On a Mixed Equation Involving Prabhakar Fractional Order Integral-Differential Operators

In this short note, we would like to give the main parts of the investigation of direct and inverse problems for mixed equations involving regularised Prabhakar fractional order integral-differential operator. Namely, we present the explicit solution of the Cauchy problem for an ordinary differential equation with the Prabhakar fractional derivative. We also have presented important statements on the bivariate Mittag-Leffler function. Namely, Euler-type integral representations and certain estimations for the bivariate Mittag-Leffler type function \(E_2(x,y)\) . The main targets are direct and inverse source problems for mixed equations involving regularised Prabhakar derivatives. We will present here the algorithm of the proof and a short description of the key steps.