Some inverse problems for time-fractional diffusion equation with nonlocal Samarskii-Ionkin type condition

Two inverse problems for time-fractional diffusion equation having a family of nonlocal boundary conditions are discussed. In first inverse problem, initial distribution is determined provided that the data at final temperaturet=Tis given. Second inverse problem addresses the recovery of temporal component of source term whenever total energy of the system is known. A bi-orthogonal system of functions is used to write the series solution by Fourier's method. The classical nature of the solution of both inverse problems is established by using the estimates of Mittag-Leffler function and by imposing some regularity conditions on given datum.