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A numerical method for Fredholm integral equations of the second kind by the IMT-type DE rules

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Abstract

In this paper, we propose a numerical method for one-dimensional Fredholm integral equations of the second kind by the IMT-type DE rules for numerical integration. We obtain our method by enhancing the DE-Nyström method by replacing the DE rule used for discretizing the integral operator with the IMT-type DE rules. It is free of the difficulty of parameter tuning, that is, the problem of choosing the mesh size of the DE rule for the given number of unknowns as in the DE-Nyström method. Numerical examples show that it is competitive with the DE-Nyström method.

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Notes

  1. Exactly speaking, the intrinsic error given in [3] is defined by the error of the rule applied to a function of the form \(f(x)=(1-x^2)^{\alpha -1}\) \((\,\alpha >0\,)\).

  2. In our computation, \(\epsilon _N\) is approximately evaluated by computing the maximal value of the error on 1000 equidistant points on the interval.

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Acknowledgements

The author would like to thank the reviewers for the comments and Enago ( www.enago.jp) for the English language review.

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  1. Department of Computer and Network Engineering, Graduate School of Informatics and Engineering, The University of Electro-Communications, 1-5-1, Chofu-Ga-Oka, Chofu, 182-8686, Japan

    Hidenori Ogata

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  1. Hidenori Ogata
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Correspondence to Hidenori Ogata.

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Ogata, H. A numerical method for Fredholm integral equations of the second kind by the IMT-type DE rules. Japan J. Indust. Appl. Math. 38, 715–729 (2021). https://doi.org/10.1007/s13160-021-00457-z

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  • DOI: https://doi.org/10.1007/s13160-021-00457-z

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