Numerical approximation of a one-dimensional space fractional advection-dispersion equation with boundary layer

Computers and Mathematics with Applications

Volumn 56, Issue 7, 2008, Pages 1808-1819

  Roop, John Paul ^a  

^a NORTH CAROLINA A AND T STATE UNIVERSITY   (United States)

Author keywords

Boundary layers; Finite element method; Fractional advection dispersion equation; Fractional derivatives; Fractional diffusion

Indexed keywords

ADVECTION-DISPERSION EQUATION (ADE); CONVECTION DIFFUSION EQUATIONS; FINITE ELEMENT COMPUTATIONS; NUMERICAL APPROXIMATIONS; ONE-DIMENSIONAL; SINGULARLY PERTURBED;

EID: 48049106196     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2008.04.025     Document Type: Article

References (28)

1. Benson D., Wheatcraft S., and Meerschaeert M. The fractional order governing equations of Lévy motion. Water Resour. Res. 36 (2000) 1413-1423
2. Chaves A. A fractional diffusion eqution to describe Lévy flights. Phys. Lett. A 239 (1998) 13-16
3. Molz F., Fix G., and Lu S. A physical interpretation for the fractional derivative in Lévy diffusion. Appl. Math. Lett. 15 (2002) 907-911
4. J. Boggs, L. Beard, W. Waldrop, Transport of tritium and four organic compounds during a natural-gradient experiment (MADE-2), Technical Report EPRI TR-101998, Elec. Power Res. Inst., Pleasant Hill, Calif., 1993
5. Benson D., Schumer R., Meerschaeert M., and Wheatcraft S. Fractional dispersion, Lévy motion and the made tracer tests. Transp. Porous Media 42 (2001) 211-240
6. Benson D., Wheatcraft S., and Meerschaeert M. Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000) 1403-1412
7. Carreras B., Lynch V., and Zaslavsky G. Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models. Phys. Plasmas 8 12 (2001) 5096-5103
8. Zaslavsky G., Stevens D., and Weitzner H. Self-similar transport in incomplete chaos. Phys. Rev. E 48 3 (1993) 1683-1694
9. Podlubny I. Fractional Differential Equations (1999), Academic Press, New York
10. Meerschaert M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2004) 65-77
11. Ervin V., Heuer N., and Roop J. Numerical approximation of a time dependent, non-linear, fractional order diffusion equation. SIAM J. Numer. Anal. 45 (2005) 572-591
12. Ervin V., and Roop J. Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differential Equations 22 (2006) 558-576
13. d. Numer. Methods P.D.E. 23 (2007) 256-281
14. Fix G., and Roop J. Least squares finite element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48 (2004) 1017-1033
15. J. Roop, Variational solution of the fractional advection dispersion equation, Ph.D. Thesis, Clemson University, 2004
16. 2. J. Comput. Appl. Math. 193 (2005) 243-268
17. Biler P., Funaki T., and Woyczynski W. Fractal Burgers equations. J. Differential Equations 148 (1998) 9-46
18. John V., Maubach J., and Tobiska L. Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78 (1997) 165-188
19. Verfurth R. A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (1996), Wiley-Teubner, Chichester, UK
20. Friedrichs K. The mathematical structure of the boundary layer problem. In: von Mises R., and Friedrichs K. (Eds). Fluid Dynamics (1941), Brown Univ., Providence, RI 171-174 Reprinted by Springer-Verlag, New York, 1971
21. Cheng W., and Temam R. Numerical approximation of one-dimensional stationary diffusion equations with boundary layers. Comput. & Fluids 31 (2002) 453-466
22. Cheng W., Temam R., and Wang X. New approximation algorithms for a class of partial differential equations displaying boundary layer behavior. Methods Appl. Anal. 7 (2000) 363-390
23. Jung C. Numerical approximation of two-dimensional convection-diffusion equations with boundary layers. Numer. Methods Partial Differential Equations 21 (2005) 623-648
24. Oldham K., and Spanier J. The Fractional Calculus (1974), Academic Press, New York
25. Samko S., Kilbas A., and Marichev O. Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach, New York
26. Grisvard P. Singularities in Boundary Value Problems (1992), Springer-Verlag, New York
27. Brenner S., and Scott L. The Mathematical Theory of Finite Element Methods (1994), Springer-Verlag, New York
28. I. Podlubny, M. Kacenak, Matlab implementation of the Mittag-Leffler function. Available online: http://www.mathworks.com, 2005