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Shifted Chebyshev collocation with CESTAC-CADNA-based instability detection for nonlinear Volterra–Hammerstein integral equations.

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Title: Shifted Chebyshev collocation with CESTAC-CADNA-based instability detection for nonlinear Volterra–Hammerstein integral equations.
Authors: Remili, Walid1 (AUTHOR) walid.remili@univ-msila.dz, Noeiaghdam, Samad1,2,3 (AUTHOR) snoei@hnas.ac.cn
Source: Mathematics & Computers in Simulation. Aug2026, Vol. 246, p60-77. 18p.
Subjects: Chebyshev approximation, Integral equations, Iterative methods (Mathematics), Numerical analysis, Discretization methods, Error analysis in mathematics
Abstract: This paper introduces a high-order numerical method for the solution of nonlinear Volterra–Hammerstein integral equations (NVHIEs) with smooth and weakly singular kernels, based on the collocation approach. The proposed method employs a collocation scheme with shifted Chebyshev polynomials (SCPs), combined with an appropriate variable transformation, to reduce the integral equation to a nonlinear algebraic system. We rigorously analyze the convergence properties of the collocation method, establishing its theoretical validity and proving a specific convergence rate of O (N 3 / 4 − m) , which highlights the rigor and efficiency of the approach. To ensure reliable error control and stability, we integrate the CESTAC (Contrôle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, providing a unified framework that identifies numerical instabilities (self-validation, mathematical, branching, and intrinsic) while also determining the optimal step size, optimal approximation, and optimal error. Several numerical examples are presented and compared with existing methods to illustrate the enhanced efficiency and accuracy of our approach. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics & Computers in Simulation is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Shifted Chebyshev collocation with CESTAC-CADNA-based instability detection for nonlinear Volterra–Hammerstein integral equations.
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  Data: <searchLink fieldCode="AR" term="%22Remili%2C+Walid%22">Remili, Walid</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> walid.remili@univ-msila.dz</i><br /><searchLink fieldCode="AR" term="%22Noeiaghdam%2C+Samad%22">Noeiaghdam, Samad</searchLink><relatesTo>1,2,3</relatesTo> (AUTHOR)<i> snoei@hnas.ac.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+%26+Computers+in+Simulation%22">Mathematics & Computers in Simulation</searchLink>. Aug2026, Vol. 246, p60-77. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Chebyshev+approximation%22">Chebyshev approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Integral+equations%22">Integral equations</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Discretization+methods%22">Discretization methods</searchLink><br /><searchLink fieldCode="DE" term="%22Error+analysis+in+mathematics%22">Error analysis in mathematics</searchLink>
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  Label: Abstract
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  Data: This paper introduces a high-order numerical method for the solution of nonlinear Volterra–Hammerstein integral equations (NVHIEs) with smooth and weakly singular kernels, based on the collocation approach. The proposed method employs a collocation scheme with shifted Chebyshev polynomials (SCPs), combined with an appropriate variable transformation, to reduce the integral equation to a nonlinear algebraic system. We rigorously analyze the convergence properties of the collocation method, establishing its theoretical validity and proving a specific convergence rate of O (N 3 / 4 − m) , which highlights the rigor and efficiency of the approach. To ensure reliable error control and stability, we integrate the CESTAC (Contrôle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, providing a unified framework that identifies numerical instabilities (self-validation, mathematical, branching, and intrinsic) while also determining the optimal step size, optimal approximation, and optimal error. Several numerical examples are presented and compared with existing methods to illustrate the enhanced efficiency and accuracy of our approach. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematics & Computers in Simulation is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/j.matcom.2026.01.029
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 18
        StartPage: 60
    Subjects:
      – SubjectFull: Chebyshev approximation
        Type: general
      – SubjectFull: Integral equations
        Type: general
      – SubjectFull: Iterative methods (Mathematics)
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Discretization methods
        Type: general
      – SubjectFull: Error analysis in mathematics
        Type: general
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      – TitleFull: Shifted Chebyshev collocation with CESTAC-CADNA-based instability detection for nonlinear Volterra–Hammerstein integral equations.
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            NameFull: Remili, Walid
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            NameFull: Noeiaghdam, Samad
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            – D: 01
              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
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              Value: 246
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