# 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)

### 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 language English 243-252 10 Journal of Mathematical Analysis and Applications 423 1 2 Oct 2014 https://doi.org/10.1016/j.jmaa.2014.09.067 Published - 1 Mar 2015

### Fingerprint

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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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.",
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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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AU - McKee, S.

AU - Cuminato, J. A.

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N2 - 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.

AB - 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.

KW - Mittag-Leffler function

KW - non-local diffusion

KW - two-dimensional Volterra integral equation

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