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

- Zhonibek Zhumaev,
- Durdimurod Durdiev

## 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 = x_{0}
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.

24 Sep 2020Submitted to *Mathematical Methods in the Applied Sciences* 25 Sep 2020Submission Checks Completed

25 Sep 2020Assigned to Editor

08 Oct 2020Reviewer(s) Assigned

04 Nov 2020Review(s) Completed, Editorial Evaluation Pending

08 Nov 2020Editorial Decision: Revise Major

19 Nov 20201st Revision Received

19 Nov 2020Submission Checks Completed

19 Nov 2020Assigned to Editor

24 Nov 2020Reviewer(s) Assigned

25 Nov 2020Review(s) Completed, Editorial Evaluation Pending

25 Nov 2020Editorial Decision: Accept