TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 681707 10.1155/2014/681707 681707 Research Article A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation http://orcid.org/0000-0003-0349-1100 Gong Chunye 1,2,3 Bao Weimin 1,2 Tang Guojian 1 Jiang Yuewen 4 Liu Jie 3 Atangana A. Noutchie S. C. O. Secer A. 1 College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China nudt.edu.cn 2 Science and Technology on Space Physics Laboratory Beijing 100076 China 3 School of Computer Science National University of Defense Technology Changsha 410073 China nudt.edu.cn 4 Department of Engineering Science University of Oxford Oxford OX2 0ES UK ox.ac.uk 2014 1932014 2014 21 12 2013 20 02 2014 19 3 2014 2014 Copyright © 2014 Chunye Gong et al. 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.

The computational complexity of one-dimensional time fractional reaction-diffusion equation is O ( N 2 M ) compared with O ( N M ) 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: (1)    0 C D t α u ( x , t ) + μ u ( x , t ) u ( x , 0 ) = 2 u ( x , t ) x 2 + K f ( x , t ) , ( 0 < α < 1 ) , u ( x , 0 ) = g ( x ) , x [ 0 , x R ] , u ( 0 , t ) = u ( x R , t ) = 0 , t [ 0 , T ] on a finite domain 0 x x R and 0 t T . The μ > 0 and K are constants. If α 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 f ( t ) in the Caputo sense is defined as  (2)    0 C D t α f ( t ) = 1 Γ ( 1 - α ) 0 t f ( ξ ) ( t - ξ ) α d ξ , ( 0 < α < 1 ) .

If f ( t ) is continuous bounded derivatives in [ 0 , T ] for every T > a , we can get (3) 0 D t α f ( t ) = lim ξ 0 , n ξ = t ξ α i = 0 n ( - 1 ) i ( α i ) 0 D t α f ( t ) = f ( 0 ) t - α Γ ( 1 - α ) + 1 Γ ( 1 - α ) 0 t f ( ξ ) ( t - ξ ) α d ξ .

Define τ = T / N , h = x R / ( M + 1 ) , t n = n τ , and x i = 0 + i h for 0 n N ,  0 i M + 1 . Define u i n , f i n , and g i as the numerical approximation to u ( x i , t n ) , f ( x i , t n ) , and g ( x i ) . We can get  (4)    0 C D t α u ( x , t ) | x i t n = 1 τ Γ ( 1 - α )    0 C D t α u ( x , t ) | x i t n = × [ b 0 u i n - k = 1 n - 1 ( b n - k - 1 - b n - k ) u i k - b n - 1 u i 0 ]    0 C D t α u ( x , t ) | x i t n = + ( τ 2 - α ) , where 1 i M , n 1 , and (5) b l = τ 1 - α 1 - α [ ( l + 1 ) 1 - α - l 1 - α ] , l 0 .

By using center difference scheme for 2 u ( x , t ) / x 2 , we can get (6) 2 u ( x , t ) x 2 | x i t n = 1 h 2 ( u i + 1 n - 2 u i n + u i - 1 n ) + ( h 2 ) .

The implicit finite difference approximation for (1) is (7) 1 τ Γ ( 1 - α ) [ b 0 u i n - k = 1 n - 1 ( b n - k - 1 - b n - k ) u i k - b n - 1 u i 0 ] + μ u i n = u i + 1 n - 2 u i n + u i - 1 n h 2 + K f i n .

Define s = 2 / h 2 + b 0 τ - 1 / Γ ( 1 - α ) + μ , U n = ( u 1 n , u 2 n , , u M n ) T , F n = ( f 1 n , f 2 n , , f M n ) T , and r l as (8) r l = b l - b l + 1 s .

Equation (7) evolves as (9) A U n = k = 1 n - 1 r n - 1 - k U k + b n - 1 U 0 + K F n , where matrix A is a tridiagonal matrix, defined by (10) A M × M = ( s - 1 h 2 - 1 h 2 s - 1 h 2 · · · · · - 1 h 2 - 1 h 2 s )

Because μ > 0 , b 0 > 0 , the elements of matrix A satisfy | s | > | - 1 / h 2 | + | - 1 / h 2 | . This means that matrix A is strictly diagonally dominant.

2.2. Computational Challenge

In order to get U n , 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.

The constant vector multiplications are V = b n - 1 U 0 , V k = r n - 1 - k U k , and V ′′ = K F n .

The vector vector additions are V = V + k = 1 n - 1 V k + V ′′ .

After solving tridiagonal linear system A U n + 1 = V , we get U n + 1 .

The Thomas algorithm for tridiagonal systems needs 5 M multiplications and 3 M additions. The computational complexity of A U n = V is O ( M ) . The total computation of (9) is determined by k = 1 n - 1 r n - 1 - k U k , which means ( n - 1 ) M multiplications and ( n - 2 ) M additions for each time step; (11) n = 1 N ( 2 n M - 3 M ) = O ( N 2 M ) . The computational complexity of (1) is O ( N 2 M ) , while the computational complexity of classical one-dimensional reaction-diffusion equation is only O ( N M ) . 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 Ω = [ 0 , x R ] = p 0 , p 1 , , p M can be divided into two subdomains Ω a and Ω b . There are M + 1 grid points for Ω . Ω a = { p 0 , p 1 , , p m , p m + 1 } and Ω b = { p m , p m + 1 , , p M } , where 0 < m < M . The global physical boundary is defined in (1). Ω a (the right boundary of Ω a ) and Ω b (the left boundary of Ω b ) 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 U Ω i n is given as initial guess U Ω i n - 1 , where i = a , b . The better approximation of U Ω i n can be obtained iteratively. During each iteration, which is inside of a time step, the time fractional equation is solved in the subdomain Ω a , using the approximation of the previous iteration from Ω b on Ω a as follows: (12)    0 C D t α u ( x , t ) + μ u ( x , t ) u ( x , 0 ) = 2 u ( x , t ) x 2 + K f ( x , t ) , ( 0 < α < 1 ) , u ( x , 0 ) = g ( x ) , x [ 0 , x R ] , u ( x , 0 ) = u b , m + 1 , previous n , x on Ω a , u ( 0 , t n ) = 0 , where u b , m + 1 , previous n stands for the previous solution of grid point p m + 1 in subdomain Ω b . The better approximation U a , new n is obtained. U a , new n is defined as { u a , 1 , new n , , u a , m , new n } . U a , previous n is defined as { u a , 1 , previous n , , u a , m , previous n } . The definitions for U b , previous n and U a , new n are similar.

Then, we solve the time fractional equation within the subdomain of Ω b , using the approximation of the previous iteration from Ω a on Ω b as follows: (13)    0 C D t α u ( x , t ) + μ u ( x , t ) u ( x R , t n ) = 2 u ( x , t ) x 2 + K f ( x , t ) , ( 0 < α < 1 ) , u ( x , 0 ) = g ( x ) , x = x R u ( x , 0 ) = u a , m , previous n , x on Ω b u ( x R , t n ) = 0 , where u a , m , previous n stands for the previous solution of grid point p m in subdomain Ω a .

The two local time fractional equations in Ω a and Ω b are connected by the artificial boundary condition. The artificial boundary condition on the internal boundary Ω a of subdomain Ω a is provided by u b , m + 1 , previous n from subdomain Ω b , and vice versa. The approximation u a , m , previous n and u b , m + 1 , previous n 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 Ω can be decomposed into a set of P subdomains { Ω p } p = 1 P with Ω = p = 1 P Ω p . For time step n , Ω 1 has one global boundary x = 0 and one artificial inner boundary Ω 1 , b . Ω P has one global boundary x = x R and one artificial inner boundary Ω P , a . The Ω p ( 1 < p < P ) has two artificial inner boundaries Ω P , a and Ω p , b . Ω p Ω p + 1 Φ means that the neighboring subdomains have explicit overlap.

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

<bold>Algorithm 1: </bold>Domain decomposition algorithm for time fractional reaction-diffusion equation.

(1) [ h !]   input: M , P , x R , T , μ , K , ϵ et al.

Output: U

(2) I M / P

(3) Allocate memory space U I , P N , A I , 3 , P , F I , P N , V I , P 3 et al.

(4) Init matrices U , A , F , b , r et al.

(5) Declare local variables δ P , δ

(6) V 1 I , 1 P 1 3 0

(7) Get U 1 I , 1 P 0 with initial boundary.

(8) t o t a l I t e r a t i o n 0

(9) for n = 0   to   N - 1 step by 1do

(10)  for p = 1   to   P step by 1do

(11)    U 1 I , p n + 1 b n U 1 I , p 0 + K F 1 I , p n + 1

(12)    for k = 1   to   n step by 1do

(13)      U 1 I , p n + 1 U 1 I , p n + U 1 I , p k r n - k

(14)    δ 1.0

(15)    l o c a l I t e r a t i o n 0

(16)    while δ > ϵ do

(17)     for p = 1   to   P step by 1do

(18)       V 1 I , p 2 U 1 I , p n + 1

(19)       if p > 1 then

(20)       V 1 , p 2 V 1 , p 2 + V I , p - 1 1 / h 2

(21)       if p < P then

(22)       V I , p 2 V I , p 2 + V I , p + 1 1 / h 2

(23)     for p = 1   to   P step by 1do

(24)        solve A 1 I , 3 , p V 1 I , p 3 = V 1 I , p 2

(25)         δ p = max { | V i , p 1 - V i , p 3 | i = 1 I }

(26)         δ = max { δ , δ p }

(27)      l o c a l I t e r a t i o n l o c a l I t e r a t i o n + 1

(28)    t o t a l I t e r a t i o n t o t a l I t e r a t i o n + c o u n t

(29)    for p = 1   to P step by 1do

(30)      U 1 I , p n + 1 V 1 I , p 3

(31) Output the information

(32) Free memory space

In Algorithm 1, there are some fast algorithms to solve the tridiagonal matrix A 1 I , 3 , p V 1 I , p 3 = V 1 I , p 2 , such as Thomas algorithm. ϵ is a threshold, such as 1 0 - 6 . The signal l o c a l I t e r a t i o n 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): (14)    0 C D t α u ( x , t ) + μ u ( x , t ) u ( x , 0 ) = 2 u ( x , t ) x 2 + K f ( x , t ) , ( 0 < α < 1 ) , u ( x , 0 ) = 0 , x ( 0,2 ) , u ( 0 , t ) = u ( 2 , t ) = 0 with μ = 1 , K = 1 , and (15) f ( x , t ) = 2 Γ ( 2.3 ) x ( 2 - x ) t 1.3 + x ( 2 - x ) t 2 + 2 t 2 .

The exact solution of (14) is (16) u ( x , t ) = x ( 2 - x ) t 2 .

With ϵ = 1 0 - 6 , T = 1.0 , x R = 2.0 , and P = 3 , 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.

Comparing exact solution and DD algorithm.

h τ Δ
2/10 1/10 8.36 × 1 0 - 3
2/10 1/20 3.44 × 1 0 - 3
2/61 1/61 7.84 × 1 0 - 4
2/61 1/100 4.02 × 1 0 - 4
2/100 1/300 6.10 × 1 0 - 5

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 ϵ = 1 0 - 6 and P = 3 , 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.

Comparing Jacobi method and DDM.

h τ 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.

Kilicman A. Zhour Z. A. A. A. Kronecker operational matrices for fractional calculus and some applications Applied Mathematics and Computation 2007 187 1 250 265 2-s2.0-34247336205 10.1016/j.amc.2006.08.122 Zhai S. Feng X. Weng Z. New high-order compact adi algorithms for 3D nonlinear time-fractional convection-diffusion equation Mathematical Problems in Engineering 2013 2013 11 246025 10.1155/2013/246025 Atangana A. Alabaraoye E. Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4 + cells and attractor one-dimensional Keller-Segel equations Advances in Difference Equations 2013 2013 1, article 94 1 14 10.1186/1687-1847-2013-94 Liu Q. Liu F. Turner I. Anh V. Numerical simulation for the 3D seep age flow with fractional derivatives in porous media IMA Journal of Applied Mathematics 2009 74 2 201 229 10.1093/imamat/hxn044 Zhuang P. Liu F. Anh V. Turner I. Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process IMA Journal of Applied Mathematics 2009 74 5 645 667 2-s2.0-70350134071 10.1093/imamat/hxp015 Secer A. Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method Advances in Difference Equations 2012 2012 1, article 198 1 10 10.1186/1687-1847-2012-198 Atangana A. Secer A. The time-fractional coupled-Korteweg-de-Vries equations Abstract and Applied Analysis 2013 2013 8 947986 10.1155/2013/947986 Chen J. Liu F. Turner I. Anh V. The fundamental and numerical solutions of the Riesz space-fractional reaction-dispersion equation The ANZIAM Journal 2008 50 1 45 57 2-s2.0-77954685875 10.1017/S1446181108000333 Zhang X. Liu J. Wei L. Ma C. Finite element method for Grwünwald-Letnikov time-fractional partial differential equation Applicable Analysis 2013 92 10 1 12 10.1080/00036811.2012.718332 Ray S. S. 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 2008 202 2 544 549 2-s2.0-48349141442 10.1016/j.amc.2008.02.043 Li C. Zeng F. Liu F. Spectral approximations to the fractional integral and derivative Fractional Calculus and Applied Analysis 2012 15 3 383 406 10.2478/s13540-012-0028-x Chen J. An implicit approximation for the Caputo fractional reaction dispersion equation Journal of Xiamen University 2007 46 5 616 619 Schwarz H. A. Gesammelte Mathematische Abhandlungen 1972 AMS Bookstore Xu J. Zou J. Some nonoverlapping domain decomposition methods SIAM Review 1998 40 4 857 914 2-s2.0-0032290868 Chan T. F. Mathew T. P. Domain decomposition algorithms Acta Numerica 1994 3 61 143 10.1017/S0962492900002427 Gong C. Bao W. Tang G. Yang B. Liu J. An efficient parallel solution for Caputo fractional reaction-diffusion equation The Journal of Supercomputing 2014 10.1007/s11227-014-1123-z Gong C. Liu J. Chi L. Huang H. Fang J. Gong Z. GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method Journal of Computational Physics 2011 230 15 6010 6022 2-s2.0-79956143384 10.1016/j.jcp.2011.04.010 Gong C. Liu J. Huang H. Gong Z. Particle transport with unstructured grid on GPU Computer Physics Communications 2012 183 3 588 593 2-s2.0-84855469169 10.1016/j.cpc.2011.12.002 Cao X. Mo Z. Liu X. Xu X. Zhang A. Parallel implementation of fast multipole method based on JASMIN Science China Information Sciences 2011 54 4 757 766 2-s2.0-79955479276 10.1007/s11432-011-4181-3 Gong C. Bao W. Tang G. A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method Fractional Calculus and Applied Analysis 2013 16 3 654 669 Gong C. Bao W. Tang G. Jiang Y. Liu J. A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method The Scientific World Journal 2014 2014 8 219580 10. 1155/2014/219580 Gong C. Bao W. Tang G. Jiang Y. Liu J. A domain decomposition method for time fractional reaction-diffusion equation The Scientific World Journal. In press Podlubny I. Fractional Differential Equations 1999 San Diego, Calif, USA Academic Press Cai X. Overlapping domain decomposition methods Advanced Topics in Computational Partial Differential Equations 2003 33 Berlin, Germany Springer 57 95 Lecture Notes in Computational Science and Engineering 10.1007/978-3-642-18237-2_2 Mathew T. P. A. Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations 2008 61 Berlin, Germany Springer Lecture Notes in Computational Science and Engineering Yang X.-J. Liao X.-K. Lu K. Hu Q.-F. Song J.-Q. Su J.-S. The TianHe-1A supercomputer: its hardware and software Journal of Computer Science and Technology 2011 26 3 344 351 2-s2.0-79959969892 10.1007/s02011-011-1137-8