The inverse problem of determining the energy-temperature relation a(t) and the heat conduction relation k(t) functions in the one-dimensional integro–differential heat equation are investigated. The direct problem is the initial-boundary problem for this equation. The integral terms have the time convolution form of unknown kernels and direct problem solution. As additional information for solving inverse problem, the solution of the direct problem for $x = x_0$; $x = x_1$ are given. At the beginning an auxiliary problem, which is equivalent to the original problem is introduced. Then the auxiliary problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to 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 is a local existence and uniqueness theorem of inverse problem solutions.