On a certain bivariate Mittag-Leffler function analysed from a fractional-calculus point of view

Abstract

Mittag-Leffler functions of one variable play a vital role in several areas of study. Their connections with fractional calculus enable many physical processes, such as diffusion and viscoelasticity, to be efficiently modelled. Here, we consider a Mittag-Leffler function of two variables and the associated double integral operator, with the goal of establishing once again connections with fractional calculus. By working from the fractional-calculus viewpoint, it is possible to obtain many new results concerning the double integral operator, including a series formula and a bivariate chain rule. We also discover a left inverse operator, which completes this model of fractional calculus. As applications, we solve some initial value problems and use a modified Stancu-Bernstein model to approximate the image of a Holder-continuous function under the action of our double integral operator.