On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Abstract

We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

Some Boundary Value Problems With Involution for the Nonlocal Poisson Equation

“…А краевые задачи с матрицей вида Q для классического уравнения Пуассона рассмотрены в работах [2][3][4]. Кроме того, в одномерном случае аналогичные задачи для нелокальных аналогов параболических и гиперболических уравнений исследовались в работах [5][6][7][8][9][10][11][12]. Заметим, также, что в работе [13] для уравнения (1) изучена краевая задача с граничным оператором дробного порядка.…”

Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

“…In [25], an inverse problem to determine the right-hand side for a mixed type integro-differential equation with fractional order Gerasimov-Caputo operators is considered. The problem of determining the source function for a degenerate parabolic equation with the Gerasimov-Caputo operator was investigated [26]. In [27], the solvability of the nonlocal boundary problem for a mixed-type differential equation with a fractional-order operator and degeneration is studied.…”

Global existence and blow-up of solutions of the time-fractional space-involution reaction-diffusion equation

“…Furthermore, for the equations containing transformation of the spatial variable in the diffusion term, we can cite Cabada and Tojo [8], where an example that describes a concrete situation in physics is given. Note that, the direct and inverse problems for diffusion and fractional diffusion equations with involutions were studied in [4,5,12,13].…”