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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 820162, 7 pages
Solving the Caputo Fractional Reaction-Diffusion Equation on GPU
1School of Computer Science, National University of Defense Technology, Changsha 410073, China
2Science and Technology on Space Physics Laboratory, Beijing 100076, China
3College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China
4Department of Engineering Science, University of Oxford, Oxford OX2 0ES, UK
Received 1 April 2014; Accepted 27 May 2014; Published 17 June 2014
Academic Editor: Dorian Popa
Copyright © 2014 Jie Liu 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.
- F. Huang and F. Liu, “The time fractional diffusion equation and the advection-dispersion equation,” The ANZIAM Journal, vol. 46, no. 3, pp. 317–330, 2005.
- S. Chen and X. Jiang, “Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus,” Physica A: Statistical Mechanics and Its Applications, vol. 391, no. 15, pp. 3865–3874, 2012.
- R. K. Pandey, O. P. Singh, and V. K. Baranwal, “An analytic algorithm for the space-time fractional advection-dispersion equation,” Computer Physics Communications, vol. 182, no. 5, pp. 1134–1144, 2011.
- R. W. Ibrahim, “Ulam-hyers stability for cauchy fractional differential equation in the unit disk,” Abstract and Applied Analysis, vol. 2012, Article ID 613270, 10 pages, 2012.
- J. Brzdęk, N. Brillouët-Belluot, K. Ciepliński, and B. Xu, “Ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 329702, 2 pages, 2012.
- N. Brillouët-Belluot, J. Brzdk, and K. Ciepliński, “On some recent developments in ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012.
- Q. Liu, F. Liu, I. Turner, and V. Anh, “Numerical simulation for the 3D seepage flow with fractional derivatives in porous media,” IMA Journal of Applied Mathematics, vol. 74, no. 2, pp. 201–229, 2009.
- A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations with the dirichlet condition,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 696179, 15 pages, 2012.
- A. Ashyralyev and F. Dal, “Finite difference and iteration methods for fractional hyperbolic partial differential equations with the neumann condition,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 434976, 15 pages, 2012.
- 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.
- J. H. Chen, “An implicit approximation for the caputo fractional reaction-dispersion equation,” Journal of Xiamen University (Natural Science), vol. 46, no. 5, pp. 616–619, 2007 (Chinese).
- B. I. Henry and S. L. Wearne, “Fractional reaction-diffusion,” Physica A: Statistical Mechanics and Its Applications, vol. 276, no. 3-4, pp. 448–455, 2000.
- V. Gafiychuk, B. Datsko, and V. Meleshko, “Mathematical modeling of time fractional reaction-diffusion systems,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 215–225, 2008.
- H. J. Haubold, A. M. Mathai, and R. K. Saxena, “Further solutions of fractional reaction-diffusion equations in terms of the h-function,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1311–1316, 2011.
- S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “On the solutions of time-fractional reaction-diffusion equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3847–3854, 2010.
- R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Solution of generalized fractional reaction-diffusion equations,” Astrophysics and Space Science, vol. 305, no. 3, pp. 305–313, 2006.
- S. J. Pennycook, S. D. Hammond, G. R. Mudalige, S. A. Wright, and S. A. Jarvis, “On the acceleration of wavefront applications using distributed many-core architectures,” The Computer Journal, vol. 55, no. 2, pp. 138–153, 2012.
- Z. Mo, A. Zhang, X. Cao et al., “Jasmin: a parallel software infrastructure for scientific computing,” Frontiers of Computer Science in China, vol. 4, no. 4, pp. 480–488, 2010.
- R. Zhang, “A three-stage optimization algorithm for the stochastic parallel machine scheduling problem with adjustable production rates,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 280560, 15 pages, 2013.
- 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.
- Y.-X. Wang, L.-L. Zhang, W. Liu et al., “Efficient parallel implementation of large scale 3D structured grid CFD applications on the Tianhe-1A supercomputer,” Computers and Fluids, vol. 80, pp. 244–250, 2013.
- C. Xu, X. Deng, L. Zhang et al., “Parallelizing a high-order CFD software for 3D, multi-block, structural grids on the TianHe-1A supercomputer,” in Supercomputing, J. Kunkel, T. Ludwig, and H. Meuer, Eds., vol. 7905 of Lecture Notes in Computer Science, pp. 26–39, Springer, Heidelberg, Germany, 2013.
- NVIDIA Corporation, CUDA Programming Guide Version 3.1, 2010.
- 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.
- 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.
- Q. Wu, C. Yang, T. Tang, and L. Xiao, “Exploiting hierarchy parallelism for molecular dynamics on a petascale heterogeneous system,” Journal of Parallel and Distributed Computing, vol. 73, no. 12, pp. 1592–1604, 2013.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- Y. Xu and Z. He, “The short memory principle for solving abel differential equation of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4796–4805, 2011.
- 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.
- K. Diethelm, “An efficient parallel algorithm for the numerical solution of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 14, no. 3, pp. 475–490, 2011.
- 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.
- 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.
- 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, vol. 2014, Article ID 681707, 5 pages, 2014.
- S. Williams, L. Oliker, R. Vuduc, J. Shalf, K. Yelick, and J. Demmel, “Optimization of sparse matrix-vector multiplication on emerging multicore platforms,” Parallel Computing, vol. 35, no. 3, pp. 178–194, 2009.
- A. Fošner, “On the generalized Hyers-Ulam stability of module left -derivations,” Aequationes Mathematicae, vol. 84, no. 1-2, pp. 91–98, 2012.
- D. Popa, “Hyers-Ulam stability of the linear recurrence with constant coefficients,” Advances in Difference Equations, no. 2, pp. 101–107, 2005.
- R. P. Agarwal, B. Xu, and W. Zhang, “Stability of functional equations in single variable,” Journal of Mathematical Analysis and Applications, vol. 288, no. 2, pp. 852–869, 2003.