Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method

## Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method  Volume-71 Issue-2
Year of Publication : 2023
Author : Nor Syahida Mohamad, Jumat Sulaiman, Azali Saudi, Nur Farah Azira Zainal
DOI : 10.14445/22315381/IJETT-V71I2P220

How to Cite?

Nor Syahida Mohamad, Jumat Sulaiman, Azali Saudi, Nur Farah Azira Zainal, "Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method," International Journal of Engineering Trends and Technology, vol. 71, no. 2, pp. 165-173, 2023. Crossref, https://doi.org/10.14445/22315381/IJETT-V71I2P220

Abstract
The Gauss-Seidel and Successive over Relaxation methods are two classic methods frequently used to solve the system linear equation. This study has stated that the Gauss-Seidel method was a linear solver, and both iterative methods operated in column and row spaces, respectively. In addition, the modification of the GS method has transformed the Successive Over Relaxation (SOR) thus, has improvised the iteration process in terms of iteration size and computational time. Therefore, to prove those parameters of the iteration process, the first-order piecewise polynomial has been applied to the Fredholm integral equation of second kind (FIE) with collocation point with composited trapezoidal quadrature method in order to get the approximation equations of Fredholm integral. This paper has successfully derived an approximation equation of composited trapezoidal method with first-order piecewise polynomial through the process of discretization on the FIEs with the consideration of vertex-centered type on the domain solutions. As mentioned previously, the numerical experiment has been tested on the approximation equation of composited trapezoidal method with first-order piecewise polynomial by using the derived algorithm of Gauss-Seidel (GS) and Successive Over Relaxation (SOR). The results that have also been recorded included the Maximum absolute error (MAE) besides the iteration size (IC) and computational time (CT).

Keywords
Collocation, SOR, GS, Piecewise, Polynomial, Trapezoidal.

References
 Yao Lu, Lixin Shen, and Yuesheng Xu, “Integral Equation Models for Image Restoration: High Accuracy Methods and Fast Algorithms,” IOP Publishing, vol. 26 no. 4, p. 045006, 2010. Crossref, http://doi.org/10.1088/0266-5611/26/4/045006
 Yulianto, Nurhasanah Risma Yulistiani, and Gede Putra Kusuma, “Face Image Super-Resolution Using Combination of Max-FeatureMap and CMU-Net to Enhance LowResolution Face Recognition,” International Journal of Engineering Trends and Technology, vol. 70, no. 70, pp. 1-12, 2022. Crossref, https://doi.org/10.14445/22315381/IJETT-V70I3P201
 Barzkar, P. Assari, and M.A. Mehrpouya, “Application of the CAS Wavelet in Solving Fredholm-Hammerstein Integral Equations of the Second Kind with Error Analysis,” World Applied Sciences Journal, vol. 12, no. 12, pp. 1659-1704, 2012. Crossref, https://doi.org/10.5829/idosi.wasj.2012.18.12.467
 Damian Mikulski, Krzysztof Eder, and Jerzy Konarski, “Fredholm Integral Equation for the Perturbation Theory in Quantum Mechanics,” Journal of Mathematical Chemistry, vol. 52, no. 1, pp. 2317-2321, 2014. Crossref, https://doi.org/10.1007/s10910-014-0387-0
 Samsul Ariffin Abdul Karim, Faheem Khan, and Muhammad Basit, “Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis,” Symmetry, vol. 14, no. 7, p. 1343, 2022. Crossref, https://doi.org/10.3390/sym14071343
 Haitao Xu, and Liya Fan, “Numerical Solution of Volterra-Fredholm Integral Equations Based on ε-SVR Method,” Journal of Computational and Applied Mathematics, vol. 298, pp. 201-210, 2016. Crossref, https://doi.org/10.1016/j.cam.2015.12.002
 Samsul Ariffin Abdul Karim, Faheem Khan, and Muhammad Basit “Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis,” Symmetry, vol. 14, no. 7, p. 1343, 2022. Crossref, https://doi.org/10.3390/sym14071343
 N. S. Mohamad et al., “First Order Piecewise Collocation Solution of Fredholm Integral Equation Second Type Using SOR Iteration,” Lecture Notes in Electrical Engineering, vol. 724, no. 1, pp. 395–406, 2021. Crossref, https://doi.org/10.1007/978-981-33-4069-5_32
 Nasibeh Karamollahi, Mohammad Heydari, and Ghasem Barid Loghmani, “Approximate Solution of Nonlinear Fredholm Integral Equations of the Second Kind Using a Class of Hermite Interpolation Polynomials,” Mathematics and Computers in Simulation, vol. 187, pp. 414–432, 2021. Crossref, https://doi.org/10.1016/j.matcom.2021.03.015
 Wing Kam Liu, Shaofan Li, and Harold S. Park, “Eighty Years of the Finite Element Method: Birth, Evolution, and Future,” Archives of Computational Methods in Engineering, vol. 29, no. 1, pp. 4431–4453, 2021. Crossref, https://doi.org/10.1007/s11831-022-09740-9
 A. Faghih, and P. Mokhtary, “A Novel Petrov-Galerkin Method for a Class of Linear Systems of Fractional Differential Equations,” Applied Numerical Mathematics, vol. 169, pp. 396–341, 2021. Crossref, https://doi.org/10.1016/j.apnum.2021.07.012
 Najmuddin Ahmad, and Balmukund Singh, “Numerical Accuracy of Errors in Volterra Integral Equation by Using Quadrature Methods,” Malaya Journal of Matematik, vol. 9, no. 1, pp. 655-661, 2021. Crossref, https://doi.org/10.26637/MJM0901/0114
 C. V. Rao, “Improved Form of Simpson’s One-Third Rule for Finding Approximate Value of Definite Integrals by Using Trigonometric Functions,” Journal of Hunan University (Natural Sciences), vol. 48, no. 9, pp. 165-169, 2021.
 Abdel Radi Abdel Rahman Abdel Gadir, and Samia Abdallah Yagoub Ibrahim, "Comparison Between Some Analysis Solutions for Solving Fredholm Integral Equation of Second Kind," International Journal of Mathematics Trends and Technology, vol. 66, no. 6, pp. 245-260, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I6P525
 Federico Izzo, “High Order Trapezoidal Rule-Based Quadratures for Boundary Integral Methods on Non-Parametrized Surfaces,” Universitetsservice US-AB, Sweden, 2022.
 Kai Liu, Jie Yang, and Wei Shi, “A New SOR-Type Iteration Method for Solving Linear Systems,” Applied Mathematics Letters, vol. 102, p. 106104, 2020. Crossref, https://doi.org/10.1016/j.aml.2019.106104
 N. A. M. Ali et al., “The Similarity Finite Difference Solutions for Two-Dimensional Parabolic Partial Differential Equations via SOR Iteration,” Computational Science and Technology, vol. 724, no. 1, pp. 515-526, 2021. Crossref, https://doi.org/10.1007/978-981-33-4069-5_42
 Ai-Guo Wu, Hui-Jie Sun, and Ying Zhang, “An SOR Implicit Iterative Algorithm for Coupled Lyapunov Equations,” The International Federation of Automatic Control, vol. 97, pp. 38–47, 2018. Crossref, https://doi.org/10.1016/j.automatica.2018.07.021
 N. F. A. Zainal, J. Sulaiman, and M. U. Alibubin, “Application of EGSOR Iteration with Nonlocal Arithmetic Discretization Scheme for Solving Burger’s Equation,” 5th International Conference on Fundamental and Applied Science, vol. 11231, p. 012051, 2018. Crossref, https://doi.org/10.1088/1742-6596/1123/1/012051
 N. S. Mohamad, and J. Sulaiman, “The Piecewise Polynomial Collocation Method for the Solution of Fredholm Equation of Second Kind by Using AGE Iteration,” International Conference of World Engineering, Science and Technology Congress, vol. 1123, p. 012039, 2018. Crossref, https://doi.org/10.1088/1742-6596/1123/1/012039
 Harishchandra S. Ramane et al., “Numerical Solution of Fredholm Integral Equations Using Hosoya Polynomial of Path Graphs,” American Journal of Numerical Analysis, vol. 5, no. 1, pp. 11-15, 2017. Crossref, https://doi.org/10.12691/ajna-5-1-2
 Ravikiran A. Mundewadi, and Bhaskar A. Mundewadi, "Hermite Wavelet Collocation Method for the Numerical Solution of Integral and Integro - Differential Equations," International Journal of Mathematics Trends and Technology, vol. 53, no. 3, pp. 215-231, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V53P527
 N. Paradin, and Sh. Gholamtabar, “A Numerical Solution of the Linear Fredholm Integral Equations of the Second Kind,” Journal of Mathematical Extension, vol. 5, pp. 31-39, 2010.
 G. D Smith, Numerical Solution of Partial Differential Equations: Finite Difference Method, Clarendon Press: Oxford, 1985.
 A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques, Oxford: Elsevier Science, 2009.
 A Nurhakim, H M Saputra, and N Ismail, “Complementary of Quaternion Method And Boole’s Rule On IMU Sensor To Calculate Orientation Angle Of Stewart Platform,” 4th Annual Applied Science and Engineering Conference, vol. 1402, no. 3, p. 033106, 2019. Crossref, https://doi.org/10.1088/1742-6596/1402/3/033106
 Umair Khalid Qureshi et al., “Modified Quadrature Iterated Methods of Boole Rule and Weddle Rule for Solving Non Linear Equations,” Journal of Mechanics of Continua and Mathematical Sciences, vol. 16, no. 2, pp. 87-101, 2021. Crossref, https://doi.org/10.26782/jmcms.2021.02.00008
 A. F. Abdulhameed, and Q. A. Memon, “An Improved Trapezoidal Rule for Numerical Integration,” Journal of Physics, vol. 2090, p. 12104, 2021. Crossref, https://doi.org/10.1088/1742-6596/2090/1/012104
 Adel Mohamed Hassan, and Salah Mahdi Aflook, “Novel Methods to Acceleration Simpson’s 3/8 Rule,” Iraqi Academics Syndicate International Conference for Pure and Applied Sciences, vol. 1818, no. 1, p. 012200, 2021. Crossref, https://doi.org/10.1088/1742-6596/1818/1/012200
 Md. Jashim Uddin, Mir Md. Moheuddin, and Md. Kowsher, “A New Study of Trapezoidal, Simpson’s 1/3 And Simpson’s 3/8 Rules of Numerical Integral Problems,” Applied Mathematics and Sciences, An International Journal, vol. 6, no. 4, pp. 1-13, 2019. Crossref, https://doi.org/10.5121/mathsj.2019.6401
 Muhammad Mujtaba Shaikh, “Analysis of Polynomial Collocation and Uniformly Spaces Quadrature Methods for Second Kind Linear Fredholm Integral Equations- A Comparison,” Turkish Journal of Analysis and Number Theory, vol. 7, no. 4, pp. 91-97, 2019. Crossref, https://doi.org/10.12691/tjant-7-4-1