Identifying the Fractional Order and Time‐Dependent Source in a Fourth‐Order Diffusion Equation Under Samarskii–Ionkin Boundary Conditions

This paper considers an inverse problem of simultaneously identifying the fractional order and the time‐dependent source for a time‐fractional fourth‐order diffusion equation under Samarskii–Ionkin type nonlocal boundary conditions. Firstly, the regularity of the direct problem is discussed based on a non‐self‐adjoint spectral problem. Secondly, the uniqueness of the simultaneous inversion of the fractional order and the time‐dependent source is proved, an estimate for the measured data is provided, and the continuity of the forward mapping is further established. Finally, the iterative regularizing ensemble Kalman method is used to invert the fractional order and time‐dependent source of the equation simultaneously, and two numerical examples are given to illustrate the effectiveness of the method.