Pointwise Error Estimates of $L1$ Method for Multi-Singularity Problems Arising in Delay Fractional Equations

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.