Memory kernel reconstruction problems in the integro‐differential equation of rigid heat conductor

Abstract

The inverse problems of determining the energy–temperature relation α(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 problems, the solution of the direct problem for x = x0 is 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 with weighted norms, we prove the main result of the article, which is a global existence and uniqueness theorem of inverse problem solutions.