An inverse problem of finding the time-dependent diffusion coefficient from an integral condition

We consider the inverse problem of determining the time-dependent diffusivity in one-dimensional heat equation with periodic boundary conditions and nonlocal over-specified data. The problem is highly nonlinear and it serves as a mathematical model for the technological process of external guttering applied in cleaning admixtures from silicon chips. First, the well-posedness conditions for the existence, uniqueness, and continuous dependence upon the data of the classical solution of the problem are established. Then, the problem is discretized using the finite-difference method and recasts as a nonlinear least-squares minimization problem with a simple positivity lower bound on the unknown diffusivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. In order to investigate the accuracy, stability, and robustness of the numerical method, results for a few test examples are presented and discussed.