Advances in Mathematical Physics

Volume 2018, Article ID 8301656, 8 pages

https://doi.org/10.1155/2018/8301656

## On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data

^{1}Institute of Mathematics and Mathematical Modeling, 125 Pushkin Str., 050010 Almaty, Kazakhstan^{2}Al-Farabi Kazakh National University, 71 Al-Farabi Ave., 050040 Almaty, Kazakhstan^{3}South-Kazakhstan State Pharmaceutical Academy, 1 al-Farabi Sq., 160019 Shymkent, Kazakhstan

Correspondence should be addressed to Makhmud A. Sadybekov; zk.htam@vokebydas

Received 28 February 2018; Accepted 21 May 2018; Published 12 June 2018

Academic Editor: Pavel Kurasov

Copyright © 2018 Makhmud A. Sadybekov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

#### 1. Introduction and Statement of the Problem

The problems that imply the determination of coefficients or the right-hand side of a differential equation (together with its solution) are commonly referred to as inverse problems of mathematical physics. In this paper we consider one family of problems implying the determination of the density distribution and of heat sources from given values of initial and final distributions. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The mathematical statement of such problems leads to an inverse problem for the heat equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable.

In this paper, we will consider the inverse problem close to that investigated in [1, 2]. Together with the solution it is necessary to find an unknown right-hand side of the equation. The equation contains the usual time derivative and an involution with respect to the spatial variable. In contrast to [1], we investigate the problem under nonlocal boundary conditions with respect to the spatial variable. The conditions for overdetermination are initial and final states.

The second of the main differences in the investigated inverse problem being studied is that the unknown function enters, both in the right-hand side of the equation and in the conditions of the initial and final overdetermination.

Let us consider a problem of modeling the thermal diffusion process which is close to that described in the report of Cabada and Tojo [2], where the example that describes a concrete situation in physics is given. Consider a closed metal wire (length ) wrapped around a thin sheet of insulation material in the manner shown in Figure 1.