- 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 2012 (2012), Article ID 596184, 25 pages
Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term
Department of Mathematics Education, Seoul National University, Seoul 151-748, Republic of Korea
Received 26 April 2012; Revised 6 July 2012; Accepted 17 July 2012
Academic Editor: Bashir Ahmad
Copyright © 2012 Y. J. Choi and S. K. Chung. 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.
- K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
- A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.
- A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002.
- R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.
- R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004.
- K. B. Oldham and J. Spanier, The Fractional Calculus, Dover Publications, New York, NY, USA, 2002.
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- B. Baeumer, M. Kovács, and M. M. Meerschaert, “Numerical solutions for fractional reaction-diffusion equations,” Computers & Mathematics with Applications, vol. 55, no. 10, pp. 2212–2226, 2008.
- W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008.
- Z. Q. Deng, V. P. Singh, and L. Bengtsson, “Numerical solution of fractional advection-dispersion equation,” Journal of Hydraulic Engineering, vol. 130, no. 5, pp. 422–431, 2004.
- Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533–1552, 2007.
- F. Liu, A. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, pp. 209–219, 2004.
- B. Baeumer, M. Kovács, and M. M. Meerschaert, “Fractional reproduction-dispersal equations and heavy tail dispersal kernels,” Bulletin of Mathematical Biology, vol. 69, no. 7, pp. 2281–2297, 2007.
- M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65–77, 2004.
- M. M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation,” Journal of Computational Physics, vol. 211, no. 1, pp. 249–261, 2006.
- M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006.
- V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks, “Numerical methods for the solution of partial differential equations of fractional order,” Journal of Computational Physics, vol. 192, no. 2, pp. 406–421, 2003.
- H. W. Choi, S. K. Chung, and Y. J. Lee, “Numerical solutions for space fractional dispersion equations with nonlinear source terms,” Bulletin of the Korean Mathematical Society, vol. 47, no. 6, pp. 1225–1234, 2010.
- W. Deng, “Numerical algorithm for the time fractional Fokker-Planck equation,” Journal of Computational Physics, vol. 227, no. 2, pp. 1510–1522, 2007.
- W. Deng and C. Li, “Finite difference methods and their physical constraints for the fractional Klein-Kramers equation,” Numerical Methods for Partial Differential Equations, vol. 27, no. 6, pp. 1561–1583, 2011.
- T. A. M. Langlands and B. I. Henry, “The accuracy and stability of an implicit solution method for the fractional diffusion equation,” Journal of Computational Physics, vol. 205, no. 2, pp. 719–736, 2005.
- F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 12–20, 2007.
- P. Zhuang, F. Liu, V. Anh, and I. Turner, “New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 46, no. 2, pp. 1079–1095, 2008.
- V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2006.
- J. P. Roop, “Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 243–268, 2006.
- 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.
- D. Braess, Finite Elements, Cambridge University Press, Cambridge, UK, 2nd edition, 2001.
- S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1994.
- Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
- S. Tang and R. O. Weber, “Numerical study of Fisher's equation by a Petrov-Galerkin finite element method,” Australian Mathematical Society B, vol. 33, no. 1, pp. 27–38, 1991.