Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations
Year: 2017
Author: Chunlong Sun, Gongsheng Li, Xianzheng Jia
Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 6 : pp. 1525–1546
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2016-0170
Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 6 : pp. 1525–1546
Published online: 2017-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Multi-term time-fractional diffusion multivariate Mittag-Leffler function backward problem ill-posedness numerical inversion.