### Journals Information

Mathematics and Statistics Vol. 10(1), pp. 246 - 250
DOI: 10.13189/ms.2022.100123
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## Newton-PKSOR with Quadrature Scheme in Solving Nonlinear Fredholm Integral Equations

Labiyana Hanif Ali 1,*, Jumat Sulaiman 1, Azali Saudi 2
1 Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu, Sabah - 88400, MALAYSIA
2 Faculty of Computing and Informatics, Universiti Malaysia Sabah, Kota Kinabalu, Sabah - 88400, MALAYSIA

ABSTRACT

In this study, we applied Newton method with a new version of KSOR, called PKSOR to form NPKSOR in solving nonlinear second kind Fredholm integral equations. A new version of KSOR is an update to the KSOR method with two relaxation parameters. The properties of KSOR helps in enlargement of the solution domain so the relaxation parameter can take the value . With PKSOR, the relaxation parameter in KSOR is treated into two different relaxation paramaters as and which resulting lower number of iteration compared to the KSOR method. By combining the Newton method with PKSOR, we intend to from more efficient method to solve the nonlinear Fredholm integral equations. The discretization part of this study is done using first-order quadrature scheme to develop a nonlinear system. We formulate the solution of the nonlinear system using the given approach by reducing it to a linear system and then solving it using iterative methods to obtain an approximate solution. Furthermore, we compare the results of the proposed methods with NKSOR and NGS methods on three examples. Based on our findings, the NPKSOR method is more efficient than NKSOR and NGS methods. By implementing the NPKSOR method, we can boost the convergence rate of the iteration by considering two relaxation parameters, resulting in a lower number of iteration and computational time.

KEYWORDS
PKSOR, NPKSOR, Nonlinear Fredholm Integral Equations, Newton's Method, Quadrature Scheme

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Labiyana Hanif Ali , Jumat Sulaiman , Azali Saudi , "Newton-PKSOR with Quadrature Scheme in Solving Nonlinear Fredholm Integral Equations," Mathematics and Statistics, Vol. 10, No. 1, pp. 246 - 250, 2022. DOI: 10.13189/ms.2022.100123.

(b). APA Format:
Labiyana Hanif Ali , Jumat Sulaiman , Azali Saudi (2022). Newton-PKSOR with Quadrature Scheme in Solving Nonlinear Fredholm Integral Equations. Mathematics and Statistics, 10(1), 246 - 250. DOI: 10.13189/ms.2022.100123.