On the recovery of a time dependent diffusion coefficient for a space fractional diffusion equation

An inverse problem of recovering a time dependent diffusion coefficient for a space-fractional diffusion equation has been considered. The space fractional derivative of order 1<α<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\alpha < 2$$\end{document} is defined in the sense of Caputo. Due to an over-determination condition of integral type, we construct a mapping. Under certain conditions on the given data and application of Banach fixed point theorem ensured the unique local existence of the solution, moreover local solution is proved to be classical. The global existence of the solution of the inverse problem is shown by using Schauder fixed point theorem. Examples are also provided to support our analysis.