Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations

This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations. The inversion problem is of instability, but it is uniquely solvable based on the solution's expression for the forward problem and estimation to the multivariate Mittag-Leffler function. From view point of optimality, solving the inversion problem is transformed to minimizing a cost functional, and existence of a minimum is proved by the weakly lower semi-continuity of the functional. Furthermore, the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions, and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.