Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation

In this paper, we consider the Levy-Feller fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order @[email protected]?(0,2](@a 1) and skewness @q (|@q|=

[1]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[2]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[3]  Gianni De Fabritiis,et al.  Discrete random walk models for symmetric Lévy–Feller diffusion processes , 1999 .

[4]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[5]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[6]  I. Turner,et al.  Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation , 2005 .

[7]  J. Crank.  Tables of Integrals , 1962 .

[8]  B. Achar,et al.  A random walk simulation of fractional diffusion , 2004 .

[9]  P. M. Lee,et al.  Random Walks and Random Environments: Volume 1: Random Walks , 1995 .

[10]  Massimiliano Giona,et al.  Fractional diffusion equation for transport phenomena in random media , 1992 .

[11]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[12]  Fawang Liu,et al.  Numerical simulation for solute transport in fractal porous media , 2004 .

[13]  Francesco Mainardi,et al.  Approximation of Levy-Feller Diffusion by Random Walk , 1999 .

[14]  Fawang Liu,et al.  Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method , 2007, J. Comput. Phys..

[15]  R. Gorenflo,et al.  Discrete random walk models for space-time fractional diffusion , 2002, cond-mat/0702072.

[16]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[17]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[18]  Marie-Christine Néel,et al.  Space-fractional advection-diffusion and reflective boundary condition. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[20]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[21]  M. Shlesinger,et al.  Beyond Brownian motion , 1996 .

[22]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[23]  I. Podlubny.  Fractional differential equations , 1998 .

[24]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[25]  Liliana Di Pietro,et al.  Fractional diffusion and reflective boundary condition , 2006 .

[26]  Fawang Liu,et al.  Implicit difference approximation for the time fractional diffusion equation , 2006 .

[27]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .

[28]  Joshua Kiddy K. Asamoah,et al.  Fractal–fractional age-structure study of omicron SARS-CoV-2 variant transmission dynamics , 2022, Partial Differential Equations in Applied Mathematics.

[29]  Fawang Liu,et al.  Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends , 2005 .

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

[31]  R. Laha.  Probability Theory , 1979 .

[32]  Fred J. Molz,et al.  A physical interpretation for the fractional derivative in Levy diffusion , 2002, Appl. Math. Lett..

[33]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[34]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[35]  Rudolf Gorenflo,et al.  Fully discrete random walks for space-time fractional diffusion equations , 2003, Signal Process..