Inverse Coefficient Problem for a Time-Fractional Diffusion Equation in the Bounded Domain
Abstract
In this paper, an inverse problem of determining a time-dependent source coefficient in a one-dimensional time-fractional diffusion equation is investigated with initial-boundary and overdetermination conditions. Before, equivalent auxiliary problem is obtained for direct problem. By Fourier method this auxiliary problem is reduced to equivalent integral equation. Then, using estimates of the Mittag-Leffler function and generalized singular Gronwall inequalities, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness are proven. Also the stability estimate is obtained.