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

 

Abstrak

Persamaan pembezaan-kamiran adalah penting untuk memodelkan fenomena dunia sebenar dalam fizik, kejuruteraan dan biologi. Kertas 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: Beza terhingga; Fredholm; lelaran separuh sapuan; min kuadratik; Peraturan Simpson; persamaan pembezaan-kamiran; trapezoid komposit

 

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