Mixed wave-diffusion-wave equation: solvability of an initial-boundary problem

Abstract

We analyze the unique solvability of the direct problem for the wave-diffusion-wave equation, considering initial, boundary, and transmission conditions. This problem is associated with a mathematical model of gas flow in a channel, governed by a combination of wave and diffusion equations. Our primary method of investigation is the separation of variables, with proof of the uniform convergence of the resulting series as a central aspect of our study. To ensure this uniform convergence, we impose specific conditions on the given functions. A key component of the solution is the bivariate Mittag-Leffler-type function, whose properties play a crucial role in our analysis. Although the operator used in the spatial variables is relatively simple, our approach remains applicable to more general operators and domains.