Inverse problem on determining two kernels in integro-differential equation of heat flow

Abstract

We study the inverse problem on determining the energy-temperature relation $\chi(t)$ and the heat conduction relation $k(t)$ functions in the one-dimensional integro-differential heat equation. The direct problem is an initial-boundary value problem for this equation with the Dirichlet boundary conditions. The integral terms involve the time convolution of unknown kernels and a direct problem solution. As an additional information for solving inverse problem, the solution of the direct problem for $x=x_0$ and $x=x_1$ is given. We first introduce an auxiliary problem equivalent to the original one. Then the auxiliary problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to the unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, we prove the main result of the article, which a local existence and uniqueness theorem for the inverse problem.