Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations (Advances in Applied Mathematics and Mechanics, 2021)
This article deals with numerical inversion for the initial distribution in themulti-term time-fractional diffusion equation using final observations. The inversionproblem is of instability, but it is uniquely solvable based on the solution's expressionfor the forward problem and estimation to the multivariate Mittag-Leffler function.From view point of optimality, solving the inversion problem is transformed to minimizinga cost functional, and existence of a minimum is proved by the weakly lowersemi-continuity of the functional. Furthermore, the homotopy regularization algorithmis introduced based on the minimization problem to perform numerical inversions,and the inversion solutions with noisy data give good approximations to the exactinitial distribution demonstrating the efficiency of the inversion algorithm.