### 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 |
---|---|

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 |

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### Keywords

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

### Cite this

*Journal of Mathematical Analysis and Applications*,

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

}

*Journal of Mathematical Analysis and Applications*, vol. 423, no. 1, pp. 243-252. https://doi.org/10.1016/j.jmaa.2014.09.067

**Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation.** / McKee, S.; Cuminato, J. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation

AU - McKee, S.

AU - Cuminato, J. A.

PY - 2015/3/1

Y1 - 2015/3/1

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

UR - http://www.scopus.com/inward/record.url?scp=84922895006&partnerID=8YFLogxK

UR - https://www.sciencedirect.com/journal/journal-of-mathematical-analysis-and-applications

U2 - 10.1016/j.jmaa.2014.09.067

DO - 10.1016/j.jmaa.2014.09.067

M3 - Article

VL - 423

SP - 243

EP - 252

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -