Finite domain anomalous spreading consistent with first and second laws

Communications in Nonlinear Science and Numerical Simulation

Volumn 15, Issue 11, 2010, Pages 3455-3470

  Valko P P ^a   Zhang, X H ^b  

^a Department of Electrical and Computer Engineering   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Caputo derivative; Collocation; Finite differences; Laplace transform; Riesz derivative; Space time fractional differential equation

Indexed keywords

DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; THERMODYNAMICS;

ANOMALOUS SPREADING; CAPUTO DERIVATIVES; COLLOCATION; FINITE DIFFERENCE; FINITE DOMAINS; FIRST LAW; FRACTIONAL DIFFERENTIAL EQUATIONS; RIESZ DERIVATIVE; SPACE-TIME FRACTIONAL DIFFERENTIAL EQUATION; SPACETIME;

LAPLACE TRANSFORMS;

EID: 77952243763     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2009.12.001     Document Type: Article

References (40)

1. Schneider W.R., and Wyss W. Fractional diffusion and wave equations. J Math Phys 30 (1989) 134-144
2. Shlesinger M.F., Zaslavsky G.M., and Klafter J. Strange kinetics. Nature (London) 363 (1993) 31-37
3. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7 (1997) 1461-1477
4. Berkowitz B., and Scher H. On characterization of anomalous dispersion in porous media. Water Resour Res 31 (1995) 1461-1466
5. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. The fractional-order governing equation of Levy motion. Water Resour Res 36 (2000) 1403-1412
6. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. Application of a fractional advection-dispersion equation. Water Resour Res 36 (2000) 1413-1423
7. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339 (2000) 1
8. Metzler R., and Klafter J. The restaurant at the end of the random walk. Recent developments in the description of anomalous transport by fractional dynamics. J Phys A 37 (2004) R161-R208
9. Zaslavsky G.M. Chaos, fractional kinetics and anomalous transport. Phys Rep 371 6 (2002) 461-580
10. Chechkin A.V., Gonchar V.Y., Metzler R., and Klafter J. Fundamentals of Lévy flight processes. Adv Chem Phys 133 (2006) 439-496
11. Zhang Y., Benson D.A., and Reeves D.M. Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv Water Resour 32 4 (2009) 561-581
12. Montroll E.W., and Weiss G.H. Random walks on lattices II. J Math Phys 6 (1965) 167-181
13. Cushman J.H. On diffusion in fractal media. Water Resour Res 27 4 (1991) 643-644
14. Barkai E. CTRW pathways to the fractional diffusion equation. Chem Phys 284 (2002) 13-27
15. Gorenflo R., Mainardi F., Moretti D., Pagnini G., and Paradisi P. Discrete random walk models for space-time fractional diffusion. Chem Phys 284 1-2 (2002) 521-541
16. Paradisi P., Cesari R., Mainardi F., and Tampieri F. Fractional Fick's law for non-local transport processes. Phys A: Stat Mech Appl 293 1-2 (2001) 130-142
17. Néel M.-C., Abdennadher A., and Joelson M. Fractional Fick's law: the direct way. J Phys A: Math Theor 40 29 (2007) 8299-8314 [art. 007]
18. Caputo M. Linear models of dissipation whose Q is almost frequency independent-II. Geophys J.R. Astro Soc 13 (1967) 529-539
19. Saichev A.I., and Zaslavsky G.M. Fractional kinetic equations: solutions and applications. Chaos 7 (1997) 753-764
20. Chechkin A.V., Gonchar V., Gorenflo R., Korabel N., and Sokolov I.M. Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights. Phys Rev E 78 (2008) 1-13 [art. 021111]
21. Montroll E.W., and Scher H. Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries. J Stat Phys 9 2 (1973) 101-135
22. Mainardi F., Luchko Y., and Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus Appl Anal 4 (2001) 153-192
23. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and applications of fractional differential equations. In: Mill J. (Ed). North-Holland mathematical studies vol. 204 (2006), North-Holland, Amsterdam
24. Piryatinska A., Saichev A.I., and Woyczynski W.A. Models of anomalous diffusion: the subdiffusive case. Physica A 349 (2005) 375-420
25. Podlubny I., Chechkin A., Skovranek T., Chen Y., and Vinagre Jara B.M. Matrix approach to discrete fractional calculus. II: Partial fractional differential equations. J Comp Phys 228 8 (2009) 3137-3153 10.1016/j.jcp.2009.01.014
26. Chechkin A.V., Metzler R., Gonchar V.Y., Klafter J., and Tanatarov L.V. First passage and arrival time densities for Levy flights and the failure of the method of images. J Phys A - Math Gen 36 (2003) L537-L544
27. Metzler R., Chechkin A.V., Gonchar V.Yu., and Klafter J. Some fundamental aspects of Levy flights. Chaos Soliton Fractals 34 (2007) 129-142 10.1016/j.chaos.2007.01.055
28. Podlubny I. Fractional differential equation (1999), Academic Press, New York
29. Zhang Y., Benson D.A., Meerschaert M.M., and LaBolle E.M. Space-fractional advection-dispersion equations with variable parameters: diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data. Water Resour Res 43 5 (2007) [art. W05439]
30. del-Castillo-Negrete D., Gonchar V., and Chechkin A.V. Fluctuation-driven directed transport in the presence of Lvy flights. Phys A: Stat Mech Appl 387 27 (2008) 6693-6704
31. Zhang X., Lv M., Crawford J.W., and Young I.M. 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
32. Scalas E. The application of continuous-time random walks in finance and economics. Phys A: Stat Mech Appl 362 2 (2006) 225-239
33. Krepysheva N., Pietro L.Di., and Néel M.-C. Fractional diffusion and reflective boundary condition. Physica A 368 (2006) 355-361
34. Krepysheva N., Pietro L.Di., and Néel M.-C. Space fractional advection-diffusion and reflective boundary conditions. Phys Rev E 73 021104 (2006) 1-9
35. Calvo I., Carreras B.A., Sanchez R., and van Milligen B.Ph. Continuous time random walks in periodic systems: fluid limit and fractional differential equations on the circle. J Phys A: Math Theor 40 (2007) 13511
36. van Milligen B.Ph., Calvo I., and Sánchez R. Continuous time random walks in finite domains and general boundary conditions: some formal considerations. J Phys A: Math Theor 41 21 (2008) [art. 215004]
37. Zoia A., Rosso A., and Kardar M. Fractional Laplacian in bounded domains. Phys Rev E - Stat Nonlinear Soft Matter Phys 76 2 (2007) 1-13 art. 021116
38. Valkó P.P., and Abate J. Comparison of sequence accelerators for the Gaver method of numerical laplace transform inversion. Comp Math Appl 48 (2004) 629-636 10.1016/j.camwa.2002.10.017
39. Abate J., and Valkó P.P. Multi-precision Laplace transform inversion. Int J Numer Meth Eng 60 (2004) 979-993 10.1002/nme.995
40. Mathematica v7.0 http://www.wolfram.com [Last visited: August 15, 2009].