A second order explicit finite difference method for the fractional advection diffusion equation

Computers and Mathematics with Applications

Volumn 64, Issue 10, 2012, Pages 3141-3152

  Sousa, Ercília ^a  

^a UNIVERSITY OF COIMBRA   (Portugal)

Author keywords

Finite differences; Fractional advection diffusion; Integro differential equations

Indexed keywords

ADVECTION - DIFFUSION; ADVECTION DIFFUSION EQUATION; ADVECTION-DIFFUSION PROBLEM; EXPLICIT FINITE DIFFERENCE METHOD; FINITE DIFFERENCE; HOMOGENEOUS BOUNDARY CONDITION; NUMERICAL RESULTS; SECOND ORDERS; SOURCE TERMS; TIME DISCRETIZATION;

INTEGRODIFFERENTIAL EQUATIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

FINITE DIFFERENCE METHOD;

EID: 84868209623     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2012.03.002     Document Type: Article

References (27)

1. R. Metzler, and J. Klafter The random walk's guide to anomalous diffusion: a fractional dynamics approach Phys. Rep. 339 2000 1 77
2. R. Metzler, and J. Klafter Accelerating Brownian motion: a fractional dynamics approach to fast diffusion Europhys. Lett. 51 2000 492 498
3. A.S. Chaves A fractional diffusion equation to describe Lévy flights Phys. Lett. A 239 1998 13 16 (Pubitemid 128392702)
4. D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert Application of a fractional advection-dispersion equation Water Resour. Res. 36 2000 1403 1412 (Pubitemid 30334241)
5. D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert The fractional order governing equation of Lévy motion Water Resour. Res. 36 2000 1413 1423 (Pubitemid 30334242)
6. M.M. Meerschaert, and C. Tadjeran Finite difference approximations for fractional advection-dispersion flow equations J. Comput. Appl. Math. 172 2004 65 77 (Pubitemid 39204390)
7. E. Sousa Finite difference approximations for a fractional advection diffusion problem J. Comput. Phys. 228 2009 4038 4054
8. 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 205 213 (Pubitemid 43174872)
9. X. Zhang, L. Mouchao, 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 Resour. 30 2007 1205 1217 (Pubitemid 46323434)
10. X. Li, and C. Xu A space-time spectral method for the time fractional diffusion equation SIAM J. Numer. Anal. 47 2009 2108 2131
11. W. Deng Finite element method for the space and time fractional Fokker-Planck equation SIAM J. Numer. Anal. 47 2008 204 226
12. V.J. Ervin, H. Norbert, and J.P. Roop Numerical approximation of a time dependent nonlinear, space-fractional diffusion equation SIAM J. Numer. Anal. 45 2007 572 591 (Pubitemid 351599168)
13. Q. Huang, G. Huang, and H. Zhan A finite element solution for the fractional advection-dispersion equation Adv. Water Resour. 31 2008 1578 1589
14. C. Li, Z. Zhao, and Y. Chen Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion Comput. Math. Appl. 62 2011 855 875
15. R 2 J. Comput. Appl. Math. 193 2006 243 268
16. H. Zhang, L. Fawang, and V. Anh Galerkin finite element approximation of symmetric space-fractional partial differential equations Appl. Math. Comput. 217 2010 2534 2545
17. Y.Y. Zheng, C.P. Li, and Z.G. Zhao A fully discrete discontinuous Galerkin method for nonlinear fractional Fokker-Planck equation Math. Probl. Eng. 2010 Article ID 279038
18. P.D. Lax, and B. Wendroff Difference schemes for hyperbolic equations with high order of accuracy Commun. Pur. Appl. Math. 17 1964 381 398
19. K. Diethelm The analysis of fractional differential equations Lecture Notes in Math. 2004
20. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential equations 2006 Elsevier
21. K.B. Oldham, and J. Spanier The Fractional Calculus 1974 Dover
22. M.D. Ortigueira Fractional calculus for scientists and engineers Lecture Notes in Electrical Engineering vol. 84 2011
23. I. Podlubny Fractional Differential Equations 1999 Academic Press San Diego
24. S.G. Samko, A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives: Theory and Applications 1993 Gordon and Breach Science Publishers
25. E. Sousa Numerical approximations for fractional diffusion equations via splines Comput. Math. Appl. 62 2011 938 944
26. E. Sousa, C. Li, A weighted finite difference method for the fractional diffusion based on the Riemann-Liouville derivative, 2011. arXiv:1109.2345v1 [math.NA] (submitted for publication).
27. E. Sousa On the edge of stability analysis Appl. Numer. Math. 59 2009 1322 1336