An inverse problem for 1D fractional integro-differential wave equation with fractional time derivative

This paper is devoted to obtaining a unique solution to an inverse problem for a one-dimensional time-fractional integro-differential equation. First, we consider the direct problem, and the unique existence of the weak solution is established, after that, the smoothness conditions for the solution are obtained. Secondly, we study the inverse problem of determining the unknown coefficient and kernel, and the well-posedness of this inverse problem is proved. The local existence and global uniqueness results are based on the Fourier method, fractional calculus, properties of the Mittag-Leffler function, and Banach fixed point theorem in a suitable Sobolev space.