Problem of determining two relaxation functions in the integro -
differential equation of rigid heat conductor
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.