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arxiv: 1907.07308 · v1 · pith:FJ4H4QMKnew · submitted 2019-07-17 · 🧮 math.NA · cs.NA

Corrections on A numerical method for solving nonlinear Volterra--Fredholm integral equations

Ngo Thanh Binh , Khuat Van Ninh This is my paper

Pith reviewed 2026-05-24 20:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Volterra-Fredholm integral equationsnonlinear integral equationsnumerical discretizationcorrectionsnumerical analysis
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The pith

Corrections refine the discretization of a nonlinear Volterra-Fredholm integral equation into a tighter form while leaving the original main results unchanged.

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

The paper supplies targeted corrections to an earlier numerical method for solving the nonlinear integral equation that combines a Volterra term over [a,t] with a Fredholm term over [a,b]. These corrections adjust how the equation is turned into its discrete counterpart so that the approximation becomes tighter and more accurate. The authors state that the changes leave the principal theorems and conclusions of the original article intact. A reader would care because small errors in discretization can accumulate in numerical solutions of such mixed integral equations, and the note offers a precise fix without requiring a full rewrite of the theory.

Core claim

The corrections transform equation (1.1), x(t) plus the Volterra integral of K1(t,s,x(s)) from a to t plus the Fredholm integral of K2(t,s,x(s)) from a to b equals g(t), into a discretized form that is tighter and more accurate than the version appearing in the 2019 article, without affecting the main results.

What carries the argument

The corrected discretization procedure applied to the mixed Volterra-Fredholm integral equation (1.1).

If this is right

  • Numerical solutions obtained with the corrected scheme satisfy the original error bounds and convergence statements.
  • The method can be implemented by replacing only the discretization step in existing code based on the 2019 article.
  • Test problems used to illustrate the original method remain valid demonstrations once the corrected discretization is applied.

Where Pith is reading between the lines

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

  • Readers who implemented the 2019 algorithm may need to update only the part that converts the continuous equation to discrete form.
  • The note implies that the original discretization contained an identifiable looseness that can be removed without altering the underlying convergence analysis.

Load-bearing premise

The corrections really leave the original main results unaffected.

What would settle it

Direct comparison of the numerical output produced by the corrected discretization against the output of the original discretization on the same test problems from the 2019 paper.

read the original abstract

Some corrections are made in our article, which was published in Appl. Anal. Optim. Vol. 3 (2019), No. 1, 103--127. These corrections are intended to transform the equation \eqref{eq:1.1} \begin{equation}\label{eq:1.1} x(t) + \int\limits_a^t {K_1(t,s,x(s)) ds} + \int\limits_a^b {K_2(t,s,x(s)) ds} = g(t),\;\,a \le t \le b \tag{1.1} \end{equation} into a discretized form in a tighter and more accurate way without affecting the main results of the article.

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.

Referee Report

1 major / 0 minor

Summary. This short correction note to the authors' 2019 paper in Appl. Anal. Optim. states that unspecified corrections have been made to the discretization of the nonlinear Volterra-Fredholm equation (1.1) so that it becomes tighter and more accurate, while asserting that the original paper's main results (convergence, error estimates, existence) remain unaffected.

Significance. Correction notes serve a useful archival purpose when they clearly document changes and confirm that theoretical conclusions are preserved. If the corrections were explicitly listed and shown to be local to the discretization step, the note would be a modest but positive contribution to the record of the original numerical method. The current text supplies neither the corrections nor any verification, so the significance is limited to a statement of intent.

major comments (1)
  1. [Abstract] Abstract (and entire manuscript): the central claim that the corrections 'transform the equation (1.1) into a discretized form in a tighter and more accurate way without affecting the main results' is unsupported. No explicit form of the corrected discretization, no comparison with the original scheme, and no re-derivation or numerical check confirming that convergence rates and existence results are unchanged are provided. This directly affects the load-bearing assertion that the theoretical conclusions survive the fix.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and entire manuscript): the central claim that the corrections 'transform the equation (1.1) into a discretized form in a tighter and more accurate way without affecting the main results' is unsupported. No explicit form of the corrected discretization, no comparison with the original scheme, and no re-derivation or numerical check confirming that convergence rates and existence results are unchanged are provided. This directly affects the load-bearing assertion that the theoretical conclusions survive the fix.

    Authors: We agree that the present short note does not exhibit the explicit corrected discretization, nor does it contain a side-by-side comparison or a re-derivation of the error estimates. In a revised version we will insert the original and corrected quadrature formulas, note the precise (local) modification to the discretization operator, and add a short paragraph confirming that the convergence proof and existence argument remain unchanged because they depend only on the Lipschitz properties of the kernels and the uniform boundedness of the quadrature weights, both of which are preserved by the correction. revision: yes

Circularity Check

0 steps flagged

No derivation presented; correction note only

full rationale

This document is a short correction note. It states that changes to the discretization of (1.1) are tighter yet leave the original paper's convergence, error estimates and existence results unchanged. No derivation chain, ansatz, fitted parameter, or self-citation load-bearing step appears in the note itself. The assertion is a factual claim about the scope of the fixes, not a reduction of any result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new axioms, parameters, or entities introduced as this is a correction to discretization in a prior work.

pith-pipeline@v0.9.0 · 5649 in / 784 out tokens · 14466 ms · 2026-05-24T20:36:24.997588+00:00 · methodology

discussion (0)

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Reference graph

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