On the numerical solution of space-time fractional diffusion models

Computers and Fluids

Volumn 46, Issue 1, 2011, Pages 33-39

  Hanert, Emmanuel ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Fractional derivatives; Mittag Leffler functions; Pseudo spectral method; Space time fractional diffusion equation

Indexed keywords

DISCRETIZATIONS; FINITE DIFFERENCE; FINITE-ELEMENT; FRACTIONAL DERIVATIVES; HIGH-ORDER; MITTAG-LEFFLER FUNCTIONS; MODEL SOLUTION; NUMERICAL SCHEME; NUMERICAL SOLUTION; PSEUDOSPECTRAL METHODS; SPACE DISCRETIZATIONS; SPACE-TIME FRACTIONAL DIFFUSION EQUATION; SPECTRAL EXPANSIONS; TIME AND SPACE; TIME FRACTIONAL DIFFUSION EQUATION; TIME-FRACTIONAL DIFFUSION;

FUNCTIONS; PARTIAL DIFFERENTIAL EQUATIONS; SPECTROSCOPY;

DIFFUSION;

EID: 79954926078     PISSN: 00457930     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compfluid.2010.08.010     Document Type: Article

References (47)

1. Richardson L.F. Atmospheric diffusion shown on a distance-neighbour graph. Proc Roy Soc Lond 1926, 110:709-737.
2. Richardson L.F., Stommel H. Note on eddy diffusion in the sea. J Meteorol 1948, 5:238-240.
3. Stommel H. Horizontal diffusion due to oceanic turbulence. J Marine Res 1949, 8:199-225.
4. Okubo A. Oceanic diffusion diagrams. Deep Sea Res 1971, 18:789-802.
5. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 2000, 339:1-77.
6. Metzler R., Klafter J. The restaurant at the end of the random walk: recent development in the description of anomalous transport by fractional dynamics. J Phys A 2004, 37:161-208.
7. Chaves A.S. A fractional diffusion equation to describe Lévy flights. Phys Lett A 1998, 239:13-16.
8. Pachepsky Y., Timlin D., Rawls W. Generalized Richards' equation to simulate water transport in unsaturated soils. J Hydrol 2003, 272:3-13.
9. Berkowitz B., Cortis A., Dentz M., Scher H. Modelling non-Fickian transport in geological formations as a continuous time random walk. Rev Geophys 2006, 44:RG2003.
10. Deng Z.Q., de Lima J.L.M.P., de Lima M.I.P., Singh V.P. A fractional dispersion model for overland solute transport. Water Resour Res 2006, 42:W03416.
11. Huang G., Huang Q., Zhan H. Evidence of one-dimensional scale-dependent fractional advection-dispersion. J Contam Hydrol 2006, 85:53-71.
12. Kim S., Kavvas M.L. Generalized Fick's law and fractional ADE for pollution transport in a river: detailed derivation. J Hydrol Eng 2006, 11(1):80-83.
13. del Castillo Negrete D., Carreras B.A., Lynch V.E. Fractional diffusion in plasma turbulence. Phys Plasmas 2004, 11(8):3854-3864.
14. del Castillo Negrete D., Carreras B.A., Lynch V.E. Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Phys Rev Lett 2005, 94(065003).
15. Scalas E., Gorenflo R., Mainardi F. Fractional calculus and continuous-time finance. Phys A: Stat Mech Appl 2000, 284:376-384.
16. Mainardi F., Raberto M., Gorenflo R., Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Phys A: Stat Mech Appl 2000, 287:468-481.
17. Gorenflo R., Mainardi F., Scalas E., Raberto R. Fractional calculus and continuous-time finance III: the diffusion limit. Mathematical finance 2010, 171-180. Birkhauser. M. Kohlmann, S. Tang (Eds.).
18. Cartea A., del Castillo Negrete D. Fractional diffusion models of option prices in markets with jumps. Phys A: Stat Mech Appl 2007, 374(2):749-763.
19. Djordjević V.D., Jarić J., Fabry B., Fredberg J.J., Stamenović D. Fractional derivatives embody essential features of cell rheological behavior. Ann Biomed Eng 2003, 31(6):692-699.
20. Ding Y., Ye H. A fractional-order differential equation model of HIV infection of CD4+ T-cells. Math Comput Model 2009, 50(3-4):386-392.
21. Brockmann D., Hufnagel L., Geisel T. The scaling laws of human travel. Nature 2006, 439:462-465.
22. Meerschaert M.M., Tadjeran C. Finite difference approximations for fractional advection-diffusion flow equations. J Comput Appl Math 2004, 172:65-77.
23. Meerschaert M.M., Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 2006, 56(1):80-90.
24. Tadjeran C., Meerschaert M.M., Scheffler H.-P. A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 2006, 213:205-213.
25. Lin Y., Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 2007, 225(2):1533-1552.
26. Podlubny I., Chechkin A., Skovranek T., Chen Y., Jara B.M.V. Matrix approach to discrete fractional calculus II: partial fractional differential equations. J Comput Phys 2009, 228:3137-3153.
27. Fix G.J., Roop J.P. Least square finite-element solution of a fractional order two-point boundary value problem. Comput Math Appl 2004, 48:1017-1033.
28. Roop J.P. Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J Comput Appl Math 2006, 193:243-268.
29. Podlubny I. Fractional differential equations. Mathematics in science and engineering 1999, vol. 198. Academic Press.
30. Ford N.J., Simpson A.C. The numerical solution of fractional differential equations: speed versus accuracy. Numer Algor 2001, 26(4):333-346.
31. Hanert E. A comparison of three Eulerian numerical methods for fractional-order transport models. Environ Fluid Mech 2010, 10:7-20. 10.1007/s10652-009-9145-4.
32. Li X., Xu C. A space-time spectral method for the time fractional diffusion equation. SIAM J Numer Anal 2009, 47:2108-2131.
33. Montroll E.W., Weiss G.H. Random walks on lattices. II. J Math Phys 1965, 6:167-181.
34. Sornette D. Critical phenomena in natural sciences 2006, Springer.
35. Fulger D., Scalas E., Germano G. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys Rev E 2008, 77(1):021122.1-021122.7.
36. Chambers J.M., Mallows C.L., Stuck B.W. A method for simulating stable random variables. J Am Stat Assoc 1976, 71:340-344.
37. Kozubowski T.J., Rachev S.T. Univariate geometric stable laws. J Comput Anal Appl 1999, 1(2):177-217.
38. Li C., Deng W. Remarks on fractional derivatives. Appl Math Comput 2007, 187:777-784.
39. Metzler R., Nonnenmacher T.F. Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation. Chem Phys 2002, 284:67-90.
40. Boyd J.P. Chebyshev and Fourier spectral methods 2001, Dover Publications. 2nd ed.
41. Mainardi F., Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes. J Comput Appl Math 2000, 118:283-299.
42. Metzler R., Klafter J. From stretched exponential to inverse power-law: fractional dynamics, Cole-Cole relaxation processes, and beyond. J Non-Cryst Solids 2002, 305(1-3):81-87.
43. Fornberg B. A practical guide to pseudospectral methods 1996, Cambridge University Press, Cambridge.
44. Lynch V.E., Carreras B.A., del Castillo Negrete D., Ferreiras-Mejias K.M., Hicks H.R. Numerical methods for the solution of partial differential equations of fractional order. J Comput Phys 2003, 191:406-421.
45. Fornberg B., Flyer N., Russell J.M. Comparisons between pseudospectral and radial basis function derivative approximations. IMA J Numer Anal 2010, 30(1):149-172.
46. Flyer N., Fornberg B. Radial basis functions: developments and applications to planetary scale flows. Comput Fluids 2010, 46:23-32.
47. Laub A.J. Matrix analysis for scientists and engineers 2005, SIAM.