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Linked References

1. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, vol. 18 of Springer Series in Synergetics, Springer, Berlin, Germany, 2nd edition, 1989. View at MathSciNet
2. 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. View at Publisher · View at Google Scholar
3. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Lévy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413–1423, 2000. View at Publisher · View at Google Scholar
4. J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Physics Reports, vol. 195, no. 4-5, pp. 127–293, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
5. R. Metzler and J. Klafter, “Fractional Fokker-Planck equation: dispersive transport in an external force field,” Journal of Molecular Liquids, vol. 86, no. 1, pp. 219–228, 2000. View at Publisher · View at Google Scholar
6. G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
7. F. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209–219, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
8. R. Gorenflo and F. Mainardi, “Random walk models for space-fractional diffusion processes,” Fractional Calculus & Applied Analysis, vol. 1, no. 2, pp. 167–191, 1998. View at Zentralblatt MATH · View at MathSciNet
9. C. W. Chow and K. L. Liu, “Fokker-Planck equation and subdiffusive fractional Fokker-Planck equation of bistable systems with sinks,” Physica A, vol. 341, no. 1–4, pp. 87–106, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
11. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
12. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
13. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
14. C.-M. Chen, F. Liu, and V. Anh, “A Fourier method and an extrapolation technique for Stokes' first problem for a heated generalized second grade fluid with fractional derivative,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 777–789, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
15. S. Chen, F. Liu, P. Zhuang, and V. Anh, “Finite difference approximations for the fractional Fokker-Planck equation,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 256–273, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
16. 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, New York, NY, USA, 1999. View at MathSciNet
17. A. Schot, M. K. Lenzi, L. R. Evangelista, L. C. Malacarne, R. S. Mendes, and E. K. Lenzi, “Fractional diffusion equation with an absorbent term and a linear external force: exact solution,” Physics Letters A, vol. 366, no. 4-5, pp. 346–350, 2007. View at Publisher · View at Google Scholar
18. E. K. Lenzi, L. C. Malacarne, R. S. Mendes, and I. T. Pedron, “Anomalous diffusion, nonlinear fractional Fokker-Planck equation and solutions,” Physica A, vol. 319, pp. 245–252, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
19. P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical treatment for the fractional Fokker-Planck equation,” The ANZIAM Journal, vol. 48, pp. C759–C774, 2006-2007. View at MathSciNet
20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
21. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 11 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974. View at MathSciNet
22. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
23. R. Lin and F. Liu, “Fractional high order methods for the nonlinear fractional ordinary differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 856–869, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
24. 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. View at Publisher · View at Google Scholar · View at MathSciNet
25. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
26. 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, 2008-2009. View at Publisher · View at Google Scholar · View at MathSciNet
27. J. I. Ramos, “Damping characteristics of finite difference methods for one-dimensional reaction-diffusion equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 607–609, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
28. D. del-Castillo-Negrete, B. A. Carreras, and V. E. Lynch, “Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach,” Physical Review Letters, vol. 91, no. 1, Article ID 018302, 4 pages, 2003. View at Publisher · View at Google Scholar
29. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
30. 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. View at Publisher · View at Google Scholar · View at PubMed · View at MathSciNet