Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

Josiah D. Cleland () and Martin A. K. Williams
Additional contact information
Josiah D. Cleland: School of Natural Sciences, Massey University, Palmerston North 4442, New Zealand
Martin A. K. Williams: School of Natural Sciences, Massey University, Palmerston North 4442, New Zealand

Mathematics, 2022, vol. 10, issue 18, 1-13

Abstract: This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations.

Keywords: generalized; fractional; diffusion; Fokker–Planck (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/18/3235/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/18/3235/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:18:p:3235-:d:908432

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().