Fractional (space–time) diffusion equation on comb-like model

Abstract In this work, a directed connection between the fractal structure and the fractional calculus has been achieved. The fractional space–time diffusion equation is derived using the comb-like structure as a background model. The solution of the obtained equation will be established for three different interesting cases.

[1]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[2]  O'Shaughnessy,et al.  Anomalous diffusion at liquid surfaces. , 1995, Physical review letters.

[3]  A. Compte,et al.  Anomalous diffusion in linear shear flows , 1997 .

[4]  V. M. Kenkre Generalization to spatially extended systems of the relation between stochastic Liouville equations and generalized master equations , 1978 .

[5]  M Muskat,et al.  THE FLOW OF HOMOGENEOUS FLUIDS THROUGH POROUS MEDIA: ANALOGIES WITH OTHER PHYSICAL PROBLEMS , 1937 .

[6]  Stapf,et al.  Proton and deuteron field-cycling NMR relaxometry of liquids in porous glasses: Evidence for Lévy-walk statistics. , 1995, Physical review letters.

[7]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[8]  P. Mazur,et al.  Non-equilibrium thermodynamics, , 1963 .

[9]  Damián H. Zanette,et al.  Macroscopic current in fractional anomalous diffusion , 1998 .

[10]  Fractional representation of Fokker–Planck equation , 2001 .

[11]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[12]  The diffusion-drift equation on comb-like structure , 2002 .

[13]  P. Rosenau,et al.  Fast and superfast diffusion processes. , 1995, Physical review letters.

[14]  Arak M. Mathai,et al.  The H-function with applications in statistics and other disciplines , 1978 .

[15]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[16]  A. Chaves,et al.  A fractional diffusion equation to describe Lévy flights , 1998 .

[17]  D. Bedeaux,et al.  On the Relation between Master Equations and Random Walks and Their Solutions , 1971 .

[18]  S. A. El-Wakil,et al.  Fractional (space-time) Fokker–Planck equation , 2001 .

[19]  V. E Arkhincheev Anomalous diffusion and charge relaxation on comb model: exact solutions , 2000 .

[20]  S. A. El-Wakil,et al.  The fractional Fokker–Planck equation on comb-like model , 2003 .

[21]  H. Scher,et al.  Time-dependent electrical transport in amorphous solids: As 2 Se 3 , 1977 .

[22]  Morton E. Gurtin,et al.  On the diffusion of biological populations , 1977 .