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On the error of the Euler scheme for approximation of solutions of nonlinear DDEs under inexact information

Paweł Przybyłowicz, Martyna Wiącek

Abstract

We analyze the behavior of the Euler method for delay differential equations under nonstandard assumptions on the right-hand-side function f, when evaluations of f are corrupted by informational noise. We provide theoretical upper bounds on the Euler discretization error and present results from the numerical experiments.

On the error of the Euler scheme for approximation of solutions of nonlinear DDEs under inexact information

Abstract

We analyze the behavior of the Euler method for delay differential equations under nonstandard assumptions on the right-hand-side function f, when evaluations of f are corrupted by informational noise. We provide theoretical upper bounds on the Euler discretization error and present results from the numerical experiments.

Paper Structure

This paper contains 19 sections, 8 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inexact information model of computation
  3. Analysis under the Global Lipschitz Condition
  4. Problem formulation
  5. Error of the Euler scheme
  6. Analysis under a Local One-Sided Lipschitz and Local Hölder Condition
  7. Problem formulation
  8. Error of the Euler scheme
  9. Numerical experiments
  10. Test equations
  11. Example 1 (metal-type drift with $\mathrm{sgn}(y)$).
  12. Example 2 (one-sided power nonlinearity in $y$ and Hölder term in $z$).
  13. Example 3 (symmetric power in $y$ and Hölder term in $z$).
  14. Example 4 (modified oscillatory delay equation).
  15. Perturbed information model
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\tau \in (0,+\infty)$, $n \in \mathbb{N}$, $\eta \in \mathbb{R}^d$, and let $f$ satisfy assumptions (E1)–(E2). There exist constants $\tilde{C}_0, \tilde{C}_1, \dots, \tilde{C}_n \in (0,+\infty)$, such that for all $N \in \mathbb{N}$ the following holds $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Empirical convergence plots for the four test equations $f_1,f_2,f_3,f_4$ and two values of $\gamma$.
  • Figure 2: Cumulative supremum errors for the four test equations $f_1,f_2,f_3,f_4$ and two values of $\gamma$.

Theorems & Definitions (19)

  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Lemma 3
  • ...and 9 more