The Inverse Problem for the Heat Equation with Two Unknown Coefficients

Abstract

We solve the simultaneous recovery of thermal conductivity and a high-frequency coefficient of a source in a one-dimensional initial-boundary value problem for the heat equation with Dirichlet boundary conditions and an inhomogeneous initial condition from some information on the partial asymptotics of a solution. We show that the coefficients can be restored from some data on the asymptotics of a solution, which is constructed and justified. This article was inspired by Denisov’s research on a variety of inverse problems without accounting for high-frequency oscillations. Also, we continue the research by Levenshtam and his students which firstly addressed the inverse problems for parabolic equations with high-frequency coefficients and developed the relevant methodology. In contrast to the previous research of the case that only the source function or its factors are unknown, we assume that the thermal conductivity and the factor of a source function are unknown simultaneously.