Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations

Sara Remogna (), Driss Sbibih and Mohamed Tahrichi
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Sara Remogna: Department of Mathematics “G. Peano”, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Driss Sbibih: LANO Laboratory, Faculty of Science, University Mohammed I, Oujda 60000, Morocco
Mohamed Tahrichi: Team ANAA, EST, LANO Laboratory, University Mohammed I, Oujda 60000, Morocco

Mathematics, 2023, vol. 11, issue 14, 1-10

Abstract: Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the use of spline quasi-interpolating operators in the space of splines of degree d and of class C d − 1 on uniform partitions of a bounded interval for the numerical solution of Urysohn integral equations, by using a superconvergent Nyström method. Firstly, we generate the approximate solution and we obtain outcomes pertaining to the convergence orders. Additionally, we examine the iterative version of the method. In particular, we prove that the convergence order is ( 2 d + 2 ) if d is odd and ( 2 d + 3 ) if d is even. In case of even degrees, we show that the convergence order of the iterated solution increases to ( 2 d + 4 ) . Finally, we conduct numerical tests that validate the theoretical findings.

Keywords: Urysohn integral equation; quasi-interpolating spline; Nyström method; superconvergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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