Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Computers and Mathematics with Applications

Volumn 67, Issue 9, 2014, Pages 1673-1681

  Chen, S ^a   Liu, F ^b   Burrage, K ^b,c  

^a QUANZHOU NORMAL UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Convergence; Fractional derivative of variable order; Fractional percolation equation; Implicit alternating direct method; Stability; Variable coefficients

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS; SOLUTE TRANSPORT; SOLVENTS; TWO DIMENSIONAL;

CONVERGENCE; DIRECT METHOD; FRACTIONAL PERCOLATION EQUATION; VARIABLE COEFFICIENTS; VARIABLE ORDER;

NUMERICAL METHODS;

EID: 84899637563     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2014.03.003     Document Type: Article

References (28)

1. H. Chou, B. Lee, and C. Chen The transient infiltration process for seepage flow from cracks Advances in Subsurface Flow and Transport: Eastern and Western Approaches III 2006
2. N. Petford, and M.A. Koenders Seepage flow and consolidation in a deforming porous medium Geophys. Res. Abstr. 5 2003 13329
3. Ji-Huan He Approximate analytical solution for seepage flow with fractional derivatives in porous media Comput. Methods Appl. Mech. Engrg. 167 1998 57 68
4. I. Podlubny Fractional Differential Equations 1999 Academic Press
5. K.P. Evans, and N. Jacob Feller semigroups obtained by variable order subordination Rev. Mat. Complut. 20 2 2007 293 307
6. R. Lin, F. Liu, V. Anh, and I. Turner Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation Appl. Math. Comput. 212 2004 435 445
7. M.D. Ruiz-Medina, V.V. Anh, and J.M. Angulo Fractional generalized random fields of variable order Stoch. Anal. Appl. 22 2 2004 775 799
8. P. Zhuang, F. Liu, V. Anh, and I. Turner Numerical methods for the variable-order fractional advection-diffusion with a nonlinear source term SIAM J. Numer. Anal. 47 3 2009 1760 1781
9. H. Sun, W. Chen, C. Li, and Y. Chen Fractional differential models for anomalous diffusion Physica A 389 2010 2719 2724
10. C. Li, and F. Zeng Finite difference methods for fractional differential equations Int. J. Bifurcation Chaos 22 4 2012 1230014
11. 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 J. Appl. Math. Comput. 191 2007 12 21
12. M.M. Meerschaert, H.P. Scheffler, and C. Tadjeran Finite difference methods for two-dimensional fractional dispersion equation J. Comput. Phys. 211 2006 249 261
13. F. Liu, V. Anh, I. Turner, and P. Zhuang Numerical simulation for solute transport in fractal porous media ANZIAM J. 45 E 2004 461 473
14. M.M. Meerschaert, and C. Tadjeran Finite difference approximations for two-sided space-fractional partial differential equations J. Appl. Numer. Math. 56 2006 80 90
15. Q. Liu, F. Liu, V. Anh, and I. Turner Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method J. Phys. Comp. 222 2007 57 70
16. F. Liu, P. Zhuang, and K. Burrage Numerical methods and analysis for a class of fractional advection-dispersion models Comput. Math. Appl. 64 2012 2990 3007
17. F. Liu, S. Chen, I. Turner, K. Burrage, and V. Anh Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term Cent. Eur. J. Phys 1110 2013 1221 1232
18. F. Liu, M.M. Meerschaert, R. McGough, P. Zhuang, and Q. Liu Numerical methods for solving the multi-term time fractional wave equations Fract. Calc. Appl. Anal. 16 1 2013 9 25
19. M.M. Meerschaert, and A. Sikorskii Stochastic Models for Fractional Calculus 2012 De Gruyter Berlin
20. Rd Numer. Methods Partial Differential Equations 22 2006 558 576
21. G.J. Fix, and J.P. Roop Least squares finite element solution of a fractional order two-point boundary value problem J. Comput. Math. Appl. 48 2004 1017 1033
22. S. Chen, F. Liu, P. Zhuang, and V. Anh Finite difference approximations for the Fokker-Planck equation Appl. Math. Model. 33 2009 256 273
23. S. Chen, F. Liu, I. Turner, and V. Anh A novel implicit finite difference methods for the one dimensional fractional percolation equation Numer. Algorithms 56 2011 517 535
24. F. Liu, C. Yang, and K. Burrage Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term J. Comput. Appl. Math. 231 2009 160 176
25. 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 J. Numer. Anal. 46 2 2008 1079 1095
26. Q. Liu, and F. Liu Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives Math. Numer. Sin. (China) 31 2 2009 179 194
27. C. Tadjeran, H.P. Scheffler, and M.M. Meerschaert A second order accurate numerical approximation for the fractional diffusion equation J. Comput. Phys. 213 1 2006 205 213
28. S. Chen, F. Liu, I. Turner, and V. Anh An implicit numerical method for the two-dimensional fractional percolation equation Appl. Math. Comput. 219 2013 4322 4331