Determination of time-dependent coefficient in time fractional heat equation

Abstract

The aim of this work is to determine the time-dependent heat coefficient in a type of inverse problem for one dimensional time-fractional heat equations defined by the Caputo operator for (0<α<1) by applying a method based on the finite difference scheme and Tikhonov’s regularization. First, a stable implicit finite difference scheme is applied to find a numerical solution to the (forward) direct problem. While the inverse problem was reformulated as a nonlinear least-square minimization problem with a simple physical bound on the unknown coefficient and solved efficiently by MATLAB routine lsqnonlin from the optimization toolbox. But the latter problem will remain ill-posed because the presence of any error in the input data will lead to a large error in the output data. So, Tikhonov’s technique is applied to obtain stable results. In the end, two numerical test examples show that the proposed method is stable, accurate, and works well with different noise levels