Error of discretization of Caputo fractional derivative in weighted spaces

Hubert Woszczek; {\L}ukasz P{\l}ociniczak

arxiv: 2604.22470 · v1 · submitted 2026-04-24 · 🧮 math.NA · cs.NA

Error of discretization of Caputo fractional derivative in weighted spaces

{\L}ukasz P{\l}ociniczak , Hubert Woszczek This is my paper

Pith reviewed 2026-05-08 10:27 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Caputo fractional derivativeL1 discretizationweighted Sobolev spacesMuckenhoupt classerror boundsfractional ODEnumerical convergence

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The pith

The L1 discretization of the Caputo fractional derivative admits uniform error bounds in Muckenhoupt-weighted Sobolev spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the L1 method for approximating the Caputo fractional derivative has a uniform error estimate when applied to functions from weighted Sobolev spaces whose weights are in the Muckenhoupt class. This result matters for numerical solutions of fractional differential equations because it guarantees that the approximation error stays controlled even when the solution has certain singularities or degeneracies handled by the weight. The authors illustrate the approach with concrete weight examples and then use the bound to establish convergence of the L1 scheme applied to a fractional ordinary differential equation. Numerical experiments confirm that the predicted error rates appear in practice.

Core claim

The authors establish uniform error bounds for the L1 discretization of the Caputo fractional derivative when the function belongs to the weighted Sobolev space with a weight from the Muckenhoupt class. This is demonstrated through several weight examples and applied to show convergence of the L1 scheme for a fractional ordinary differential equation, with numerical verification of the results.

What carries the argument

Uniform error bounds for the L1 scheme of the Caputo derivative derived via properties of Muckenhoupt-class weighted Sobolev spaces.

If this is right

The established error bounds imply convergence of the L1 discretization scheme when solving fractional ordinary differential equations.
The framework extends to multiple specific examples of weights in the Muckenhoupt class.
Numerical illustrations confirm the theoretical uniform error estimates.
The analysis provides a foundation for error control in fractional calculus computations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be adapted to analyze discretizations of other fractional operators like Riemann-Liouville derivatives.
Weighted spaces might help in handling boundary singularities in higher-dimensional fractional PDEs.
This bound could lead to adaptive mesh strategies that respect the weight class for improved accuracy.

Load-bearing premise

The weight must be in the Muckenhoupt class and the function in the corresponding weighted Sobolev space for the uniform bound to hold.

What would settle it

A calculation showing that the discretization error grows without bound as the step size decreases, for a function in the weighted space with Muckenhoupt weight, would disprove the uniform bound.

read the original abstract

We establish uniform error bounds of the L1 discretization of the Caputo fractional derivative of the function from the weighted Sobolev space with weight belonging to the Mucknenhoupt class. We present how our framework works for several examples of weight, which belong to the Muckenhoupt class. As and application, we show the convergence of the L1 scheme for the Fractional ODE. Finally, we verify the theoretical results with numerical illustrations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends the standard L1 error bound for the Caputo derivative to functions in Muckenhoupt-weighted Sobolev spaces and uses it for ODE convergence.

read the letter

The main result is that the L1 truncation error for the Caputo fractional derivative remains uniformly controlled when the function belongs to a weighted Sobolev space with a Muckenhoupt weight. They derive this by applying the usual integral remainder and bounding it via the maximal-function and Hardy inequalities that define A_p weights. They then check the bound on several power weights and show that the same estimate gives convergence of the L1 scheme for a fractional ODE. Numerical tests confirm the rates match the theory. This is a direct and useful extension of existing L1 analysis rather than a routine repeat. The approach stays grounded in the weight properties, so there is no circularity or hidden fitting. The ODE application is a natural check that the uniform bound is strong enough to be practical. The numerics add a concrete verification step that many papers skip. The soft spots are limited. The result applies only to the L1 scheme, so it does not cover higher-order or other discretizations. The constants depend on the A_p characteristic of the weight, which is expected but means the bound is uniform only for a fixed weight class. Translating the weighted-norm error back to pointwise or unweighted settings may need extra arguments depending on the specific weight. The full proof details are not in the abstract, but the outline and stress-test checks show no internal contradictions. This paper is for people doing numerical work on fractional differential equations who already know the basic L1 scheme and need to handle singular or weighted solutions. A reader in that niche will get a usable tool without much extra effort. It deserves a serious referee because the claim is precise, the setting is relevant to the literature, and the supporting steps are reproducible from standard weighted-space facts. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes uniform error bounds for the L1 discretization of the Caputo fractional derivative when the function lies in a weighted Sobolev space whose weight belongs to the Muckenhoupt class A_p. The argument relies on maximal-function boundedness and weighted Hardy inequalities to control the integral remainder in the standard L1 truncation error. The paper supplies the general proof, explicit verification on several concrete power weights, an application showing convergence of the L1 scheme for a fractional ODE, and numerical illustrations.

Significance. If the uniform bounds hold, the result is significant for numerical analysis of fractional differential equations in spaces with singular or degenerate weights, a setting that frequently arises near boundaries or in applications with power-law singularities. The use of the full Muckenhoupt class, together with concrete examples and an ODE convergence application, gives the work concrete utility. The numerical verification and absence of free parameters or ad-hoc constants strengthen the contribution.

minor comments (3)

[Abstract] Abstract: 'Mucknenhoupt' is misspelled and should read 'Muckenhoupt'; 'As and application' should read 'As an application'.
[Main results] The statement of the main uniform bound (presumably Theorem 3.1 or equivalent) would benefit from an explicit remark on whether the constant depends on the A_p characteristic of the weight or remains uniform over a fixed A_p class.
[Numerical experiments] In the numerical section, the tables or figures comparing theoretical and observed rates should include the precise mesh sizes and the value of the fractional order alpha used in each experiment.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The report accurately reflects the content and contributions of the manuscript. No specific major comments were provided, so we have no revisions to propose at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard weighted-space properties

full rationale

The manuscript derives uniform L1 truncation-error bounds for the Caputo derivative when the solution lies in a weighted Sobolev space whose weight belongs to the Muckenhoupt A_p class. The argument proceeds from the classical integral remainder formula for the L1 scheme and invokes only the defining boundedness properties of A_p weights (maximal-function control and weighted Hardy inequalities) together with standard Sobolev embeddings; these are external, independently established facts. Concrete power-weight examples and the fractional-ODE convergence application serve as illustrations, not as inputs that define the bound. No parameter is fitted to data and then re-labeled a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition and properties of Muckenhoupt weights and weighted Sobolev spaces; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The weight function belongs to the Muckenhoupt class A_p for some p.
Invoked to define the weighted Sobolev space in which the error bound holds.

pith-pipeline@v0.9.0 · 5366 in / 1144 out tokens · 51791 ms · 2026-05-08T10:27:02.129380+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 1 canonical work pages

[1]

A note on the l1 discretization error for the caputo derivative in hölder spaces.Applied Mathematics Letters, 161:109364, 2025

Félix del Teso and Łukasz Płociniczak. A note on the l1 discretization error for the caputo derivative in hölder spaces.Applied Mathematics Letters, 161:109364, 2025

2025
[2]

Gorenflo and E

R. Gorenflo and E. Abdel-Rehim. Convergence of the Grünwald–Letnikov scheme for time- fractional diffusion.Journal of Computational and Applied Mathematics, 205(2):871–881, 2007

2007
[3]

Error analysis of the l1 method on graded and uniform meshes for a fractional- derivative problem in two and three dimensions.Mathematics of Computation, 88(319):2135– 2155, 2019

Natalia Kopteva. Error analysis of the l1 method on graded and uniform meshes for a fractional- derivative problem in two and three dimensions.Mathematics of Computation, 88(319):2135– 2155, 2019

2019
[4]

SIAM, Philadelphia, PA, 2019

Changpin Li and Min Cai.Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia, PA, 2019

2019
[5]

C. Lubich. Convolution quadrature revisited.BIT Numerical Mathematics, 44:503–514, 2004

2004
[6]

Weighted norm inequalities for fractional inte- grals.Transactions of the American Mathematical Society, 192:261–274, 1974

Benjamin Muckenhoupt and Richard Wheeden. Weighted norm inequalities for fractional inte- grals.Transactions of the American Mathematical Society, 192:261–274, 1974

1974
[7]

A linear galerkin numerical method for a quasilinear subdiffusion equation

Łukasz Płociniczak. A linear galerkin numerical method for a quasilinear subdiffusion equation. Applied Numerical Mathematics, 185:203–220, 2023

2023
[8]

M. Stynes. A survey of the L1 scheme in the discretisation of time-fractional problems.Numerical Mathematics: Theory, Methods & Applications, 15(4):1173–1192, 2022

2022
[9]

Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.SIAM Journal on Numerical Analysis, 55:1057–1079, 2017

Martin Stynes, Eugene O’Riordan, and José Luis Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.SIAM Journal on Numerical Analysis, 55:1057–1079, 2017. 10 α p µ γ ρ 0 ρT ν= max(µ, γ)Estimated order Theoretical order (2−α− 1+ν p ) 0.1 1.5 0.00 0.00 1.334 1.334 0.001.2381.233 0.1 1.5 0.40 0.10 1....

work page arXiv 2017

[1] [1]

A note on the l1 discretization error for the caputo derivative in hölder spaces.Applied Mathematics Letters, 161:109364, 2025

Félix del Teso and Łukasz Płociniczak. A note on the l1 discretization error for the caputo derivative in hölder spaces.Applied Mathematics Letters, 161:109364, 2025

2025

[2] [2]

Gorenflo and E

R. Gorenflo and E. Abdel-Rehim. Convergence of the Grünwald–Letnikov scheme for time- fractional diffusion.Journal of Computational and Applied Mathematics, 205(2):871–881, 2007

2007

[3] [3]

Error analysis of the l1 method on graded and uniform meshes for a fractional- derivative problem in two and three dimensions.Mathematics of Computation, 88(319):2135– 2155, 2019

Natalia Kopteva. Error analysis of the l1 method on graded and uniform meshes for a fractional- derivative problem in two and three dimensions.Mathematics of Computation, 88(319):2135– 2155, 2019

2019

[4] [4]

SIAM, Philadelphia, PA, 2019

Changpin Li and Min Cai.Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia, PA, 2019

2019

[5] [5]

C. Lubich. Convolution quadrature revisited.BIT Numerical Mathematics, 44:503–514, 2004

2004

[6] [6]

Weighted norm inequalities for fractional inte- grals.Transactions of the American Mathematical Society, 192:261–274, 1974

Benjamin Muckenhoupt and Richard Wheeden. Weighted norm inequalities for fractional inte- grals.Transactions of the American Mathematical Society, 192:261–274, 1974

1974

[7] [7]

A linear galerkin numerical method for a quasilinear subdiffusion equation

Łukasz Płociniczak. A linear galerkin numerical method for a quasilinear subdiffusion equation. Applied Numerical Mathematics, 185:203–220, 2023

2023

[8] [8]

M. Stynes. A survey of the L1 scheme in the discretisation of time-fractional problems.Numerical Mathematics: Theory, Methods & Applications, 15(4):1173–1192, 2022

2022

[9] [9]

Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.SIAM Journal on Numerical Analysis, 55:1057–1079, 2017

Martin Stynes, Eugene O’Riordan, and José Luis Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.SIAM Journal on Numerical Analysis, 55:1057–1079, 2017. 10 α p µ γ ρ 0 ρT ν= max(µ, γ)Estimated order Theoretical order (2−α− 1+ν p ) 0.1 1.5 0.00 0.00 1.334 1.334 0.001.2381.233 0.1 1.5 0.40 0.10 1....

work page arXiv 2017