Operational calculus approach to PDE arising in QR-regularisation of ill-posed problems

Abstract

In the method of quasireversibility (QR), ill-posed problems for linear partial differential equations are replaced by well-posed problems for equations of twice-higher order. In this paper, we consider two problems: the first for the backward heat equation, and the second for a nonparabolic evolution equation. These problems are treated analytically using operational calculus intended for boundary value problems [1]. We show that an appropriate operational approach allows us to extend the Duhamel principle to treat such problems and to obtain their solutions in closed form.