Legendre superconvergent degenerate kernel and Nyström methods for nonlinear integral equations

C.  Allouch; M.  Arrai; H.  Bouda; M.  Tahrichi

doi:10.37863/umzh.v75i5.7039

C. Allouch University Mohammed I, FPN, MSC Team, LAMAO Laboratory, Nador, Morocco)
M. Arrai University Mohammed I, FPN, MSC Team, LAMAO Laboratory, Nador, Morocco
H. Bouda University Mohammed I, FPN, MSC Team, LAMAO Laboratory, Nador, Morocco
M. Tahrichi University Mohammed I, ESTO, ANAA Team, ANO Laboratory, Oujda, Morocco

DOI: https://doi.org/10.37863/umzh.v75i5.7039

Keywords: Hammerstein equation, Urysohn equation, Degenerate kernel method, Nystr¨om method, Interpolatory projection, Legendre polynomial, Superconvergence.

Abstract

UDC 517.9

We study polynomially based superconvergent collocation methods for the approximation of solutions of nonlinear integral equations. The superconvergent degenerate kernel method is chosen for approximating the solutions of Hammerstein equations, while a superconvergent Nystr\"om method is used for solving Urysohn equations. By applying interpolatory projections based on Legendre polynomials of degree $\leq n,$ we analyze the superconvergence of these methods and their iterated versions. Numerical results are presented to validate the theoretical results.

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