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

TL;DR

This paper develops a novel method using fractional calculus and optimal control to accurately identify unknown source terms in viscoelastic membranes from limited interior data.

Contribution

It introduces an optimal control approach with a coupled fractional PDE system and a numerical scheme for effective source reconstruction in viscoelastic membranes.

Findings

The method successfully reconstructs source terms from limited data.

Numerical examples demonstrate robustness and accuracy.

The approach handles the memory effect via Riemann-Liouville derivatives.

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.