, and Applications of Fractional Calculus

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 681707 | 5 pages | https://doi.org/10.1155/2014/681707

# A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation

Accepted20 Feb 2014
Published19 Mar 2014

#### Abstract

The computational complexity of one-dimensional time fractional reaction-diffusion equation is compared with for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.

#### 1. Introduction

Fractional equations can be used to describe some physical phenomenon more accurately than the classical integer order differential equation. The reaction-diffusion equations play an important role in dynamical systems of mathematics, physics, chemistry, bioinformatics, finance, and other research areas. There has been a wide variety of analytical and numerical methods proposed for fractional equations , for example, finite difference method , finite element method , Adomian decomposition method , and spectral technique . Interest in fractional reaction-diffusion equations has increased .

Domain decomposition methods (DDM) solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating it to coordinate the solution between adjacent subdomains . A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The DDM can be divided into two categories: the overlapping and nonoverlapping . Chan and Mathew  gave a survey on iterative domain decomposition techniques that had been developed for solving several kinds of partial differential equations, including elliptic, parabolic, and differential systems such as the Stokes problem and mixed formulations of elliptic problems. The problems on the subdomains are almost independent, which makes domain decomposition methods suitable for parallel computing. Parallel computing is used to solve intensive computation applications simultaneously , such as particle transport [17, 18] and fast multipole methods . It is time consuming to numerically solve fractional differential equations for long time tail. Parallel computing  can be used to overcome the computational challenge of fractional approximation. DDM will embody large potential for a parallelization of the numerical solution for fractional equations. Until today the power of DDM for approximating fractional derivatives and solving fractional differential equations has not been recognized.

This paper focuses on the Caputo fractional reaction-diffusion equation: on a finite domain and . The and are constants. If equals 1, (1) is the classical reaction-diffusion equation. The fractional derivative is in the Caputo form.

#### 2. Background

##### 2.1. Numerical Solution

The fractional derivative of in the Caputo sense is defined as 

If is continuous bounded derivatives in for every , we can get

Define , , , and for , . Define , , and as the numerical approximation to , , and . We can get  where , , and

By using center difference scheme for , we can get

The implicit finite difference approximation for (1) is

Define , , , and as

Equation (7) evolves as where matrix is a tridiagonal matrix, defined by

Because , , the elements of matrix satisfy . This means that matrix is strictly diagonally dominant.

##### 2.2. Computational Challenge

In order to get , the right-sided computation of (9) should be performed and tridiagonal linear system should be solved. There are mainly many constant vector multiplications and many vector vector additions in the right-sided computation.(1)The constant vector multiplications are , , and .(2)The vector vector additions are .(3)After solving tridiagonal linear system , we get .

The Thomas algorithm for tridiagonal systems needs multiplications and additions. The computational complexity of is . The total computation of (9) is determined by , which means multiplications and additions for each time step; The computational complexity of (1) is , while the computational complexity of classical one-dimensional reaction-diffusion equation is only . The computational cost of (11) varies linearly along the number of grid points but squares with the number of time steps.

#### 3. Domain Decomposition Method

##### 3.1. DDM with Two Subdomains

Similar to the classical alternating Schwarz method [13, 24], the domain can be divided into two subdomains and . There are grid points for . and , where . The global physical boundary is defined in (1). (the right boundary of ) and (the left boundary of ) are called artificial internal boundary.

In order to approximate the time fractional equation on the two subdomains separately, the following iterative procedure can be performed. For each time step, the right hand side of (9) is calculated at first and the is given as initial guess , where . The better approximation of can be obtained iteratively. During each iteration, which is inside of a time step, the time fractional equation is solved in the subdomain , using the approximation of the previous iteration from on as follows: where stands for the previous solution of grid point in subdomain . The better approximation is obtained. is defined as . is defined as . The definitions for and are similar.

Then, we solve the time fractional equation within the subdomain of , using the approximation of the previous iteration from on as follows: where stands for the previous solution of grid point in subdomain .

The two local time fractional equations in and are connected by the artificial boundary condition. The artificial boundary condition on the internal boundary of subdomain is provided by from subdomain , and vice versa. The approximation and may change until converged to the true solution. So, in an inner iteration of each time step, the two time fractional equations need to exchange two sets of data (send one and receive one) to update the artificial boundary conditions.

##### 3.2. A Domain Decomposition Algorithm

Section 3.1 shows the procedure of DDM for time fractional equation with two subdomains. It is not hard to extend the method of Section 3.1 to more than two subdomains. The domain can be decomposed into a set of subdomains with . For time step , has one global boundary and one artificial inner boundary . has one global boundary and one artificial inner boundary . The () has two artificial inner boundaries and . means that the neighboring subdomains have explicit overlap.

The iterative procedure for the time step is similar to Section 3.1. The current iteration of uses the data of previous iteration of its neighboring subdomains. Assuming is divisible with , the domain decomposition algorithm is shown in Algorithm 1.

 (1) [ !]   input:  et al. Output: (2) (3) Allocate memory space et al. (4) Init matrices et al. (5) Declare local variables (6) (7) Get with initial boundary. (8) (9) for    to   step by 1 do (10)  for    to   step by 1 do (11) (12)     for    to   step by 1 do (13) (14) (15) (16)     while   do (17)     for    to   step by 1 do (18) (19)        if   then (20) (21)        if   then (22) (23)     for    to   step by 1 do (24)        solve (25) (26) (27) (28) (29)     for    to  step by 1 do (30) (31) Output the information (32) Free memory space

In Algorithm 1, there are some fast algorithms to solve the tridiagonal matrix , such as Thomas algorithm. is a threshold, such as . The signal is used to count how many iterations are needed in each time step. The data exchange between neighboring iterations is shown in lines 19–22. From the view of computer science, lines 2–6, lines 7–30, and lines 31-32 are preprocessing procedure, numerical solver, and postprocessing procedure.

##### 3.3. Analysis

The presented DD algorithm updates the artificial boundary condition in a Jacobi fashion, using approximation from all the relevant neighboring subdomains from the previous iteration for each time step. A subdomain only exchanges two sets of data for one artificial boundary with its neighbor. Therefore, the subdomain solved in Algorithm 1 can be carried out almost completely independently, thus making the method inherently as parallel as the Jacobi iteration. The DD algorithm keeps the good parallelism of Jacobi iteration but needs fewer inner iterations in each time step; see Section 4. Equation (9) can be regarded as approximation of a special integer order reaction-diffusion equation. The stability and convergence analysis of integer order reaction-diffusion equation can refer to Mathew’s book .

#### 4. Numerical Example

The following Caputo fractional reaction-diffusion equation  was considered, as shown in (14): with , , and

The exact solution of (14) is

With , , , and , the comparison between exact solution and the presented DD algorithm is shown in Table 1. We can find that the DD algorithm compares well with the exact solution.

 2/10 1/10 2/10 1/20 2/61 1/61 2/61 1/100 2/100 1/300

We can replace the DDM (lines 16–27 of Algorithm 1) with Jacobi method. The Jacobi method for a time step has the same parallelism with the DD algorithm. But the Jacobi method needs more iterations. With and , the comparison between Jacobi method and the presented DD algorithm is shown in Table 2. The sum of “count” (total iterations) for all time steps is recorded. We can see that the DDM needs much less iterations than Jacobi method.

 Jacobi method DDM 2/10 1/10 741 250 2/10 1/20 1147 378 2/61 1/61 52423 3155 2/61 1/100 67164 4138 2/100 1/300 276243 11373

As a part of the future work, we would like to implement an efficient DDM for time fractional equations on parallel computer systems, for example, Tianhe-1A supercomputer .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research work is supported by the National Natural Science Foundation of China under Grant no. 11175253. The authors would like to thank the anonymous reviewers for their helpful comments as well.

1. A. Kilicman and Z. A. A. A. Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007. View at: Publisher Site | Google Scholar
2. S. Zhai, X. Feng, and Z. Weng, “New high-order compact adi algorithms for 3D nonlinear time-fractional convection-diffusion equation,” Mathematical Problems in Engineering, vol. 2013, Article ID 246025, 11 pages, 2013. View at: Publisher Site | Google Scholar
3. A. Atangana and E. Alabaraoye, “Solving a system of fractional partial differential equations arising in the model of HIV infection of ${\text{CD4}}^{\mathbf{\text{+}}}$ cells and attractor one-dimensional Keller-Segel equations,” Advances in Difference Equations, vol. 2013, no. 1, article 94, pp. 1–14, 2013. View at: Publisher Site | Google Scholar
4. Q. Liu, F. Liu, I. Turner, and V. Anh, “Numerical simulation for the 3D seep age flow with fractional derivatives in porous media,” IMA Journal of Applied Mathematics, vol. 74, no. 2, pp. 201–229, 2009. View at: Publisher Site | Google Scholar
5. P. Zhuang, F. Liu, V. Anh, and I. Turner, “Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process,” IMA Journal of Applied Mathematics, vol. 74, no. 5, pp. 645–667, 2009. View at: Publisher Site | Google Scholar
6. A. Secer, “Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method,” Advances in Difference Equations, vol. 2012, no. 1, article 198, pp. 1–10, 2012. View at: Publisher Site | Google Scholar
7. A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-Vries equations,” Abstract and Applied Analysis, vol. 2013, Article ID 947986, 8 pages, 2013. View at: Publisher Site | Google Scholar
8. J. Chen, F. Liu, I. Turner, and V. Anh, “The fundamental and numerical solutions of the Riesz space-fractional reaction-dispersion equation,” The ANZIAM Journal, vol. 50, no. 1, pp. 45–57, 2008. View at: Publisher Site | Google Scholar
9. X. Zhang, J. Liu, L. Wei, and C. Ma, “Finite element method for Grwünwald-Letnikov time-fractional partial differential equation,” Applicable Analysis, vol. 92, no. 10, pp. 1–12, 2013. View at: Publisher Site | Google Scholar
10. S. S. Ray, “A new approach for the application of Adomian decomposition method for the solution of fractional space diffusion equation with insulated ends,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 544–549, 2008. View at: Publisher Site | Google Scholar
11. C. Li, F. Zeng, and F. Liu, “Spectral approximations to the fractional integral and derivative,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 383–406, 2012. View at: Publisher Site | Google Scholar
12. J. Chen, “An implicit approximation for the Caputo fractional reaction dispersion equation,” Journal of Xiamen University, vol. 46, no. 5, pp. 616–619, 2007 (Chinese). View at: Google Scholar
13. H. A. Schwarz, Gesammelte Mathematische Abhandlungen, AMS Bookstore, 1972.
14. J. Xu and J. Zou, “Some nonoverlapping domain decomposition methods,” SIAM Review, vol. 40, no. 4, pp. 857–914, 1998. View at: Google Scholar
15. T. F. Chan and T. P. Mathew, “Domain decomposition algorithms,” Acta Numerica, vol. 3, pp. 61–143, 1994. View at: Publisher Site | Google Scholar
16. C. Gong, W. Bao, G. Tang, B. Yang, and J. Liu, “An efficient parallel solution for Caputo fractional reaction-diffusion equation,” The Journal of Supercomputing, 2014. View at: Publisher Site | Google Scholar
17. C. Gong, J. Liu, L. Chi, H. Huang, J. Fang, and Z. Gong, “GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method,” Journal of Computational Physics, vol. 230, no. 15, pp. 6010–6022, 2011. View at: Publisher Site | Google Scholar
18. C. Gong, J. Liu, H. Huang, and Z. Gong, “Particle transport with unstructured grid on GPU,” Computer Physics Communications, vol. 183, no. 3, pp. 588–593, 2012. View at: Publisher Site | Google Scholar
19. X. Cao, Z. Mo, X. Liu, X. Xu, and A. Zhang, “Parallel implementation of fast multipole method based on JASMIN,” Science China Information Sciences, vol. 54, no. 4, pp. 757–766, 2011. View at: Publisher Site | Google Scholar
20. C. Gong, W. Bao, and G. Tang, “A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method,” Fractional Calculus and Applied Analysis, vol. 16, no. 3, pp. 654–669, 2013. View at: Google Scholar
21. C. Gong, W. Bao, G. Tang, Y. Jiang, and J. Liu, “A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method,” The Scientific World Journal, vol. 2014, Article ID 219580, 8 pages, 2014. View at: Publisher Site | Google Scholar
22. C. Gong, W. Bao, G. Tang, Y. Jiang, and J. Liu, “A domain decomposition method for time fractional reaction-diffusion equation,” The Scientific World Journal. In press. View at: Google Scholar
23. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
24. X. Cai, “Overlapping domain decomposition methods,” in Advanced Topics in Computational Partial Differential Equations, vol. 33 of Lecture Notes in Computational Science and Engineering, pp. 57–95, Springer, Berlin, Germany, 2003. View at: Publisher Site | Google Scholar
25. T. P. A. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, vol. 61 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, Germany, 2008.
26. X.-J. Yang, X.-K. Liao, K. Lu, Q.-F. Hu, J.-Q. Song, and J.-S. Su, “The TianHe-1A supercomputer: its hardware and software,” Journal of Computer Science and Technology, vol. 26, no. 3, pp. 344–351, 2011. View at: Publisher Site | Google Scholar

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions. 