Control and Optimization in Applied Mathematics

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Optimal Solution of Volterra-Fredholm Integral Equations Based on the Clique and Pell-Lucas Series Collocation Method

Articles in Press

Document Type : Research Article

Authors

1 Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon, Turkiye

2 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai-602 105, Tamil Nadu, India

3 Department of Basic Sciences, Technical and Vocational University (TVU), Tehran, Iran

Abstract

This study presents a numerical technique for solving Volterra--Fredholm integral equations of the second kind. The solution is approximated via two independent polynomial bases, namely the Pell--Lucas polynomials and the Clique polynomials, each paired with a collocation discretisation. Both approaches are evaluated under two distinct sets of collocation points: standard equally spaced points and Chebyshev--Gauss--Lobatto points, the latter included for comparative purposes. Solution accuracy is assessed through the L2 and L∞ norms, the absolute error, and the root mean squared error (RMSE), while an upper bound error analysis is provided to establish convergence. Numerical experiments confirm that both methods achieve progressively higher accuracy as the degree N of the approximating polynomial series increases. Where applicable, the results are benchmarked against those reported in the existing literature. All numerical findings are presented in tabular and graphical form.

Highlights

  • A unified polynomial collocation framework is developed for Volterra-Fredholm integral equations.
  • The Pell-Lucas and Clique polynomial bases are compared under standard and Chebyshev-Gauss-Lobatto nodes.
  • An upper bound error estimate is derived and validated without requiring the exact solution.
  • Both methods achieve exponential decreases in error as the polynomial degree N increases.
  • The proposed approach outperforms several existing collocation methods reported in the literature.

Keywords

Main Subjects

References
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Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 16 June 2026
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