Coefficient inverse problem for the advection-dispersion equation with fractional derivatives

In this paper, we study the inverse problem of determining the time-dependent coefficient in a one-dimensional fractional order equation with initial-boundary conditions and over-determination conditions. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using estimates of the Mittag–Leffler function and the method of successive approximations, an estimate of the solution to the direct problem is obtained through the norm of the unknown coefficient, which will be used in the study of the inverse problem. The inverse problem is reduced to an equivalent integral equation. To solve this equation, the contraction mapping principle is used. The results of local existence and uniqueness are proved.