A Numerical Method for a Nonlocal Parameter Identification Problem

This paper addresses the inverse problem of identifying the time-dependent thermal conductivity in the one-dimensional parabolic heat equation from nonlocal overdetermination conditions.Such problems are typical in applications where only partial boundary data are available, leading to ill-posedness.We reformulate the problem after time discretization in such a way that the integral boundary condition is used to decouple the search for the thermal conductivity coefficient from the solution of the parabolic equation.Haar wavelets are used to approximate the solution of the modified Helmholtz equation obtained from time discretization.A system of algebraic equations is obtained via collocation, providing an efficient computational framework for solving the problem.