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Fractional Calculus and Applied Analysis

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A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method

1College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China

2Science and Technology on Space Physics Libratory, Beijing, 10076, China

3School of Computer Science, National University of Defense Technology, Changsha, 410073, China

© 2013 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Fractional Calculus and Applied Analysis. Volume 16, Issue 3, Pages 654–669, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: 10.2478/s13540-013-0041-8, June 2013

Publication History

Published Online:


The fractional reaction-diffusion equations play an important role in dynamical systems. Indeed, it is time consuming to numerically solve differential fractional diffusion equations. In this paper, we present a parallel algorithm for the Riesz space fractional diffusion equation. The parallel algorithm, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel matrix vector multiplication and vector vector addition. As to the authors’ knowledge, this is the first parallel algorithm for the Riesz space fractional reaction-diffusion equation. The experimental results show that the parallel algorithm is as accurate as the serial algorithm. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.3-3.4 times faster than the serial algorithm on single CPU core. The parallel efficiency of 64 processes is up to 79.39% compared with 8 processes on a distributed memory cluster system.

MSC: Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11, 60G22

Keywords: Riesz fractional differential equation; parallel algorithm; Reaction-Dispersion Equation; MPI; explicit finite difference method

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