Identifying the source term in a viscoelastic membrane with a Riemann-Liouville time derivative by the partial interior observation

Abstract

This paper studies an inverse source problem for a viscoelastic membrane, where the material's memory effect is characterized by the Riemann-Liouville fractional derivative. The problem is to recover the unknown source term from the limited interior observation data. We propose an optimal control framework to address this ill-posed inverse problem. The first-order optimality condition leads to a coupled system of forward and backward fractional partial differential equations. A numerical algorithm combining the finite element method and a conjugate gradient iterative scheme is then developed for the reconstruction of the source term. Several numerical examples are provided to demonstrate the effectiveness and robustness of the proposed method.