EIGENVALUE PROBLEM OF A WEAKLY SINGULAR COMPACT INTEGRAL OPERATOR BY DISCRETE LEGENDRE PROJECTION METHODS

Moumita Mandal; Kapil Kant; Gnaneshwar Nelakanti

doi:10.11948/20200316

2021 Volume 11 Issue 4

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Moumita Mandal, Kapil Kant, Gnaneshwar Nelakanti. EIGENVALUE PROBLEM OF A WEAKLY SINGULAR COMPACT INTEGRAL OPERATOR BY DISCRETE LEGENDRE PROJECTION METHODS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 2090-2101. doi: 10.11948/20200316

Citation:

Moumita Mandal, Kapil Kant, Gnaneshwar Nelakanti. EIGENVALUE PROBLEM OF A WEAKLY SINGULAR COMPACT INTEGRAL OPERATOR BY DISCRETE LEGENDRE PROJECTION METHODS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 2090-2101. doi: 10.11948/20200316

EIGENVALUE PROBLEM OF A WEAKLY SINGULAR COMPACT INTEGRAL OPERATOR BY DISCRETE LEGENDRE PROJECTION METHODS

1.
Department of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur 342037, India
2.
Department of Mathematics, Indian Institute of Technology, Kharagpur - 721302, India

Corresponding author: Email: moumita@iitj.ac.in (M. Mandal)

Abstract

Abstract

In this article, the discrete version of Legendre projection and iterated Legendre projection methods are considered to find the approximate eigenfunctions (eigenvalues and eigenvectors) of a weakly singular compact integral operator. Making use of a sufficiently accurate numerical quadrature rule, we establish the error bounds of the approximated eigenvalues and eigenvectors by discrete Legendre projection and iterated discrete Legendre projection methods in both L² and uniform norm. In particular, we obtain the optimal convergence rates $ \mathcal{O} (n^{-m}) $ for the eigenfunctions in iterated discrete Legendre projection method in L² and uniform norms, where n is the highest degree of the Legendre polynomial employed in the approximation and m is the smoothness of the eigenvectors. Numerical examples are presented to illustrate the theoretical results.
- Convergence rates /
- weakly singular kernels /
- discrete Projection methods /
- hyperinterpolation operator /
- legendre polynomials /
- spectral projection /
- eigenvalues
MSC: 45B05, 45G10, 65R20

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