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A note on the L1 discretization error for the Caputo derivative in Hölder spaces

Félix del Teso, Łukasz Płociniczak

Abstract

We establish uniform error bounds of the L1 discretization of the Caputo derivative of Hölder continuous functions. The result can be understood as: error = (degree of smoothness - order of the derivative). We present an elementary proof and illustrate its optimality with numerical examples.

A note on the L1 discretization error for the Caputo derivative in Hölder spaces

Abstract

We establish uniform error bounds of the L1 discretization of the Caputo derivative of Hölder continuous functions. The result can be understood as: error = (degree of smoothness - order of the derivative). We present an elementary proof and illustrate its optimality with numerical examples.

Paper Structure

This paper contains 3 sections, 2 theorems, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. The main result
  3. Numerical illustration

Key Result

Theorem 1

Let $y \in C^{k,\beta}[0,T]$ with $k=0,1$ and $0\leq \beta \leq 1$ with $k+\beta > \alpha$. Then, the truncation error of the L1 discretization eqn:L1Scheme for $t_n\in (0, T]$ satisfies where the error constant has an explicit form

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:Truncation']}
  • Remark 6