An Inverse Problem for Determining a Time-Dependent Coefficient in a Fourth-Order Fractional Equation with the Caputo Derivative

This paper investigates a direct problem and an inverse coefficient problem for a fourth-order fractional differential equation with the Caputo time derivative in a rectangular domain. The inverse problem consists of recovering an unknown time-dependent coefficient from an additional pointwise observation of the solution at a fixed interior point. For the direct problem, the solution is constructed by the Fourier method using the eigenfunctions of the corresponding biharmonic operator. The Fourier coefficients are represented in terms of the two-parameter Mittag--Leffler function, and suitable estimates are established to justify the uniform convergence of the solution series and its required derivatives. These results yield the existence, uniqueness, and a priori estimates for the classical solution of the direct problem. The additional observation condition is then used to reduce the inverse problem to a nonlinear operator equation. By applying the Banach contraction mapping principle, local existence and uniqueness of the unknown coefficient are proved for a sufficiently small time interval. A stability estimate is also obtained, showing continuous dependence of the recovered coefficient on the prescribed data.