Numerical simulations of 2D fractional subdiffusion problems

Journal of Computational Physics

Volumn 229, Issue 18, 2010, Pages 6613-6622

  Brunner, Hermann ^a   Ling, Leevan ^a   Yamamoto, Masahiro ^b  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

Adaptive greedy algorithm; Collocation; Fractional differential equations; Geometric time grids; Kansa's method; Radial basis functions

Indexed keywords

DIFFERENTIAL EQUATIONS; NUMERICAL METHODS;

ADAPTIVE GREEDY ALGORITHMS; BASE FUNCTION; COLLOCATION; FRACTIONAL DIFFERENTIAL EQUATIONS; FRACTIONAL SUBDIFFUSION; GEOMETRIC TIME GRID; KANSA'S METHOD; RADIAL BASE FUNCTION; RADIAL BASIS; TIME GRID;

RADIAL BASIS FUNCTION NETWORKS;

EID: 77954142858     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.05.015     Document Type: Article

References (37)

1. Podlubny I. Fractional differential equations. Mathematics in Science and Engineering 1999, vol. 198. Academic Press Inc., San Diego, CA.
2. X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral approximation, Commun. Comput. Phys. To appear, doi:10.4208/cicp.020709.221209a.
3. Gorenflo R., Mainardi F., Moretti D., Paradisi P. Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 2002, 29(1-4):129-143.
4. Gorenflo R., Mainardi F., Vivoli A. Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals 2007, 34(1):87-103.
5. Metzler R., Klafter J. Boundary value problems for fractional diffusion equations. Phys. A 2000, 278(1-2):107-125.
6. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000, 339(1):77.
7. Al-Khaled K., Momani S. An approximate solution for a fractional diffusion-wave equation using the decomposition method. Appl. Math. Comput. 2005, 165(2):473-483.
8. Cuesta E., Lubich C., Palencia C. Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 2006, 75(254):673-696.
9. Cuesta E., Palencia C. A numerical method for an integro-differential equation with memory in Banach spaces: qualitative properties. SIAM J. Numer. Anal. 2003, 41(4):1232-1241.
10. Deng W. Numerical algorithm for the time-fractional Fokker-Planck equation. J. Comput. Phys. 2007, 227(2):1510-1522.
11. Langlands T.A.M., Henry B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 2005, 205(2):719-736.
12. Odibat Z., Momani S. Approximate solutions for boundary-value problems of time-fractional wave equation. Appl. Math. Comput. 2006, 181(1):767-774.
13. Odibat Z., Momani S. Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives. Phys. Lett. A 2007, 369:349-358.
14. Odibat Z., Momani S. Numerical methods for nonlinear partial differential equations of fractional-order. Appl. Math. Model. 2008, 32(1):28-39.
15. Shen S., Liu F., Anh V., Turner I. Detailed analysis of a conservative difference approximation for the time-fractional diffusion equation. J. Appl. Math. Comput. 2006, 22(3):1-19.
16. Sun Z., Wu X. A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 2006, 56(2):193-209.
17. Yuste S.B. Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 2006, 216(1):264-274.
18. Zhuang P., Liu F. Implicit difference approximation for the two-dimensional space-time fractional diffusion equation. J. Appl. Math. Comput. 2007, 25(1-2):269-282.
19. Diethelm K., Ford N.J., Freed A.D., Luchko Y. Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 2005, 194(6-8):743-773.
20. Valkó P.P., Abate J. Numerical inversion of 2D Laplace transforms applied to fractional diffusion equations. Appl. Numer. Math. 2005, 53(1):73-88.
21. Diethelm K., Freed A.D. An efficient algorithm for the evaluation of convolution integrals. Comput. Math. Appl. 2006, 51(1):51-72.
22. Liu F., Zhuang P., Anh V., Turner I. A fractional-order implicit difference approximation for the space-time fractional diffusion equation. ANZIAM J. 2005, 47:C48-C68.
23. Ford N.J., Simpson A.C. The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 2001, 26(4):333-346.
24. Tang T., Trummer M.R. Boundary layer resolving pseudospectral methods for singular perturbation problems. SIAM J. Sci. Comput. 1996, 17(2):430-438.
25. Ling L., Trummer M.R. Multiquadratic collocation method with integral formulation for boundary layer problems. Comput. Math. Appl. 2004, 48(5-6):927-941.
26. Ling L., Trummer M.R. Adaptive multiquadric collocation for boundary layer problems. J. Comp. Appl. Math. 2006, 188(2):256-282.
27. Diethelm K. An algorithm for the numerical solution of differential equations of fractional-order. Elec. Transact. Numer. Anal. 1997, 5:1-6.
28. Brunner H. Collocation methods for Volterra integral and related functional differential equations. Cambridge Monographs on Applied and Computational Mathematics 2004, vol. 15. Cambridge University Press, Cambridge.
29. Brunner H., Pedas A., Vainikko G. Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 2001, 39(3):957-982.
30. Ling L., Schaback R. An improved subspace selection algorithm for meshless collocation methods. Int. J. Numer. Methods Eng. 2009, 80(13):1623-1639.
31. Ling L., Opfer R., Schaback R. Results on meshless collocation techniques. Eng. Anal. Bound. Elem. 2006, 30(4):247-253.
32. Ling L., Schaback R. Stable and convergent unsymmetric meshless collocation methods. SIAM J. Numer. Anal. 2008, 46(3):1097-1115.
33. Kwok T.O., Ling L. On convergence of a least-squares Kansa's method for the modified Helmholtz equations. Adv. Appl. Math. Mech. 2009, 1(3):367-382.
34. Lee C.-F., Ling L., Schaback R. On convergent numerical algorithms for unsymmetric collocation. Adv. Comput. Math. 2009, 30(4):339-354.
35. Luchko Y. Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 2010, 59(5):1766-1772.
36. K. Sakamoto, M. Yamamoto, Initial value/boundary-value problems for fractional diffusion-wave equations and applications to some inverse problems, (preprint).
37. Seybold H.J., Hilfer R. Numerical results for the generalized Mittag-Leffler function. Fract. Calc. Appl. Anal. 2005, 8(2):127-140.