On Erlang mixture approximations for differential equations with distributed time delays

Tobias K. S. Ritschel

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper rigorously proves kernel-level convergence (pointwise) of Erlang-mixture approximations and compares steady-state and stability criteria via the linear chain trick, but it does not establish trajectory-level error bounds or convergence of solutions as the kernel is approximated. The candidate solution fills this gap by proving L1 convergence of the Erlang-mixture kernels and deriving a Volterra-type comparison inequality that yields a uniform-in-time bound on the state error; under standard local Lipschitz and boundedness assumptions on a finite horizon, the argument is sound.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

This is a solid and timely paper: it gives a general Erlang-mixture approximation framework, a novel pointwise convergence theorem for kernels of wide generality, a clear LCT-based ODE surrogate, and useful stability comparisons. The numerical evidence is careful and compelling. The theoretical contribution would be strengthened by including an explicit trajectory-level error analysis (even under local boundedness and Lipschitz assumptions on a finite horizon), which the present manuscript does not provide. Adding such a result would tighten the link between kernel approximation accuracy and solution accuracy that is currently only observed numerically.