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

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

Author Details

Chunlong Sun

Gongsheng Li

Xianzheng Jia

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