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A note on the alternate trapezoidal quadrature method for Fredholm integral eigenvalue problems

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Abstract

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Paraná, Curitiba, PR, 81531-980, Brazil

    S. P. Oliveira & A. Ruiz de Zárate

  2. Laboratório Nacional de Computação Científica, Petrópolis, RJ, 25651-075, Brazil

    A. C. da Rocha

  3. Departamento de Ciência da Computação, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    D. G. Alfaro Vigo

Authors
  1. S. P. Oliveira
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  2. A. Ruiz de Zárate
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  3. A. C. da Rocha
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  4. D. G. Alfaro Vigo
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Correspondence to S. P. Oliveira.

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Oliveira, S., Ruiz de Zárate, A., da Rocha, A. et al. A note on the alternate trapezoidal quadrature method for Fredholm integral eigenvalue problems. Numer Algor 62, 601–614 (2013). https://doi.org/10.1007/s11075-012-9681-8

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  • DOI: https://doi.org/10.1007/s11075-012-9681-8

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