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

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.

Nonlocal spectral problem for a second-order differential equation with an involution

“…Consider an arbitrary function f (x) ∈ L 2 (−1, 1) orthogonal to all functions of system (4). Since it is orthogonal to the functions u (1) l (x), l ∈ N, we see that it coincides almost everywhere with an even function. Thus, Since systems (4) and (5) are complete, it follows that they are closed in L 2 (−1, 1); and since they correspond to mutually adjoint problems, we find that they are minimal.…”

“…u (−1) = u (1), u(−1) = βu(1), (1) where the differential expression contains an involution transformation of the independent variable in the highest derivative and the boundary conditions are nonlocal.…”

“…Throughout the following, the parameter α in problem (1) is an arbitrary number in the interval (−1, 1). The case β = 1 (when the boundary conditions of the problem are periodic) was investigated in [1]. The case when β = −1 leads to a degenerate problem.…”

“…Note that functional-differential equations similar to the equation in (1) were studied by numerous authors. The algebraic and analytic aspects of the theory of ordinary differential equations with involution were discussed in the monographs [12,13].…”

On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures

“…We solve the problem by the Fourier method. Some new variants for solving nonlocal boundary value problems by the method of separation of variables were used in our papers [29][30][31][32][33][34][35].…”