### Journals Information

**
Mathematics and Statistics Vol. 10(4), pp. 868 - 874 DOI: 10.13189/ms.2022.100418 Reprint (PDF) (249Kb) **

## Numerical Solution of Nonlinear Fredholm Integral Equations Using Half-Sweep Newton-PKSOR Iteration

**Labiyana Hanif Ali ^{1}^{,*}, Jumat Sulaiman ^{1}, Azali Saudi ^{2}, Xu Ming Ming ^{1}**

^{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**

This paper is concerned with producing an efficient numerical method to solve nonlinear Fredholm integral equations using Half-Sweep Newton-PKSOR (HSNPKSOR) iteration. The computation of numerical methods in solving nonlinear equations usually requires immense amounts of computational complexity. By implementing a Half-Sweep approach, the complexity of the calculation is tried to be reduced to produce a more efficient method. For this purpose, the steps of the solution process are discussed beginning with the derivation of nonlinear Fredholm integral equations using a quadrature scheme to get the half-sweep approximation equation. Then, the generated approximation equation is used to develop a nonlinear system. Following that, the formulation of the HSNPKSOR iterative method is constructed to solve nonlinear Fredholm integral equations. To verify the performance of the proposed method, the experimental results were compared with the Full-Sweep Newton-KSOR (FSNKSOR), Half-Sweep Newton-KSOR (HSNKSOR), and Full-Sweep Newton-PKSOR (FSNPKSOR) using three parameters: number of iteration, iteration time, and maximum absolute error. Several examples are used in this study to illustrate the efficiency of the tested methods. Based on the numerical experiment, the results appear that the HSNPKSOR method is effective in solving nonlinear Fredholm integral equations mainly in terms of iteration time compared to rest tested methods.

**KEYWORDS**

Nonlinear Fredholm Integral Equations, Newton-PKSOR, Half-sweep Newton-PKSOR, Iterative Method, Quadrature Scheme

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Labiyana Hanif Ali , Jumat Sulaiman , Azali Saudi , Xu Ming Ming , "Numerical Solution of Nonlinear Fredholm Integral Equations Using Half-Sweep Newton-PKSOR Iteration," Mathematics and Statistics, Vol. 10, No. 4, pp. 868 - 874, 2022. DOI: 10.13189/ms.2022.100418.

(b). APA Format:

Labiyana Hanif Ali , Jumat Sulaiman , Azali Saudi , Xu Ming Ming (2022). Numerical Solution of Nonlinear Fredholm Integral Equations Using Half-Sweep Newton-PKSOR Iteration. Mathematics and Statistics, 10(4), 868 - 874. DOI: 10.13189/ms.2022.100418.