  Citation: Saou, A.; Sbibih, D.; Tahrichi, M.; Barrera, D. Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations. Mathematics 2022, 10, 893. https:// doi.org/10.3390/math10060893 Academic Editor: Miroslaw Lachowicz Received: 30 January 2022 Accepted: 21 February 2022 Published: 11 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations Abdelmonaim Saou 1 , Driss Sbibih 2 , Mohamed Tahrichi 1, * and Domingo Barrera 3 1 Team ANAA, ANO Laboratory, Faculty of Sciences, University Mohammed First, Oujda 60000, Morocco; saou.abdelmonaim@gmail.com 2 Team ANTO, ANO Laboratory, Faculty of Sciences, University Mohammed First, Oujda 60000, Morocco; sbibih@yahoo.fr 3 Department of Applied Mathematics, University of Granada, Campus de Fuentenueva s/n, 18071 Granada, Spain; dbarrera@ugr.es * Correspondence: m.tahrichi@ump.ac.ma Abstract: The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree r − 1, we obtain convergence order 2r for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples. Keywords: degenerate kernel method; Nyström method; Fredholm integro-differential equation 1. Introduction Integro-differential equations emerged at the beginning of the twentieth century thanks to the work of Vito Volterra. The applications of these equations have proved worthy and effective in the fields of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, electrostatics, etc. (see [1–4] and references therein). Many numerical methods have been developed for solving integro-differential equa- tions. Each of these methods has its inherent advantages and disadvantages, and the search for easier and more accurate methods is a continuous and ongoing process. Among the existing methods in the literature, we cite the Adomian decomposition [5], homotopy anal- ysis [2], Chebyshev and Taylor collocation [6], Taylor series expansion [7,8], integral mean value [9], and decomposition method [10]. For other methods to solve integro-differential equations, see [11–14]. Recently, many authors have used spline functions for the numerical solution of integro-differential equations; in particular, a semi-orthogonal spline wavelets approxima- tion method for Fredholm integro-differential equations was proposed in [15]. In [16], the authors used a fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions. In [17], the authors ap- plied B-spline collocation method for solving numerically linear and nonlinear Fredholm and Volterra integro-differential equations, and in [18] an exponential spline method for approximating the solution of Fredholm integro-differential equation was studied. More recently, in [19] Kulkarni introduced an efficient method called modified projection method or multi-projection method to solve Fredholm integral equations of the second kind. In- spired in Kulkarni’s method, authors in [20] have introduced superconvergent Nyström and degenerate kernel methods to solve the same type of equations. Mathematics 2022, 10, 893. https://doi.org/10.3390/math10060893 https://www.mdpi.com/journal/mathematics