Determination of the flux terms in a time fractional viscoelastic equation

TL;DR

This paper addresses the inverse problem of identifying flux terms in a nonlinear time-fractional viscoelastic equation using boundary measurements, establishing well-posedness, differentiability, and proposing a numerical solution with demonstrated accuracy.

Contribution

It introduces a novel approach for flux identification in fractional viscoelastic equations, including theoretical proofs and a practical conjugate gradient algorithm.

Findings

The direct problem is well-posed with continuous dependence on flux.

The cost functional is Fréchet differentiable.

Numerical examples show the method's accuracy with noisy data.

Abstract

In this paper, we study the flux identification problem for a nonlinear time-fractional viscoelastic equation with a general source function based on the boundary measurements. We prove that the direct problem is well-posed, i.e., the solution exists, unique and depends continuously on the heat flux. Then the Fr\'echet differentiability of the cost functional is proved. The Conjugate Gradient Algorithm, based on the gradient formula for the cost functional, is proposed for numerical solution of the inverse flux problem. The numerical examples, both with noise-free and noisy data, provide a clear demonstration of the applicability and accuracy of the proposed method.