Solvability of a coefficient recovery problem for a time-fractional diffusion equation with periodic boundary and overdetermination conditions

This article investigates the inverse problem for time-fractional diffusion equations with periodic boundary conditions and integral overdetermination conditions on a rectangular domain. First, the definition of a classical solution to the problem is introduced. Using the Fourier method, the direct problem is reduced to an equivalent integral equation. The existence and uniqueness of the solution to the direct problem are established by employing estimates for the Mittag-Leffler function and generalized singular Gronwall inequalities. In the second part of the work, the inverse problem is examined. This problem is reformulated as an equivalent integral equation, which is then solved using the contraction mapping principle. Local existence and global uniqueness of the solution are rigorously proven. Furthermore, a stability estimate for the solution is derived. The study contributes to the theory of inverse problems for fractional differential equations by providing a framework for analyzing problems with periodic boundary conditions and integral overdetermination. The methods developed in this work can be applied to a wide range of problems in mathematical physics and engineering, where time-fractional diffusion models are increasingly used to describe complex phenomena.