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Discrete projection methods for Hammerstein integral equations on the half-line

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Abstract

In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{r, d\}}),\) whereas iterated discrete Galerkin/iterated discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{2r, d\}}),\) where \(n^{-1}\) is the maximum norm of the graded mesh and r denotes the order of the piecewise polynomial employed and \(d-1\) is the degree of precision of quadrature formula. We also show that iterated discrete multi-Galerkin/iterated discrete multi-collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{4r, d\}})\). Hence by choosing sufficiently accurate numerical quadrature rule, we show that the convergence rates in discrete projection and discrete multi-projection methods are preserved. Numerical examples are given to uphold the theoretical results.

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  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721 302, India

    Nilofar Nahid & Gnaneshwar Nelakanti

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  1. Nilofar Nahid
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  2. Gnaneshwar Nelakanti
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Nahid, N., Nelakanti, G. Discrete projection methods for Hammerstein integral equations on the half-line. Calcolo 57, 37 (2020). https://doi.org/10.1007/s10092-020-00386-2

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