Volume 9, Issue 6
Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations

Chunlong Sun, Gongsheng Li & Xianzheng Jia

Adv. Appl. Math. Mech., 9 (2017), pp. 1525-1546.

Published online: 2017-09

Preview Full PDF 120 1347
Export citation
  • 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 diffusion, multivariate Mittag-Leffler function, backward problem, ill-posedness, numerical inversion.

  • AMS Subject Headings

35R11, 35R30, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{AAMM-9-1525, author = {}, title = {Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2017}, volume = {9}, number = {6}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2016-0170}, url = {http://global-sci.org/intro/article_detail/aamm/10191.html} }
TY - JOUR T1 - Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1525 EP - 1546 PY - 2017 DA - 2017/09 SN - 9 DO - http://doi.org/10.4208/aamm.OA-2016-0170 UR - https://global-sci.org/intro/article_detail/aamm/10191.html KW - Multi-term time-fractional diffusion, multivariate Mittag-Leffler function, backward problem, ill-posedness, numerical inversion. AB -

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.

Chunlong Sun, Gongsheng Li & Xianzheng Jia. (2020). Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations. Advances in Applied Mathematics and Mechanics. 9 (6). 1525-1546. doi:10.4208/aamm.OA-2016-0170
Copy to clipboard
The citation has been copied to your clipboard