On generalised d'Alembert-type integral representations for damped wave equations on the quarter-plane

Abstract

We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.