Inverse Problem for the Heat Equation with Two Unknown Coefficients

Ishmeev, M. R.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4.

Abstract:
In this paper, the problem of the simultaneous recovery of the thermal diffusivity coefficient and the rapidly oscillating by time source coefficient in the one-dimensional initial-boundary problem with Dirichlet boundary conditions and the inhomogeneous initial condition for the heat equation is solved using some information on the partial asymptotics of its solution. It is shown that the coefficients can be restored from certain data on the incomplete asymptotics of the solution. The asymptotics of the solution of the direct initial-boundary value problem is preliminarily constructed and substantiated. This paper was stimulated by A. M. Denisov research works, in which a number of different inverse coefficient problems for parabolic equations were investigated, but without considering high-frequency oscillations. And it also continues the research begun by V. B. Levenshtam and his students, in which inverse problems for parabolic equations with high-frequency coefficients were considered for the first time and the methodology for solving such problems was developed. In contrast to this study, where only the source function or its multiplicators are unknown, in this paper, we assume that the thermal diffusivity coefficient and the source function multiplicator are simultaneously unknown. Note that problems with rapidly oscillating in time data simulate a number of physical phenomena and processes related with high frequency impact.

Keywords: inverse problem, heat equation, rapidly oscillating coefficients, asymptotics

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