Mixed partial differential equation: Forward problem linked with the wave-diffusion process

Abstract The main target of the present investigation is a mixed partial differential equation involving the Prabhakar integral-differential operator in the time variable. We begin by briefly describing the several physical processes in which mixed partial differential equations play a key role. Then we formulate an initial-boundary problem for the considered equation linked with the wave-diffusion process. Our primary objective is to demonstrate the unique solvability of the formulated problem under specific conditions for the given data. First, we present the explicit solution of the Cauchy problem for an ordinary differential equation with the regularized Prabhakar fractional derivative. We also present important statements on the bivariate Mittag-Leffler function, namely, Euler-type integral representations and certain estimations for the bi-variate Mittag-Leffler-type function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>E</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {E_{2}(x,y)} . Since the key tool of investigation is the method of separation of variables, most of our evaluations are linked with the proof of uniform convergence of infinite series. We impose certain conditions on given functions to provide uniform convergence of infinite series corresponding to the solution of the formulated problem.