Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations

Kiyuob Jung

TL;DR

This work develops a novel framework for solving initial-value problems $\frac{du}{dt} = F(t,u)$ using a generalized, one-dimensional specular derivative $f^{\wedge}$. It establishes two representations of $f^{\wedge}$ in terms of one-sided derivatives, proves quasi-Fermat's and quasi-Mean Value theorems in the specular context, and introduces seven derivative-based numerical schemes, highlighting SE5 as a particularly effective method. Under suitable regularity, SE5 achieves first-order consistency and, with a Lipschitz continuous right-hand side, second-order convergence, outperforming classical explicit/implicit Euler and Crank–Nicolson schemes for small step sizes. The analysis relies on auxiliary functions $\mathcal{A}$ and $\mathcal{B}$, whose properties underpin both the theoretical results and the practical SE5 scheme, extending classical differentiability tools to a generalized setting for ODE discretization.

Abstract

This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including quasi-Fermat's theorem and the quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its first-order consistency and second-order local convergence.

Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations

TL;DR

This work develops a novel framework for solving initial-value problems using a generalized, one-dimensional specular derivative . It establishes two representations of in terms of one-sided derivatives, proves quasi-Fermat's and quasi-Mean Value theorems in the specular context, and introduces seven derivative-based numerical schemes, highlighting SE5 as a particularly effective method. Under suitable regularity, SE5 achieves first-order consistency and, with a Lipschitz continuous right-hand side, second-order convergence, outperforming classical explicit/implicit Euler and Crank–Nicolson schemes for small step sizes. The analysis relies on auxiliary functions and , whose properties underpin both the theoretical results and the practical SE5 scheme, extending classical differentiability tools to a generalized setting for ODE discretization.

Abstract

This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including quasi-Fermat's theorem and the quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its first-order consistency and second-order local convergence.
Paper Structure (13 sections, 18 theorems, 137 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 13 sections, 18 theorems, 137 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Organization
  4. Analysis on specular differentiation
  5. Representations and estimates of specular derivatives
  6. Quasi-Fermat's theorem
  7. Quasi-Mean Value Theorem
  8. Numerical schemes
  9. The specular trigonometric scheme
  10. The specular Euler scheme
  11. The specular Euler scheme of Type 5
  12. Analysis on auxiliary functions
  13. The deferred proofs

Key Result

Lemma 2.1

For each $x \in \mathbb{R}$, it holds that

Figures (9)

  • Figure 1: Comparison of four numerical schemes for the equation \ref{['ode: the Dahlquist test eq']} with $u_0 = 1$, $\lambda = -3$, and $h = 0.1$.
  • Figure 2: The specular trigonometric scheme for the equation \ref{['ode: the Dahlquist test eq']} with $u_0 = 1$ and $\lambda = -3$.
  • Figure 3: Comparison of numerical schemes for the equation \ref{['ode: the Dahlquist test eq']} with $u_0 = 1$, $\lambda = 3$, and $h = 0.1$.
  • Figure 4: Comparison of nine numerical schemes for the equation \ref{['ode: the Dahlquist test eq']} with $u_0 = 1$, $\lambda = -3$, and $h = 0.1$.
  • Figure 5: Comparison of five schemes for the equation \ref{['ode: the circle equation']} with $h = 0.01$.
  • ...and 4 more figures

Theorems & Definitions (53)

  • Definition 1.1
  • Example 1
  • Lemma 2.1
  • Proof 1
  • Theorem 2.2
  • Proof 2
  • Corollary 2.3
  • Proof 3
  • Example 2
  • Remark 2.4
  • ...and 43 more