Mathematical Modeling

Inverse problem of recovering fluxes from integral data

S. G. Pyatkov, E. I. Safonov, A. A. Petrov

Yugra State University, Khanty-Mansiysk, Russian Federation

Abstract: In the article we consider a second order parabolic equation and well-posedness questions in Sobolev spaces of inverse problems of recovering the heat flux on the boundary with the use of a collection of integrals of a solution with weights over the domain. The flux is representable in the form of a finite segments of the series with unknown coefficients depending on time. Under certain conditions on the data, it is demonstrated that there exists a unique solution to the problem which depends on the data continuously. A solution has all generalized derivatives occurring into the equation summable to some power. The proof relies on a priori estimates and the contraction mapping principle. The method is constructive and allows to provide numerical methods of solving the problem. The numerical algorithm is based on the finite element methods and the method of finite differences. The results of numerical experiments are quite satisfactory and the procedure of constructing a solution is stable under small perturbations.

Keywords: inverse problem, boundary regime, parabolic equation, heat and mass transfer.

UDC: 517.956

MSC: 35R30, 35K20, 80A23

Received: 22.05.2025

Language: English

DOI: 10.14529/mmp250404