INVERSE PROBLEM FOR A PARABOLIC EQUATION WITH A TIME-DEPENDENT COEFFICIENT IN THE BOUNDARY CONDITION

Abstract

In this paper, we consider an inverse problem for a parabolic equation in which a time­dependent coefficient is in cluded in th e bo undary co nd ition. Ad ditional in formation is gi ven by an overdetermination condition that relates the solution to a given function of time. To solve the problem, the spectral method is used, which is based on the expansion of the solution in terms of eigenfunctions of the corresponding boundary value problem with homogeneous boundary conditions. Due to this expansion, the original problem is reduced to a system of ordinary differential equations for the expansion coefficients. Af ter so lving th e ob tained sy stem an d su bstituting the representation of the solution into the integral condition, the problem is reduced to a nonlinear Volterra integral equation with respect to the unknown coefficient. Fo r th e ob tained eq uation, the corresponding operator is introduced and its properties are studied. The existence and uniqueness of the solution on the interval [0 , T ] are proved under the contraction condition for the constructed operator. It is shown that the presence of an unknown coefficient in th e bo undary co ndition leads to a change in the structure of the resulting integral equation compared to classical formulations of inverse problems. The proposed approach can be applied to a wider class of inverse problems for parabolic equations with nonclassical boundary conditions.