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 - <p style="text-align: justify;">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.</p>