Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method

This paper aims to study the existence and uniqueness of a weak solution for the boundary value problem of a time fractional equation involving the Caputo fractional derivative with an integral operator. By utilizing the discretization method, we first derive some a priori estimates for the approximate solutions at the points $(x, t_j)$. We then evaluate the accuracy of the proposed method to demonstrate that the implemented sequence of $\alpha$-Rothe functions converges in a certain sense, and its limit is the solution (in a weak sense) of our problem. It must be pointed out that the constructed L1 scheme is designed to approximate the Caputo fractional derivative mentioned in the problem.