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.
Highlights
Introduction and Statement of the ProblemThe problems that imply the determination of coefficients or the right-hand side of a differential equation 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
The mathematical statement of such problems leads to an inverse problem for the heat equation, where it is required to find a solution of the problem, and its right-hand side that depends only on a spatial variable
Summary
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 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. The investigated process is reduced to the following mathematical inverse problem: Find the right-hand side f(x) of the heat equation (1) and its solution Φ(x, t) subject to the initial and final conditions (2), the boundary condition (3), and condition (4)
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