### 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 |
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Pages (from-to) | 243-252 |

Number of pages | 10 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 423 |

Issue number | 1 |

Early online date | 2 Oct 2014 |

DOIs | |

Publication status | Published - 1 Mar 2015 |

### Keywords

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

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*423*(1), 243-252. https://doi.org/10.1016/j.jmaa.2014.09.067