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)
9 Downloads (Pure)


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
Issue number1
Early online date2 Oct 2014
Publication statusPublished - 1 Mar 2015


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

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