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

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  • Remili, Walid
  • Noeiaghdam, Samad

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(N3/4−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.

Suggested Citation

  • Remili, Walid & Noeiaghdam, Samad, 2026. "Shifted Chebyshev collocation with CESTAC-CADNA-based instability detection for nonlinear Volterra–Hammerstein integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 246(C), pages 60-77.
    • Handle: RePEc:eee:matcom:v:246:y:2026:i:c:p:60-77
    DOI: 10.1016/j.matcom.2026.01.029
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    References listed on IDEAS

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    1. Ashpazzadeh, Elmira & Chu, Yu-Ming & Hashemi, Mir Sajjad & Moharrami, Mahsa & Inc, Mustafa, 2022. "Hermite multiwavelets representation for the sparse solution of nonlinear Abel’s integral equation," Applied Mathematics and Computation, Elsevier, vol. 427(C).
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