Direct and Inverse Problems for the Heat Equation with a Dynamic type Boundary Condition

TL;DR

This paper studies the heat equation with a dynamic boundary condition, proving existence and uniqueness of solutions, and explores an inverse problem to determine a time-dependent coefficient from integral data.

Contribution

It introduces a generalized Fourier method for solving the direct problem and addresses the inverse problem of coefficient identification from overdetermined data.

Findings

Existence and uniqueness of classical solutions established.

Solution depends continuously on the data.

Inverse problem of coefficient recovery is formulated and analyzed.

Abstract

This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition.