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

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McKee, S. and Cuminato, J. A. (2015) Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation. Journal of Mathematical Analysis and Applications, 423 (1). pp. 243-252. ISSN 0022-247X (https://doi.org/10.1016/j.jmaa.2014.09.067)

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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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