An inverse problem for two-dimensional equations of finding the thermal conductivity of the initial distribution

Abstract

The inverse problem of finding the initial distribution has been studied on the basis of formulas for the solution of the first initial-boundary value problem for the inhomogeneous two-dimentional heat equation. The uniqueness of the solution of the direct initial-boundary value problem has proved with the completeness of the eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. The existence theorem for solving direct initial boundary value problem has been proved. Inverse problem has been investigated on the basis of the solution of direct problem, a criterion for the uniqueness of the inverse problem of finding the initial distribution has been proved. The existence of the inverse problem solution has been equivalently reduced to Fredholm integral equation of the first kind.

Keywords

• heat equation
• first initial-boundary value problem
• inverse problem
• spectral method
• uniqueness
• existence
• integral equation