We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering. Keywords Mittag-Leffler functions • Fractional integrals • Fractional derivatives • Fractional differential equations • Bivariate Mittag-Leffler functions ρ α,β (x) and E α,β,γ (x), the "two-parameter" and "three-parameter" Mittag-Leffler functions, being defined by power series similar to the Communicated by Roberto Garrappa.
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 Hölder‐continuous function under the action of our double integral operator.
In this paper, we give an analytical solution of a fractional wave equation for a vibrating string with Caputo time fractional derivatives. We obtain the exact solution in terms of three parameter Mittag-Leffler function. Furthermore, some examples of the main result are exhibited.
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