- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 493406, 15 pages
Numerical Algorithms for the Fractional Diffusion-Wave Equation with Reaction Term
Department of Mathematics, Shanghai University, Shanghai 200444, China
Received 24 April 2013; Revised 24 June 2013; Accepted 24 June 2013
Academic Editor: Juan J. Trujillo
Copyright © 2013 Hengfei Ding and Changpin Li. 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.
- E. Barkai, R. Metzler, and J. Klafter, “From continuous time random walks to the fractional Fokker-Planck equation,” Physical Review E, vol. 61, no. 1, pp. 132–138, 2000.
- D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., Springer, New York, NY, USA, 1997.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- A. I. Saichev and G. M. Zaslavsky, “Fractional kinetic equations: solutions and applications,” Chaos, vol. 7, no. 4, pp. 753–764, 1997.
- S. B. Yuste, L. Acedo, and K. Lindenberg, “Reaction front in an reactionsubdiffusion process,” Physical Review E, vol. 69, no. 3, part 2, Article ID 036126, 2004.
- S. B. Yuste and K. Lindenberg, “Subdiffusion-limited reactions,” Physical Review Letters, vol. 87, no. 11, Article ID 118301, 2001.
- M. Cui, “Compact finite difference method for the fractional diffusion equation,” Journal of Computational Physics, vol. 228, no. 20, pp. 7792–7804, 2009.
- C.-M. Chen, F. Liu, I. Turner, and V. Anh, “A Fourier method for the fractional diffusion equation describing sub-diffusion,” Journal of Computational Physics, vol. 227, no. 2, pp. 886–897, 2007.
- C. P. Li, A. Chen, and J. J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011.
- F. Liu, C. Yang, and K. Burrage, “Numerical method and analytic technique of the modified anomalous subdiffusion equation with a nonlinear source term,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 160–176, 2009.
- E. Sousa, “Finite difference approximations for a fractional advection diffusion problem,” Journal of Computational Physics, vol. 228, no. 11, pp. 4038–4054, 2009.
- E. Sousa, “Numerical approximations for fractional diffusion equations via splines,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 938–944, 2011.
- H. Wang and K. Wang, “An alternating-direction finite difference method for two-dimensional fractional diffusion equations,” Journal of Computational Physics, vol. 230, no. 21, pp. 7830–7839, 2011.
- S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862–1874, 2005.
- V. J. Ervin, N. Heuer, and J. P. Roop, “Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 45, no. 2, pp. 572–591, 2007.
- G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications, vol. 48, no. 7-8, pp. 1017–1033, 2004.
- J. P. Roop, “Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in ,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 243–268, 2006.
- C. P. Li, Z. G. Zhao, and Y. Q. Chen, “Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 855–875, 2011.
- H. Zhang, F. Liu, and V. Anh, “Galerkin finite element approximation of symmetric space-fractional partial differential equations,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2534–2545, 2010.
- Y. Y. Zheng, C. P. Li, and Z. G. Zhao, “A fully discrete discontinuous Galerkin method for nonlinear fractional Fokker-Planck equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 279038, 26 pages, 2010.
- Q. Yang, F. Liu, and I. Turner, “Numerical methods for fractional partial differential equations with Riesz space fractional derivatives,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 200–218, 2010.
- Q. Yang, F. Liu, and I. Turner, “Computationally efficient numerical methods for time and space fractional Fokker-Planck equations,” Physica Scripta, vol. 2009, Article ID 014026, 2009.
- C. P. Li, F. H. 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.
- X. Li and C. Xu, “A space-time spectral method for the time fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 2108–2131, 2009.
- H. F. Ding and C. P. Li, “Mixed spline function method for reaction-subdiusion equations,” Journal of Computational Physics, vol. 242, pp. 103–123, 2013.
- F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons and Fractals, vol. 7, no. 9, pp. 1461–1477, 1996.
- J. Q. Murillo and S. B. Yuste, “An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form,” Journal of Computational and Nonlinear Dynamics, vol. 6, no. 2, 6 pages, 2011.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- D. K. Salkuyeh, “On the finite difference approximation to the convection-diffusion equation,” Applied Mathematics and Computation, vol. 179, no. 1, pp. 79–86, 2006.
- M. Li, T. Tang, and B. Fornberg, “A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations,” International Journal for Numerical Methods in Fluids, vol. 20, no. 10, pp. 1137–1151, 1995.
- L. Su, W. Wang, and Q. Xu, “Finite difference methods for fractional dispersion equations,” Applied Mathematics and Computation, vol. 216, no. 11, pp. 3329–3334, 2010.