Approximation of unknown sources in a time fractional PDE by the optimal ones and their reconstruction

Abstract

In this paper, our focus is on studying a geometric inverse source problem that is governed by two-dimensional time-fractional subdiffusion. The problem involves determining the shape and location of the unknown source's geometrical support from boundary measurements of its associated potential. Firstly, we prove the uniqueness of the inverse problem. In the second phase, we propose a novel reconstruction method that utilizes the coupled complex boundary method (CCBM) to solve the identification problem. The main idea of this method is to approximate the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions. Next, we utilize the imaginary part of the solution throughout the domain to construct a shape cost function, which we then minimize with respect to ball-shaped sources by using a Newton-type topological derivative method to reconstruct the geometrical support of the unknown source.