Inverse problems of recovering lower-order coefficients from boundary integral data

TL;DR

This paper investigates inverse problems for second-order parabolic equations, focusing on recovering lower-order coefficients from boundary integral data, with an emphasis on existence, uniqueness, stability, and constructive solution methods.

Contribution

It introduces a new approach to solve inverse coefficient problems using boundary integrals, fixed point theorem, and a priori estimates, applicable to numerical algorithms.

Findings

Proved existence and uniqueness of solutions.

Established stability estimates for inverse problems.

Developed a constructive method for solution approximation.

Abstract

Under consideration are mathematical models of heat and mass transfer. We study inverse problems of recovering lower-order coefficients in a second order parabolic equation. The coefficients are representable in the form of a finite segments of the series whose coefficients depending on time are to be determined. The linear case is also considered. The overdetermination conditions are the integrals over the boundary of the domain of a solution with weights. The main attention is paid to existence, uniqueness, and stability estimates for solutions to inverse problems of this type. The problem is reduced to an operator equation which is studied with the use of the fixed point theorem and a priori estimates. A solution has all generalized derivatives occurring into the equation summable to some power. The method of the proof is constructive and it can be used for developing new numerical algorithms of solving the problem.