Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

SIAM Journal on Numerical Analysis

Volumn 47, Issue 3, 2009, Pages 1760-1781

  Zhuang, P ^a   Liu, F ^b,c   Anh, V ^b   Turner, I ^b  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c SOUTH CHINA UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Extrapolation method; Finite difference methods; Fractional derivative of variable order; Method of lines; Nonlinear fractional advection diffusion equation; Stability and convergence

Indexed keywords

ADVECTION; DIFFUSION; EXTRAPOLATION; FINITE DIFFERENCE METHOD; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; TRANSFER MATRIX METHOD;

ADVECTION-DIFFUSION EQUATION; EXTRAPOLATION METHODS; METHOD OF LINES; STABILITY AND CONVERGENCE; VARIABLE ORDER;

NUMERICAL METHODS;

EID: 84907893973     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/080730597     Document Type: Article

References (34)

1. A. Benassi, S. Jaffard, and D. Roux, Gaussian processes and pseudodifferential elliptic operators, Rev. Mat. Iberoamericana, 13 (1997), pp. 19-89.
2. K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland, New York, 1989.
3. D. del Castillo-Negrete, B. A. Carreras, and V. E. Lynch, Front dynamics in reactiondiffusion systems with Lévy flghts: A fractional diffusion approach, Phys. Rev. Lett., 91 (2003), 018302.
4. K. P. Evans and N. Jacob, Feller semigroups obtained by variable order subordination, Rev. Mat. Comput., 20 (2007), pp. 293-307.
5. W. Feller, On a generalization of Marcel Riesz' potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matmatiska Seminarium, (1952), pp 73-81.
6. G. Huang, Q. Huang, and H. Zhan, Evidence of one-dimensional scale-dependent fractional advection-dispersion, J. Contaminant Hydrology, 85 (2006), pp. 53-71.
7. M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation I, Fract. Calc. Appl. Anal., 8 (2005), pp. 323-341.
8. M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation II-with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9 (2006), pp. 333-349.
9. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966.
10. N. Jacob and H-G. Leopold, Pseudo differential operators with variable order of differentiation generating Feller semigroup, Integr. Equ. Oper. Th., 17 (1993), pp. 544-553.
11. K. Kikuchi and A. Negoro, On Markov processes generated by pseudodifferentail operator of variable order, Osaka J. Math., 34 (1997), pp. 319-335.
12. V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc., 80, pp. 725-768.
13. T. Komatsu, On stable-like processes, in 7th Japan-Russian Symposium on Probability Theory and Mathematical Statitics, S. Watanabe, M. Fukushima, Y.V. Prohorov, and A.N. Shiryaev, eds., World Scientific, River Edge, NJ, 1996, pp. 210-219.
14. H. G. Leopold, Embedding of function spaces of variable order of differentiation, Czechoslovak Math. J., 49 (1999), pp. 633-644.
15. R. Lin, F. Liu, V. Anh, and I. Turner, Stability and convergence of a new explicit finitedifference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 2009, to appear.
16. F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2004), pp. 209-219.
17. 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, Appl. Math. Comput., 191 (2007), pp. 12-20.
18. Q. Liu, F. Liu, I. Turner, and V. Anh, Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method, J. Comput. Phys., 222 (2007), pp. 57-70.
19. C. F. Lorenzo and T. T. Hartley, Initialization, Conceptualization, and Application in the Generlized Fractional Calculus, NASA/TP-1998-208415, Lewis Research Center, OH, 1998.
20. C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), pp. 57-98.
21. L. Zhao, The matrix forms of the Hölder and Minkowski inequalities, J. Chengdu Univ. Nat. Sci. Ed., 29 (1990), pp. 10-14.
22. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65-77.
23. C. Tadjeran, M. M. Meerschaert, and H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), pp. 205-213.
24. S. Momani, An algorithm for solving the fractional convection-diffusion equation with nonlinear source term, Commun. Nonlinear Sci. Numer. Simul., 12 (2007), pp. 1283-1290.
25. R. Peltier and J. Lévy Véhel, Multifractional Brownian Motion: Definition and Preliminary Results, INRIA Research Report 2645, INRIA, France, 1995.
26. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
27. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Krieger Publishing, Malabar, FL, 1994.
28. J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in ℝ2, J. Comput. Appl. Math., 193 (2006), pp. 243-268.
29. M. D. Ruiz-Medina, V. V. Anh, and J. M. Angulo, Fractional generalized random fields of variable order, Stoch. Anal. Appl., 22 (2004), pp. 775-799.
30. R. Schumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, Eulerian derivation of the fractional advection-dispersion equation, J. Contaminant Hydrology, 48 (2001), pp. 69-88.
31. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985.
32. D. Yu and H. Tan, Numerical Methods of Differential Equations, Science Publisher, Beijing, 2003.
33. X. Zhang, M. Lu, J. W. Crawford, and I. M. Young, The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives, Adv. Water Resources, 30 (2007), pp. 1205-1217.
34. Z. Yong, D. A. Benson, M. M. Meerschaert, and H. P. Scheffler, On using random walks to solve the space-fractional advection-dispersion equations, J. Stat. Phys., 123 (2006), pp. 89-110.