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

Part of: Miscellaneous topics - Partial differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  28 November 2017

Chunlong Sun ,
Gongsheng Li  and
Xianzheng Jia
Chunlong Sun
Affiliation:
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China
Gongsheng Li*
Affiliation:
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
Xianzheng Jia
Affiliation:
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
*
*Corresponding author. Email:ligs@sdut.edu.cn (G. S. Li)
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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.

Keywords

Multi-term time-fractional diffusionmultivariate Mittag-Leffler functionbackward problemill-posednessnumerical inversion

MSC classification

Secondary: 35R11: Fractional partial differential equations 35R30: Inverse problems 65M06: Finite difference methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1525 - 1546
Copyright
Copyright © Global-Science Press 2017 

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