Determining flux terms in a time fractional model

In this paper, we investigate the flux identification problem for a nonlinear time-fractional viscoelastic equation with a general source function, using boundary measurements. We first establish the well-posedness of the direct problem by proving the existence, uniqueness, and continuous dependence of the solution on the heat flux. Next, we demonstrate the Fréchet differentiability of the cost functional, providing a theoretical foundation for solving the inverse problem. To efficiently reconstruct the unknown flux, we develop a Conjugate Gradient Algorithm based on the derived gradient formula. Finally, we validate the effectiveness and robustness of our approach through numerical experiments, including both noise-free and noisy data, confirming its accuracy and practical applicability.