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Adaptive spectral solution method for Fredholm integral equations of the second kind

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Abstract

The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation method without dividing the domain of the integral equation, as usual, into the sub intervals. Existence and uniqueness of solutions are discussed. Convergence rates and error estimates are given for the numerical scheme in both \(L^{\infty }\)-norm and \(L^{2}\)-norm. Finally, several numerical examples are provided to show that the proposed method is preferable to its classical predecessor and some other existing approaches.

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Acknowledgements

The authors are thankful to the reviewers for giving valuable comments and suggestions.

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  1. Department of mathematics, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arreridj, El Anasser, 34030, Algeria

    Issam Abdennebi & Azedine Rahmoune

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  1. Issam Abdennebi
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  2. Azedine Rahmoune
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Contributions

The first author, Issam Abdennebi, wrote the programs and performed the main computation. The corresponding author, Azedine Rahmoune, conceptualized the manuscript and developed the methodology. Both authors contributed to the analysis and writing the manuscript.

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Correspondence to Azedine Rahmoune.

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Abdennebi, I., Rahmoune, A. Adaptive spectral solution method for Fredholm integral equations of the second kind. Numer Algor (2023). https://doi.org/10.1007/s11075-023-01671-1

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