Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

TL;DR

This paper analyzes the accuracy of the Euler scheme for delay differential equations when the right-hand side function has nonstandard regularity, including local Hölder continuity, and provides detailed error estimates and numerical results.

Contribution

It introduces a novel error analysis for the Euler scheme applied to DDEs with nonstandard assumptions on the function's regularity, including local Hölder continuity.

Findings

Error bounds for Euler scheme under nonstandard assumptions

Numerical experiments confirming theoretical results

Extension of classical analysis to less regular functions

Abstract

We deal with approximation of solutions of delay differential equations (DDEs) via the classical Euler algorithm. We investigate the pointwise error of the Euler scheme under nonstandard assumptions imposed on the right-hand side function $f$. Namely, we assume that $f$ is globally of at most linear growth, satisfies globally one-side Lipschitz condition but it is only locally H\"older continuous. We provide a detailed error analysis of the Euler algorithm under such nonstandard regularity conditions. Moreover, we report results of numerical experiments.