A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation

Journal of Applied Mathematics and Computing

Volumn 30, Issue 1-2, 2009, Pages 219-236

  Chen, Chang Ming ^a   Liu, F ^b  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Convergence; Fourier analysis; Fractional advection diffusion equation; Solvability; Stability; Three dimension

Indexed keywords

EID: 63949084174     PISSN: 15985865     EISSN: None     Source Type: Journal    
DOI: 10.1007/s12190-008-0168-7     Document Type: Article

References (39)

1. 1. V.V. Anh N.N. Leonenko 2001 Spectral analysis of fractional kinetic equations with random data J. Stat. Phys. 104 1349 1387 1034.82044 10.1023/A:1010474332598 1859007 Anh, V.V., Leonenko, N.N.: Spectral analysis of fractional kinetic equations with random data. J. Stat. Phys. 104, 1349–1387 (2001)
2. 2. B. Baeumer M.M. Meerschaert D.A. Benson S.W. Wheatsraft 2001 Subordinated advection-dispersion equation for contaminant transport Water Resour. Res. 37 6 1543 1550 10.1029/2000WR900409 Baeumer, B., Meerschaert, M.M., Benson, D.A., Wheatsraft, S.W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37 (6), 1543–1550 (2001)
3. 3. V. Balakrishnan 1985 Anomalous diffusion in one dimension Physica A 132 569 580 0654.60065 10.1016/0378-4371(85)90028-7 Balakrishnan, V.: Anomalous diffusion in one dimension. Physica A 132, 569–580 (1985)
4. 4. E. Barkai R.J. Silbey 2000 Fractional Kramers equation J. Phys. Chem. B 104 3866 3874 10.1021/jp993491m Barkai, E., Silbey, R.J.: Fractional Kramers equation. J. Phys. Chem. B 104, 3866–3874 (2000)
5. 5. D.A. Benson S.W. Wheatcraft M.M. Meerschaert 2000 Application of a fractional advection-dispersion equation Water Resour. Res. 36 6 1403 1412 10.1029/2000WR900031 Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (6), 1403–1412 (2000)
6. 6. E. Brown E. Wu W. Zipfel W. Webb 1999 Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery Biophys. J. 77 2837 2849 10.1016/S0006-3495(99)77115-8 Brown, E., Wu, E., Zipfel, W., Webb, W.: Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery. Biophys. J. 77, 2837–2849 (1999)
7. 7. C.-M. Chen F. Liu K. Burrage 2007 A Fourier method for the fractional diffusion equation describing sub-diffusion J. Comput. Phys. 227 886 897 1165.65053 10.1016/j.jcp.2007.05.012 2442379 Chen, C.-M., Liu, F., Burrage, K.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)
8. 8. C.-M. Chen F. Liu I. Turner V. Anh 2007 Implicit difference approximation of the Galilei invariant fractional advection diffusion equation ANZIAM J. 48 C775 C789 2366328 (CTAC2006) Chen, C.-M., Liu, F., Turner, I., Anh, V.: Implicit difference approximation of the Galilei invariant fractional advection diffusion equation. ANZIAM J. 48, C775–C789 (2007) (CTAC2006)
9. 9. C.-M. Chen F. Liu K. Burrage 2008 Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation Appl. Math. Comput. 198 754 769 1144.65057 10.1016/j.amc.2007.09.020 2405817 Chen, C.-M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)
10. 10. A. Compte 1997 Continuous time random walks on moving fluids Phys. Rev. E 55 6821 6831 10.1103/PhysRevE.55.6821 Compte, A.: Continuous time random walks on moving fluids. Phys. Rev. E 55, 6821–6831 (1997)
11. 11. A. Compte M.O. Caceres 1998 Fractional dynamics in random velocity fields Phys. Rev. Lett. 81 3140 3143 10.1103/PhysRevLett.81.3140 Compte, A., Caceres, M.O.: Fractional dynamics in random velocity fields. Phys. Rev. Lett. 81, 3140–3143 (1998)
12. 12. J.H. Cushman T.R. Cinn 2000 Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux Water Resour. Res. 36 3763 3766 10.1029/2000WR900261 Cushman, J.H., Cinn, T.R.: Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux. Water Resour. Res. 36, 3763–3766 (2000)
13. 13. M. Giona H.E. Roman 1992 Fractional diffusion equation for transport phenomena in random media Physica A 185 87 97 10.1016/0378-4371(92)90441-R Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in random media. Physica A 185, 87–97 (1992)
14. 14. R. Gorenflo A. Iskenderov Yu. Luchko 2000 Mapping between solutions of fractional diffusion-wave equations Fract. Calc. Appl. Math. 3 1 75 86 1033.35161 1743407 Gorenflo, R., Iskenderov, A., Luchko, Yu.: Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Math. 3 (1), 75–86 (2000)
15. 15. R. Gorenflo F. Mainardi D. Moretti G. Pagnini P. Paradisi 2002 Discrete random walk models for space-time fractional diffusion Chem. Phys. 284 521 541 10.1016/S0301-0104(02)00714-0 Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284, 521–541 (2002)
16. 16. B. Henry S. Wearne 2000 Fractional reaction-diffusion Physica A 276 448 455 10.1016/S0378-4371(99)00469-0 1780373 Henry, B., Wearne, S.: Fractional reaction-diffusion. Physica A 276, 448–455 (2000)
17. 17. B. Henry S. Wearne 2002 Existence of turning instabilities in a two-species fractional reaction-diffusion system SIAM J. Appl. Math. 62 3 870 887 1103.35047 10.1137/S0036139900375227 1897726 Henry, B., Wearne, S.: Existence of turning instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math. 62 (3), 870–887 (2002)
18. 18. F. Huang F. Liu 2005 The time fractional diffusion and advection-dispersion equation ANZIAM J. 46 317 330 1072.35218 10.1017/S1446181100008282 2124926 Huang, F., Liu, F.: The time fractional diffusion and advection-dispersion equation. ANZIAM J. 46, 317–330 (2005)
19. 19. T.A.M. Langlands B.I. Henry 2005 The accuracy and stability of an implicit solution method for the fractional diffusion equation J. Comput. Phys. 205 719 736 1072.65123 10.1016/j.jcp.2004.11.025 2135000 Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)
20. 20. F. Liu V. Anh I. Turner 2004 Numerical solution of the space fractional Fokker-Planck equation J. Comput. Appl. Math. 166 209 219 1036.82019 10.1016/j.cam.2003.09.028 2057973 Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)
21. 21. F. Liu V. Anh I. Turner P. Zhuang 2004 Numerical simulation for solute transport in fractal porous media ANZIAM J. 45 5 461 473 2180945 Liu, F., Anh, V., Turner, I., Zhuang, P.: Numerical simulation for solute transport in fractal porous media. ANZIAM J. 45 (5), 461–473 (2004)
22. 22. F. Liu S. Shen V. Anh I. Turner 2005 Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation ANZIAM J. 46 E 488 504 2182183 Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46 (E), 488–504 (2005)
23. 23. Q. Liu F. Liu I. Turner V. Anh 2007 Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method J. Comput. Phys. 222 57 70 1112.65006 10.1016/j.jcp.2006.06.005 2298036 Liu, Q., Liu, F., Turner, I., Anh, V.: Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method. J. Comput. Phys. 222, 57–70 (2007)
24. 24. F. Liu P. Zhuang V. Anh I. Turner K. Burrage 2007 Stability and convergence of difference methods for the space-time fractional advection-diffusion equation Appl. Math. Comput. 191 12 20 10.1016/j.amc.2006.08.162 2385498 Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)
25. 25. Ch. Lubich 1986 Discretized fractional calculus SIAM J. Math. Anal. 17 704 719 0624.65015 10.1137/0517050 838249 Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)
26. 26. F. Mainardi 1996 The fundamental solutions for the fractional diffusion-wave equation Appl. Math. Lett. 9 6 23 28 0879.35036 10.1016/0893-9659(96)00089-4 1419811 Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9 (6), 23–28 (1996)
27. 27. M. Meerschaert C. Tadjeran 2004 Finite difference approximations for fractional advection-dispersion flow equations J. Comput. Appl. Math. 172 65 77 1126.76346 10.1016/j.cam.2004.01.033 2091131 Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
28. 28. R. Metzler J. Klafter 2000 Boundary value problems for fractional diffusion equations Physica A 278 107 125 10.1016/S0378-4371(99)00503-8 1763650 Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000)
29. 29. R. Metzler J. Klafter 2000 The random walk’s guide to anomalous diffusion: a fractional dynamics approach Phys. Rep. 339 1 77 0984.82032 10.1016/S0370-1573(00)00070-3 1809268 Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
30. 30. R. Metzler J. Klafter I.M. Sokolov 1998 Anomalous transport in external fields: Continuous time random walksand fractional diffusion equations extended Phys. Rev. E 58 1621 1633 10.1103/PhysRevE.58.1621 Metzler, R., Klafter, J., Sokolov, I.M.: Anomalous transport in external fields: Continuous time random walksand fractional diffusion equations extended. Phys. Rev. E 58, 1621–1633 (1998)
31. 31. R. Metzler E. Barkai J. Klafter 1999 Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach Phys. Rev. Lett. 82 3563 3567 10.1103/PhysRevLett.82.3563 Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999)
32. 32. W.R. Schneider W. Wyss 1989 Fractional diffusion and wave equations J. Math. Phys. 30 134 144 0692.45004 10.1063/1.528578 974464 Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)
33. 33. K. Seki M. Wojcik M. Tachiya 2003 Fractional reaction-diffusion equation J. Chem. Phys. 119 2165 2174 10.1063/1.1587126 Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165–2174 (2003)
34. 34. S. Shen F. Liu 2005 Error analysis of an explicit finite difference approximation for the space fractional diffusion ANZIAM J. 46 E 871 887 2181905 Shen, S., Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion. ANZIAM J. 46 (E), 871–887 (2005)
35. 35. P. Smith I. Morrison K. Wilson N. Fernandez R. Cherry 1999 Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking Biophys. J. 76 3331 3344 10.1016/S0006-3495(99)77486-2 Smith, P., Morrison, I., Wilson, K., Fernandez, N., Cherry, R.: Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking. Biophys. J. 76, 3331–3344 (1999)
36. 36. W. Wyss 1986 Fractional diffusion equation J. Math. Phys. 27 2782 2785 0632.35031 10.1063/1.527251 861345 Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)
37. 37. S.B. Yuste L. Acedo 2005 An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations SIAM J. Numer. Anal. 42 5 1862 1874 1119.65379 10.1137/030602666 2139227 Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42 (5), 1862–1874 (2005)
38. 38. P. Zhang F. Liu 2006 Implicit difference approximation for the time fractional diffusion equation J. Appl. Math. Comput. 22 3 87 99 10.1007/BF02832039 2323899 Zhang, P., Liu, F.: Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 (3), 87–99 (2006)
39. 39. P. Zhuang F. Liu V. Anh I. Turner 2008 New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation SIAM J. Numer. Anal. 46 2 1079 1095 10.1137/060673114 2383224 05549714 Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46 (2), 1079–1095 (2008)