Mathematics and Statistics 11(4): 646-653, 2023 http://www.hrpub.org DOI: 10.13189/ms.2023.110405 A Piecewise Linear Collocation with Closed Newton Cotes Scheme for Solving Second Kind Fredholm Integral Equation (FIE) via Half-Sweep SOR Iteration Nor Syahida Mohamad 1,* , Jumat Sulaiman 1 , Azali Saudi 2 , Nur Farah Azira Zainal 1 1 Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Malaysia 2 Faculty of Computing and Informatics, Uiversiti Malaysia Sabah, Malaysia Received February 26, 2023; Revised May 9, 2023; Accepted June 11, 2023 Cite This Paper in the Following Citation Styles (a): [1] Nor Syahida Mohamad, Jumat Sulaiman, Azali Saudi, Nur Farah Azira Zainal , "A Piecewise Linear Collocation with Closed Newton Cotes Scheme for Solving Second Kind Fredholm Integral Equation (FIE) via Half-Sweep SOR Iteration," Mathematics and Statistics, Vol. 11, No. 4, pp. 646 - 653, 2023. DOI: 10.13189/ms.2023.110405. (b): Nor Syahida Mohamad, Jumat Sulaiman, Azali Saudi, Nur Farah Azira Zainal (2023). A Piecewise Linear Collocation with Closed Newton Cotes Scheme for Solving Second Kind Fredholm Integral Equation (FIE) via Half- Sweep SOR Iteration. Mathematics and Statistics, 11(4), 646 - 653. DOI: 10.13189/ms.2023.110405. Copyright©2023 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract In this paper, an efficient and reliable algorithm has been established to solve the second kind of FIE based on the lower-order piecewise polynomial and the lower-order quadrature method, namely Half-sweep Composite Trapezoidal (HSCT), which was used to discretize any integral term. Furthermore, due to the benefit of the complexity reduction technique via the half-sweep iteration concept presented from previous studies based on the cell-centered approach, this paper attempts to derive an HSCT piecewise linear collocation approximation equation generated from the discretization process of the proposed problem by considering the distribution of node points with vertex-centered type. Using half-sweep collocation node points over the linear collocation approximation equation, we could construct a system of HSCT linear collocation approximation equations, whose coefficient matrix is huge- scale and dense. Furthermore, to attain the piecewise linear collocation solution of this linear system, we considered the efficient algorithm of the Half-Sweep Successive Over- Relaxation (HSSOR) iterative method. Therefore, several numerical experiments of the proposed iterative methods have been implemented by solving three tested examples, and the obtained results that were based on three parameters, namely iteration quantity, accomplished time, and maximum absolute error, were recorded and compared against other two iterations, namely Full-Sweep Gauss- Seidel (FSGS) and Half-Sweep Gauss-Seidel (HSGS). Keywords Piecewise Linear Polynomial, Closed Newton Cotes Scheme, Gauss-Seidel (GS), Collocation Approach, Successive over Relaxation (SOR) 1. Introduction This section makes a quick review of several numerical solutions for solving integral equations. Some methods have been regularly used as a combination approach to the integral equation: Chebyshev, Galerkin, collocation, and Lagrange interpolation [1-4]. Interpolation is one of the standard methods that have been implemented in many areas of studies to derive the function from a given discrete data set. The purpose of interpolation is to find a formula over the node points in the data function itself [5]. Nevertheless, the collocation method is the easiest to handle and is less complex than others [6]. Besides, one of the most appealing, recently popular methods is the quadrature methods of Newton-cotes rules [7, 8]. We shall briefly explain some methods that are commonly used by recent studies. Based on one of the studies, a quadrature scheme, the trapezoidal method has especially been used in solving the integration equation by generating the linear system equation [9]. Based on the outcome of the study, by operating the equation of a linear system by using a