An inverse problem of identifying the coefficient in a nonlinear time-fractional diffusion equation

The paper deals with an inverse problem of identifying parameters in a nonlinear subdiffusion model from a final observation. The nonlinear subdiffusion model involves a Caputo fractional derivative of order α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} in time. Such problem has important application in a large field of applied science. To treat our model, we first study the regularity of the solution for the direct problem by means of Mittag–Leffler functions. Second, to study our inverse parameter problem, we reformulate it into an optimal control one with a Least Squares cost function and we establish the existence of the optimal solution. Third, we show the uniqueness and stability with respect to the data of our inverse problem based on the optimality conditions of the considered functional and the regularity of the solution for the direct problem. Finally, we present some numerical experiments using descent gradient algorithm.