Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations

Angstmann, C. N. and Donnelly, I. C. and Henry, B. I. and Langlands, T. A. M. and Straka, P. (2015) Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations. SIAM Journal on Applied Mathematics, 75 (4). pp. 1445-1468. ISSN 0036-1399

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Abstract

Continuous time random walks, which generalize random walks by adding a stochastic time between jumps, provide a useful description of stochastic transport at mesoscopic scales. The continuous time random walk model can accommodate certain features, such as trapping, which are not manifest in the standard macroscopic diffusion equation. The trapping is incorporated through a waiting time density, and a fractional diffusion equation results from a power law waiting time. A generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force. Here we have derived the master equations for continuous time random walks with space- and time-dependent forcing for two cases: when the force is evaluated at the start of the waiting time and at the end of the waiting time. The differences
persist in low order spatial continuum approximations; however, the two processes are shown to be governed by the same Fokker–Planck equations in the diffusion limit. Thus the fractional Fokker–Planck equation with space- and time-dependent forcing is robust to these changes in the underlying stochastic process.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Access to Published version in accordance with the copyright policy of the publisher.
Faculty / Department / School: Current - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences
Date Deposited: 21 Apr 2016 03:05
Last Modified: 13 Jul 2016 05:28
Uncontrolled Keywords: continuous time random walk; fractional Fokker-Planck equation; anomalous diffusion; generalized master equation; limit theorems; fractional calculus
Fields of Research : 01 Mathematical Sciences > 0104 Statistics > 010406 Stochastic Analysis and Modelling
01 Mathematical Sciences > 0101 Pure Mathematics > 010110 Partial Differential Equations
01 Mathematical Sciences > 0102 Applied Mathematics > 010299 Applied Mathematics not elsewhere classified
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Funding Details:
Identification Number or DOI: 10.1137/15M1011299
URI: http://eprints.usq.edu.au/id/eprint/27939

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