Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

Journal of Computational and Applied Mathematics

Volumn 231, Issue 1, 2009, Pages 160-176

  Liu, F ^a,c   Yang, C ^b   Burrage, K ^c,d  

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^b XIAMEN UNIVERSITY   (China)

^c UNIVERSITY OF QUEENSLAND   (Australia)

^d UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Energy method; Implicit difference method; Modified anomalous subdiffusion equation; Nonlinear source terms; Stability and convergence

Indexed keywords

ENERGY METHOD; IMPLICIT DIFFERENCE METHOD; MODIFIED ANOMALOUS SUBDIFFUSION EQUATION; NONLINEAR SOURCE TERMS; STABILITY AND CONVERGENCE;

DIFFERENCE EQUATIONS; FLOW FIELDS;

NONLINEAR EQUATIONS;

EID: 67349231192     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2009.02.013     Document Type: Article

References (47)

1. Sokolov I., and Klafter J. From diffusion to anomalous diffusion: A century after Einstein's Brownian motion. Chaos 15 (2005) 026103
2. Brown E., Wu E., Zipfel W., and Webb W. Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery. Biophys. J. 77 (1999) 2837-2849
3. Feder T., Brust-Mascher I., Slattery J., Baird B., and Webb W. Constrained diffusion or immobile fraction on cell surfaces: A new interpretation. Biophys. J. 70 (1996) 2767-2773
4. R. Ghosh, Mobility and clustering of individual low density lipoprotein receptor molecules on the surface of human skin fibroblasts, Ph.D. Thesis, Cornell University, Ithaca, NY, 1991
5. Ghosh R., and Webb W. Automated detection and tracking of individual and clustered cell surface low density lipoprotein receptor molecules. Biophys. J. 66 (1994) 1301-1318
6. Sheets E., Lee G., Simson R., and Jacobson K. Transient confinement of a glycosyl phosphatidylinositol-anchored protein in the plasma membrane. Biochemistry 36 (1997) 12449-12458
7. J. Slattery, Lateral mobility of FcRI on rat basophilic leukaemia cells as measured by single particle tracking using a novel bright fluorescent probe, Ph.D. Thesis, Cornell University, Ithaca, NY, 1991
8. Smith P., Morrison I., Wilson K., Fernandez N., and Cherry R. Anomalous diffusion of major histocompatability complex class I molecules on HeLa cells determined by single particle tracking. Biophys. J. 76 (1999) 3331-3344
9. Simson R., Yang B., Moore S., Doherty P., Walsh F., and Jacobson K. Structural mosaicism on the submicron scale in the plasma membrane. Biophys. J. 74 (1998) 297-308
10. Saxton M. Anomalous diffusion due to obstacles: A Monte Carlo Study. Biophys. J. 66 (1994) 394-401
11. Saxton M. Anomalous subdiffusion in fluorescence photobleaching recovery: A Monte Carlo study. Biophys. J. 81 (2001) 2226-2240
12. Saxton M. Anomalous diffusion due to binding: A Monte Carlo study. Biophys. J. 70 (1996) 1250-1262
13. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
14. Yuste S.B., and Lindenberg K. Subdiffusion limited reactions. Chem. Phys. 284 (2002) 169-180
15. Yuste S.B., Acedo L., and Lindenberg K. Reaction front in an A + B → C reaction-subdiffusion process. Phys. Rev. E 69 (2004) 036126
16. Seki K., Wojcik M., and Tachiya M. Fractional reaction-diffusion equation. J. Chem. Phys. 119 (2003) 2165-2174
17. Tan W.C., and Masuoka T. Stokes' first problem for a second grade fluid in a porous half-space with heated boundary. Internat. J. Non-Linear Mech. 40 (2005) 515-522
18. Chen C., Liu F., and Anh V. A Fourier method and an extrapolation technique for Stokes' first problem for a heated generalized second grade fluid with fractional derivative. J. Comp. Appl. Math. 223 2 (2009) 777-789
19. Zaslavsky G. Fractional kinetic equation for Hamiltonian chaos Chaotic advection tracer dynamics and turbulent dispersion. Physica D 76 (1994) 110-122
20. Metzler R., and Klafter J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37 (2004) R161-R208
21. Del-Castillo-Negrete D., Carreras B.A., and Lynch V.E. Front dynamics in reaction-diffusion systems with levy flights: A fractional diffusion approach. Phys. Rev. Lett. 91 (2003) 018302
22. Henry B., and Wearne S. Fractional reaction-diffusion. Physica A 276 (2000) 448-455
23. Seki K., Wojcik M., and Tachiya M. Fractional reaction-diffusion equation. J. Chem. Phys. 119 4 (2003) 2165-2170
24. Seki K., Wojcik M., and Tachiya M. Recombination kinetics in subdiffusive media. J. Chem. Phys. 119 14 (2003) 7525-7533
25. Chechkin A.V., Gorenflo R., Sokolov I.M., and Gonchar V.Yu. Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6 3 (2003) 259-279
26. Langlands T.A.M. Solution of a modified fractional diffusion equation. Physica A 367 (2006) 136-144
27. Sokolov I.M., Chechkin A.V., and Klafter J. Distributed-order fractional kinetics. Acta. Phys. Polon. B 35 (2004) 1323
28. Mainardi F., Raberto M., Gorenflo R., and Scalas E. Fractional calculus and continuous-time finance II: The waiting-time distribution. Physica A 287 (2000) 468-481
29. Raberto M., Scalas E., and Mainardi F. Waiting-times and returns in high frequency financial data: An empirical study. Physica A 314 (2002) 749-755
30. Scalas E., Gorenflo R., and Mainardi F. Fractional calculus and continuous-time finance. Physica A 284 (2000) 376-384
31. E. Scalas, R. Gorenflo, F. Mainardi, M. Mantelli, M. Raberto, Anomalous waiting times in high-frequency financial data, 2003. http://arxiv.org/abs/cond-mat/0310305i
32. Scalas E. Five years of continuous-time random walks in econophysics. In: Namatame A. (Ed). Proceedings of WEHIA 2004, Kyoto (2004)
33. Scalas E., Gorenflo R., Mainardi F., and Meerschaert M.M. Speculative option valuation and the fractional diffusion equation. In: Sabatier J., and Tenreiro Machado J. (Eds). Proceedings of the IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux (2004) 234-238
34. Scalas E. The application of continuous-time random walks in finance and economics. Physica A 362 2 (2006) 225-239
35. Masoliver J., and Montero M. Continuous-time random-walk model for financial distributions. Phys. Rev. E 67 (2003) 021112
36. J. Masoliver, M. Montero, J. Perello, G. Weiss, The CTRW in finance: Direct and inverse problems, 2003. http://arxiv.org/abs/cond-mat/0308017i
37. Masoliver J., Montero M., Perello J., and Weiss G.H. The continuous time random walk formalism in financial markets. J. Econ. Behav. Org. 61 (2006) 577-591
38. Liu F., Anh V., and Turner I. Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166 (2004) 209-219
39. Meerschaert M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2004) 65-77
40. 2. J. Comput. Appl. Math. 193 1 (2006) 243-268
41. Liu F., Zhuang P., Anh V., Turner I., and Burrage K. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. J. Appl. Math. Comput. 191 (2007) 12-21
42. Yu Q., Liu F., Anh V., and Turner I. Solving linear and nonlinear space-time fractional reaction-diffusion equations by Adomian decomposition method. Internat. J. Numer. Methods Eng. 74 (2008) 138-158
43. Yuste S.B., and 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 (2005) 1862-1874
44. Langlands T.A.M., and Henry B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205 (2005) 719-736
45. Chen C.-M., Liu F., Turner I., and Anh V. Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227 (2007) 886-897
46. Zhuang P., Liu F., Anh V., and Turner I. New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46 2 (2008) 1079-1095
47. B. Baeumer, M.K. Mark, M. Meerschaert, Fractional reaction-diffusion equation for species growth and dispersal, J. Math. Biol. (2007) (in press). http://www.maths.otago.ac.nz/~mcubed/JMBseed.pdf