Inverse Problems for a Two-Dimensional Heat Equation with Unknown Right-Hand Side

Abstract

Based on the solution of the first initial-boundary value problem for an inhomogeneous two-dimensional heat equation, we state and study inverse problems, whose right-hand sides contain unknown factors depending on spatial and time variables. Preliminarily, we explicitly construct a solution to the direct initial-boundary-value problem. We prove the uniqueness of the solution to direct and inverse problems, making use of the completeness property of the system of eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. We also establish existence theorems for solutions of inverse problems and construct solutions explicitly.