Sains Malaysiana 54(9)(2025): 2301-2313

http://doi.org/10.17576/jsm-2025-5409-16

 

A Novel Variant of Weighted Quadratic Mean Iterative Methods for Fredholm
Integro-Differential Equations

(Varian Novel Kaedah Lelaran Min Kuadratik Berwajaran untuk Persamaan
Integro-Differential Fredholm)

 

NG WEI LI¹ , ELAYARAJA ARUCHUNAN^2,* & ZAILAN SIRI¹

 

¹Institute of Mathematical Sciences, Universiti Malaya, 50603 Kuala Lumpur, Malaysia
²Department of Decision Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia

 

Received: 24 February 2025/Accepted: 10 July 2025

 

Abstract

Integro-differential equations are critical for modelling real-world phenomena in physics, engineering, and biology. This paper introduces a Quadratic Mean iterative method to solve dense linear systems derived from the discretization of second- and fourth-order Fredholm integro-differential equations (FIDEs). The solution of the FIDEs is approximated using finite difference, composite trapezoidal, and composite Simpson’s 1/3 and 3/8 schemes. The quadratic mean iterative method then solves the discretized system with different mesh sizes. As the resulting systems are large, a complexity reduction approach is implemented on the quadratic mean method to develop the half-sweep quadratic mean iterative method. The newly proposed iterative method includes a novel theorem, comprehensive proofs, and a detailed convergence analysis. The numerical results indicate that the quadratic mean method significantly outperforms the Gauss-Seidel iterative method in terms of efficiency, making it a promising solution for FIDEs.

 

Keywords: Fredholm integro-differential equations; quadratic mean; half-sweep iteration; finite difference; composite trapezoidal; Composite Simpson’s rules

 

Abstrak

Persamaan pembezaan-kamiran adalah penting untuk memodelkan fenomena dunia sebenar dalam fizik, kejuruteraan dan biologi. Dalam jurnal ini memperkenalkan kaedah lelaran Purata Kuadratik untuk menyelesaikan sistem linear tumpat yang diperoleh daripada membahagikan persamaan integro-pembezaan Fredholm tertib kedua dan keempat (FIDEs) kepada bahagian kecil. Penyelesaian FIDEs dianggarkan menggunakan perbezaan terhingga, trapezoid komposit dan skema 1/3 dan 3/8 komposit Simpson. Kemudian, kaedah lelaran purata kuadratik digunakan untuk menyelesaikan persamaan anggaran dengan saiz mesh yang berbeza. Memandangkan sistem yang akan diselesaikan adalah besar, pendekatan pengurangan kerumitan dilaksanakan pada kaedah purata kuadratik untuk membentuk kaedah lelaran purata kuadratik separuh sapuan. Kaedah lelaran yang baharu dicadangkan termasuk teorem novel, bukti komprehensif, dan analisis penumpuan terperinci. Keputusan berangka menunjukkan bahawa kaedah purata kuadratik dengan ketara mengatasi kaedah lelaran Gauss-Seidel dari segi kecekapan, menjadikannya penyelesaian terbaik untuk FIDEs.

 

Kata kunci: Persamaan pembezaan-kamiran; Fredholm; min kuadratik; lelaran separuh sapuan; beza terhingga; trapezoid komposit; Peraturan Simpson

 

REFERENCES

Aihara, K., Ozaki, K. & Mukunoki, D. 2024. Mixed-precision conjugate gradient algorithm using the groupwise update strategy. Japan Journal Industrial Applied Mathematics 41: 837-855. https://doi.org/10.1007/s13160-024-00644-8

Aruchunan, E. & Sulaiman, J. 2011. Half-sweep conjugate gardient method for Solving first order linear fredholm integro-differential equations. Australian Journal of Basic and Applied Sciences 2: 38-43.

Aruchunan, E & Sulaiman, J. 2013. Half-sweep quadrature-difference schemes with iterative method in solving linear fredholm integro-differential equations. Progress in Applied Mathematics. 2: 11-21.

Aruchunan, E., Khajohnsaksumeth, N. & Wiwatanapataphee, B. 2016. A new algorithm of geometric mean for solving high-order fredholm integro-differential equations. 723-729. 10.1109/DASC-PICom-DataCom-CyberSciTec.2016.128.

Aruchunan, E., Muthuvalu, M., Chew, J., Siri, Z. & Sulaiman, J. 2022. Examination of half-sweep closed newton–cotes quadrature schemes in solving dense system. 10.1007/978-3-030-79606-8_26.

Aruchunan, E. & Sulaiman, J. 2010. Numerical solution of second-order linear fredholm integro-differential equation using generalized minimal residual (GMRES) method. American Journal of Applied Sciences, Science Publication 7 (6): pp.780-783.

Aruchunan, E. & Sulaiman, J. 2010. Repeated Simpson 3/8 and backward difference schemes to solve first order integro-differential equation. National Seminar of Science and Computer (SKSKM 2010). ISBN: 978-967-363- 153-7.

Aruchunan, E. 2016. The New Variants of Modified Weighted Mean Iterative Methods for Fredholm Integro-Differential Equations (Doctoral dissertation, Curtin University).

Aruchunan, E., Wu, Y., Wiwatanapataphee, B. & Jitsangiam, P. 2015. A new variant of arithmetic mean iterative method for fourth order integro-differential equations solution. In 2015 3rd International Conference on Artificial Intelligence, Modelling and Simulation (AIMS). pp. 82-87. IEEE. doi: 10.1109/AIMS.2015.24.

Aruchunan, E. et al. 2022. Examination of half-sweep closed newton–cotes quadrature schemes in solving dense system. in: Abdul Karim, S.A., Shafie, A. (eds) Towards Intelligent Systems Modeling and Simulation. Studies in Systems, Decision and Control, vol 383. Springer, Cham. https://doi.org/10.1007/978-3-030-79606-8_26

Aruchunan, E., Muthuvalu, M.S., Sulaiman, J., Koh, W.S. & Akhir, K. M. 2014. An iterative solution for second order linear Fredholm integro-differential equations. Malaysian Journal of Mathematical Sciences 8(2): 157-170.

Aruchunan, E. & Sulaiman, J. 2012. Comparison of closed repeated newton-cotes quadrature schemes with half-sweep iteration concept in solving linear fredholm integro-differential equations. International Journal of Science and Engineering Investigations
1 (9): pp. 90-96.

Avudainayagam, A. & Vani, C. 2000. Wavelet-Galerkin method for integro-differential equations. Applied Mathematics and Computation. 32: 247–254.

Benzi, M. & Dayar, T. 1995. The arithmetic mean method for finding the stationary vector of Markov chains. International Journal of Parallel, Emergent and Distributed Systems 6(1): pp. 25-37, 1995.

Cai, F., Xiao, J. & Xiang, Z.H. Block SOR two-stage iterative methods for solution of symmetric positive definite linear systems. Proceedings of the 3rd International Conference on Advanced Computer Theory and Engineering, August 20-22, 2010, Chengdu, China. 378-382.2010.

Connolly, C.I., Burns, J.B., Weiss, R. 1990. Path planning using Laplace’s equation. Proceedings of the IEEE International Conference on Robotics and Automation: 2102–2106.

Darania, P. & Ebadian, A. 2007. A method for the numerical solution of the integro-differential equations. Applied Mathematics and Computation. 188(1): 657-668.

Filiz, A. 2000. Numerical solution of some Volterra integral equations, PhD Thesis, University of Manchester.

Jin S., Ming Z., George E.K. & Zhenya Y. 2024. Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations. Journal of Computational Physics Volume 505 ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2024.112917.

Katuri, S. & Maroju, P. 2025. A new approach for solving fuzzy non-linear equations using higher order iterative method. Scientific Report 15: 12972. https://doi.org/10.1038/s41598-025-97612-0

Muthuvalu, M.S., Aruchuan, E. & Sulaiman, J. 2013. Solving first kind linear Fredholm integral equations with semi-smooth kernel using 2-point half-sweep block arithmetic mean method. AIP Conference Proceedings vol. 1557: 350-354.

Muthuvalu, M.S. & Sulaiman, J., 2011. Half-sweep arithmetic mean method with composite trapezoidal scheme for solving linear fredholm integral equations. Applied Mathematics and Computation 217(12): 5442-5448..

Ortega, J. 1973. Numerical analysis: a Second Course. Mathematics of Computation 27(123): 669. https://doi.org/10.2307/2005671

Rathinasamy, A., & Balachandran, K. 2008. Mean-square stability of milstein method for linear hybrid stochastic delay integro-differential equations. Nonlinear Analysis: Hybrid Systems. 2(4): 1256-1263.

Ruggiero, V. & Galligani, E. 1990. An iterativemethod for large sparse systems on a vector computer. Computers and Mathematics with Applications 20:25-28.

Sahimi, M.S., Ahmad, A. & Bakar, A.A. 1993. The iterative alternating decomposition explicit (IADE) method to solve the heat conduction equation. International Journal of Computer Mathematics 47: 219-229.

Sezer, M. & Kaynak, M. 1977. Chebyshev polynomial solutions of linear differential equations. International Journal of Mathematical Education in Science and Technology 14: 607–618.

Sezer, M. 1996. A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials. International Journal of Mathematical Education in Science and Technology 27(4): 821–834.

Sulaiman, J., Othman, M., Yaacob, N. & Hasan. M.K. 2006. Half Sweep Geometric Mean (HSGM) method using fourth-order finite difference scheme for two-point boundary problems. In Proceedings of the First International Conference on Mathematics and Statistics. June 19-21. Bandung, Indonesia, pp. 25-33.

Ullah, M. A. 2015. Numerical integration and a proposed rule. American Journal of Engineering Research (AJER) 4(9): 120-123.

Yuhe, R. & Zhang, B. & Qiao, H. 1999. A simple Taylor-series expansion method for a class of second kind integral equations. Journal of Computational and Applied Mathematics 110: 15-24. doi: 10.1016/S0377-0427(99)00192-2.

Zhao, J. & Corless, R.M. 2006. Compact finite difference method for integro-differential equations. Applied Mathematics and Computation 177: 325-328. doi: 10.1016/j.amc.2005.11.007.

 

*Corresponding author; email: elayarajah@um.edu.my