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Volume 14, Issue 4
Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations

Dakang Cen, Hui Liang & Seakweng Vong

East Asian J. Appl. Math., 14 (2024), pp. 820-840.

Published online: 2024-09

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  • Abstract

Error estimates of $L1$ scheme for delay fractional equations are derived by discrete Laplace transform method. Theoretical result shows that the convergence order is ${\rm min}\{(k+1)α, 1\}$ at $(k\tau)^+,$ where $k ∈ \mathbb{N},$ $\tau$ is delay factor, $α ∈ (0, 1)$ is the order of Caputo fractional derivative. At the points without derivative discontinuities, first order convergence is achieved. The uniqueness of the inverse problem, the reaction coefficient, and the delay factor are established by employing asymptotic expansions and the monotonicity of the Mittag-Leffler function. An inversion algorithm based on the Tikhonov regularization method is given.

  • AMS Subject Headings

34A08

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-14-820, author = {Cen , DakangLiang , Hui and Vong , Seakweng}, title = {Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2024}, volume = {14}, number = {4}, pages = {820--840}, abstract = {

Error estimates of $L1$ scheme for delay fractional equations are derived by discrete Laplace transform method. Theoretical result shows that the convergence order is ${\rm min}\{(k+1)α, 1\}$ at $(k\tau)^+,$ where $k ∈ \mathbb{N},$ $\tau$ is delay factor, $α ∈ (0, 1)$ is the order of Caputo fractional derivative. At the points without derivative discontinuities, first order convergence is achieved. The uniqueness of the inverse problem, the reaction coefficient, and the delay factor are established by employing asymptotic expansions and the monotonicity of the Mittag-Leffler function. An inversion algorithm based on the Tikhonov regularization method is given.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2023-168.180923}, url = {http://global-sci.org/intro/article_detail/eajam/23439.html} }
TY - JOUR T1 - Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations AU - Cen , Dakang AU - Liang , Hui AU - Vong , Seakweng JO - East Asian Journal on Applied Mathematics VL - 4 SP - 820 EP - 840 PY - 2024 DA - 2024/09 SN - 14 DO - http://doi.org/10.4208/eajam.2023-168.180923 UR - https://global-sci.org/intro/article_detail/eajam/23439.html KW - Delay fractional equation, multi-singularity problem, $L1$ method, pointwise error estimate, simultaneous inversion of multi-parameters. AB -

Error estimates of $L1$ scheme for delay fractional equations are derived by discrete Laplace transform method. Theoretical result shows that the convergence order is ${\rm min}\{(k+1)α, 1\}$ at $(k\tau)^+,$ where $k ∈ \mathbb{N},$ $\tau$ is delay factor, $α ∈ (0, 1)$ is the order of Caputo fractional derivative. At the points without derivative discontinuities, first order convergence is achieved. The uniqueness of the inverse problem, the reaction coefficient, and the delay factor are established by employing asymptotic expansions and the monotonicity of the Mittag-Leffler function. An inversion algorithm based on the Tikhonov regularization method is given.

Cen , DakangLiang , Hui and Vong , Seakweng. (2024). Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations. East Asian Journal on Applied Mathematics. 14 (4). 820-840. doi:10.4208/eajam.2023-168.180923
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