Sains Malaysiana 44(1)(2015): 139–146

Quarter-Sweep Iteration Concept on Conjugate Gradient Normal Residual Method

via Second Order Quadrature - Finite Difference Schemes for Solving Fredholm Integro-Differential Equations

(Konsep Lelaran Sapuan Suku ke atas Kaedah Kecerunan Konjugat Sisa Biasa menerusi Kesukuan Peringkat Kedua - Beza Terhingga bagi Menyelesaikan Persamaan Integro-pembezaan Fredholm)

ELAYARAJA ARUCHUNAN¹*, MOHANA SUNDARAM MUTHUVALU² & JUMAT SULAIMAN³

¹Department of Mathematics and Statistics, Faculty of Science and Engineering

Curtin University, Perth WA6845, Australia

²Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS

31750 Tronoh, Perak, Malaysia

³Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu,

Sabah, Malaysia

Received: 14 August 2012/Accepted: 30 September 2014

ABSTRACT

In this paper, we have examined the effectiveness of the quarter-sweep iteration concept on conjugate gradient normal residual (CGNR) iterative method by using composite Simpson's (CS) and finite difference (FD) discretization schemes in solving Fredholm integro-differential equations. For comparison purposes, Gauss- Seidel (GS) and the standard or full- and half-sweep CGNR methods namely FSCGNR and HSCGNR are also presented. To validate the efficacy of the proposed method, several analyses were carried out such as computational complexity and percentage reduction on the proposed and existing methods.

Keywords: Conjugate gradients normal residual method; linear Fredholm integro-differential equations; quarter-sweep iteration

ABSTRAK

Dalam kertas ini, kami telah menganalisis keberkesanan konsep lelaran sapuan suku ke atas kaedah lelaran kecerunan konjugat sisa biasa (CGNR) dengan menggunakan komposit Simpson's (CS) dan beza terhingga (FD) dalam menyelesaikan persamaan integro-pembezaan Fredholm. Bagi tujuan perbandingan, Gauss-Seidel (GS) dan kaedah CGNR biasa atau penuh dan separuh sapuan iaitu FSCGNR dan HSCGNR juga turut dibincangkan. Bagi mengesahkan keberkesanan kaedah yang dicadangkan, beberapa analisis seperti kekompleksan pengiraan dan pengurangan peratusan untuk kedua-dua kaedah yang dicadangkan dan sedia ada telah dijalankan.

Kata kunci: Kaedah lelaran kecerunan konjugat sisa biasa; lelaran sapuan suku; persamaan integro-pembezaan linear Fredholm

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*Corresponding author; email: earuchunan@yahoo.com