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First Steps in Random WalksFrom Tools to Applications$
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J. Klafter and I. M. Sokolov

Print publication date: 2011

Print ISBN-13: 9780199234868

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199234868.001.0001

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Fractional diffusion and Fokker‐Planck equations for subdiffusion

Fractional diffusion and Fokker‐Planck equations for subdiffusion

Chapter:
(p.80) 6 Fractional diffusion and Fokker‐Planck equations for subdiffusion
Source:
First Steps in Random Walks
Author(s):

J. Klafter

I.M. Sokolov

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199234868.003.0006

This chapter continues the discussion of generalized master equations for CTRW and concentrates on the case of power‐law waiting time distributions lacking the mean. In this case the corresponding generalized Fokker‐Planck and diffusion equations can be represented in the form containing derivatives of fractional order in time. The corresponding mathematical tools are introduced and discussed in some detail. The chapter then concentrates on the methods of solution of such equations (following the patterns of the ones used for customary equations) and discusses the properties of the corresponding solutions. Additional attention is paid to Mittag‐Leffler functions, a class of special functions appearing inevitably in solutions for subdiffusion.

Keywords:   generalized Fokker‐Planck and diffusion equations, fractional derivatives, Mittag‐Leffler functions

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