Mathematics and Statistics Vol. 11(4), pp. 646 - 653
DOI: 10.13189/ms.2023.110405
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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

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)

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[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). APA Format:
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.