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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]
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Database: Engineering Source
Description
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]
ISSN:03784754
DOI:10.1016/j.matcom.2026.01.029