A characteristic difference method for the variable-order fractional advection-diffusion equation

Journal of Applied Mathematics and Computing

Volumn 42, Issue 1-2, 2013, Pages 371-386

  Shen, S ^a   Liu, F ^b   Anh, V ^b   Turner, I ^b   Chen, J ^c  

^a HUAQIAO UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c JIMEI UNIVERSITY   (China)

Author keywords

Characteristic finite difference method; Fractional advection diffusion equation; Stability and convergence; Variable order fractional order derivative

Indexed keywords

ADVECTION-DIFFUSION EQUATION; CHARACTERISTIC DIFFERENCE; CHARACTERISTIC METHOD; FINITE DOMAINS; FRACTIONAL ORDER DERIVATIVES; NONLINEAR SOURCE TERM; NUMERICAL EXPERIMENTS; STABILITY AND CONVERGENCE;

FINITE DIFFERENCE METHOD; NANOSTRUCTURED MATERIALS;

ADVECTION;

EID: 84880039717     PISSN: 15985865     EISSN: None     Source Type: Journal    
DOI: 10.1007/s12190-012-0642-0     Document Type: Article

References (36)

1. Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M.: The fractional order governing equation of Lévy motion. Water Resour. Res. 36, 1413-1423 (2000)
2. Chaves, A.: Fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13-16 (1998)
3. Chen, C.-M.; Liu, F.; Anh, V.; Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740-1760 (2010)
4. Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12, 692-703 (2003)
5. Douglas, J.; Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871-885 (1982)
6. d. Numer. Methods Partial Differ. Equ. 23, 256-281 (2007)
7. Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, 309-321 (1990)
8. Golbabai, A.; Sayevand, K.: On generalized fractional flux advection-dispersion equation and Caputo derivative. Int. J. Math. Comput. Sci. 2, 425-430 (2011)
9. Gorenflo, R.; Iskenderov, A.; Luchko, Yu.: Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Math. 3(1), 75-86 (2000)
10. Jacob, N.; Leopold, H.-G.: Pseudo differential operators with variable order of differentiation generating Feller semigroup. Integral Equ. Oper. Theory 17, 544-553 (1993)
11. Kikuchi, K.; Negoro, A.: On Markov processes generated by pseudodifferential operator of variable order. Osaka J. Math. 34, 319-335 (1997)
12. Komatsu, T.: On stable-like processes. In: Watanabe, S.; Fukushima, M.; Prohorov, Y.V.; Shiryaev, A.N. (eds.) 7th Japan-Russian Symposium on Probability Theory and Mathematical Statistics, pp. 210-219. World Scientific, Singapore (1996)
13. Leopold, H.G.: Embedding of function spaces of variable order of differentiation. Czechoslov. Math. J. 49, 633-644 (1999)
14. Lin, R.; Liu, F.; Anh, V.; Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212, 435-445 (2009)
15. Liu, F.; Anh, V.; Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209-219 (2004)
16. Lorenzo, C.F.; Hartley, T.T.: Initialization, conceptualization, and application in the generalized fractional calculus. NASA Technical Publication 98-208415, NASA, Lewis Research Center (1998)
17. Lorenzo, C.F.; Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57-98 (2002)
18. Meerschaert, M.M.; Sikorskii, A.: Stochastic Models for Fractional Calculus. de Gruyter Studies in Mathematics, vol. 43. de Gruyter, Berlin (2012)
19. Meerschaert, M.M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65-77 (2004)
20. Meerschaert, M.M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80-90 (2006)
21. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
22. Raberto, M.; Scalas, E.; Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749-755 (2002)
23. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J. Comput. Appl. Math. 193, 243-268 (2006)
24. Ruiz-Medina, M.D.; Anh, V.V.; Angulo, J.M.: Fractional generalized random fields of variable order. Stoch. Anal. Appl. 22, 775-799 (2004)
25. Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21, 213-236 (1995)
26. Samko, S.G.; Ross, B.: Integration and differentiation to a variable fractional order. Integral Transforms Spec. Funct. 1, 277-300 (1993)
27. Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Wheatcraft, S.W.: Eulerian derivation of the fractional advection-dispersion equation. J. Contam. Hydrol. 48, 69-88 (2001)
28. Simth, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Clarendon, Oxford (1985)
29. Soon, C.M.; Coimbra, F.M.; Kobayashi, M.H.: The variable viscoelasticity oscillator. Ann. Phys. 14, 378-389 (2005)
30. Sousa, E.: Finite difference approximations for a fractional advection diffusion problem. J. Comput. Phys. 228, 4038-4054 (2009)
31. Su, L.; Wang, W.; Wang, H.: A characteristic difference method for the transient fractional convection-diffusion equations. Appl. Numer. Math. 61, 946-960 (2011)
32. Sun, H.; Chen, W.; Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A 388, 4586-4592 (2009)
33. Wang, K.; Wang, H.: A fast characteristic finite difference method for fractional advection-diffusion equations. Adv. Water Resour. 34(7), 810-816 (2011)
34. Yong, Z.; Benson, D.A.; Meerschaert, M.M.; Scheffler, H.-P.: On using random walks to solve the space-fractional advection-dispersion equations. J. Stat. Phys. 123, 89-110 (2006)
35. Zhou, L.; Selim, H.M.: Application of the fractional advection-dispersion equation in porous media. Soil Sci. Soc. Am. J. 67, 1079-1084 (2003)
36. Zhuang, P.; Liu, F.; Anh, V.; Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760-1781 (2009)