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

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  Preview Text (McKee-Cuminato-JMAA-2015-Nonlocal-diffusion-a-Mittag-Leffler-function-and-a-two-dimensional-Volterra) McKee_Cuminato_JMAA_2015_Nonlocal_diffusion_a_Mittag_Leffler_function_and_a_two_dimensional_Volterra.pdf Final Published Version Download (283kB)| Preview

## 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.

• Item type: Article 67553 DateEvent1 March 2015Published2 October 2014Published Online Mittag-Leffler function, non-local diffusion, two-dimensional Volterra integral equation, Mathematics, Analysis, Applied Mathematics Science > Mathematics Faculty of Science > Mathematics and Statistics Pure Administrator 11 Apr 2019 13:35 23 Apr 2021 02:43 https://strathprints.strath.ac.uk/id/eprint/67553