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

Authors:
S. P. Oliveira;A. Ruiz De Zárate;A. C. Rocha;D. G. Alfaro Vigo
Affiliations:
Departamento de Matemática, Universidade Federal do Paraná, Curitiba, Brazil 81531-980;Departamento de Matemática, Universidade Federal do Paraná, Curitiba, Brazil 81531-980;Laboratório Nacional de Computação Científica, Petrópolis, Brazil 25651-075;Departamento de Ciência da Computação, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
Venue:
Numerical Algorithms
Year:
2013

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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.