Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation

S. McKee, J. A. Cuminato

Research output: Contribution to journalArticle

3 Citations (Scopus)
7 Downloads (Pure)

Abstract

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.

Original languageEnglish
Pages (from-to)243-252
Number of pages10
JournalJournal of Mathematical Analysis and Applications
Volume423
Issue number1
Early online date2 Oct 2014
DOIs
Publication statusPublished - 1 Mar 2015

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Nonlocal Diffusion
Mittag-Leffler Function
Volterra Integral Equations
Analytic Solution
Integral equations
Integral Equations
Existence and Uniqueness Theorem
Diffusion Problem
Series Solution
Singular Integral Equation
Entire Function
Forcing
Lemma
Fluxes
kernel
Output
Form
Class

Keywords

  • Mittag-Leffler function
  • non-local diffusion
  • two-dimensional Volterra integral equation

Cite this

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Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation. / McKee, S.; Cuminato, J. A.

In: Journal of Mathematical Analysis and Applications, Vol. 423, No. 1, 01.03.2015, p. 243-252.

Research output: Contribution to journalArticle

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