A simplified approach for calculating the moments of action for linear reaction-diffusion equations

Ellery, Adam, Simpson, Matthew, McCue, Scott W., & Baker, Ruth (2013) A simplified approach for calculating the moments of action for linear reaction-diffusion equations. Physical Review E (PRE), 88, 054102.

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Abstract

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time under-approximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.

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ID Code: 63886
Item Type: Journal Article
Refereed: Yes
Keywords: Critical time, Reaction-diffusion, Laplace transform, steady state, transient
DOI: 10.1103/PhysRevE.88.054102
ISSN: 1550-2376
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Biological Mathematics (010202)
Divisions: Current > Institutes > Institute of Health and Biomedical Innovation
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Funding:
Copyright Owner: Copyright 2013 American Physical Society
Deposited On: 03 Nov 2013 22:21
Last Modified: 16 Sep 2014 05:32

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