Sains Malaysiana 51(12)(2022): 4125-4144

http://doi.org/10.17576/jsm-2022-5112-20

Numerical Approach for Delay Volterra Integro-Differential Equation

(Pendekatan Berangka Bagi Penyelesaian Persamaan Pembezaan Lengah-Kamilan Volterra)

NUR AUNI BAHARUM¹, ZANARIAH ABDUL MAJID^1,2,*, NORAZAK SENU^1,2 & HALIZA ROSALI^1,2

¹Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

²Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

Diserahkan: 12 Mei 2022/Diterima: 29 Ogos 2022

Abstract

The delay integro-differential equation for the Volterra type has been solved by using the two-point multistep block (2PBM) method with constant step-size. The proposed block method of order three is formulated using Taylor expansion and will simultaneously approximate the numerical solution at two points. The 2PBM method is developed by combining the predictor and corrector formulae in the PECE mode. The predictor formulae are explicit, while the corrector formulae are implicit. The algorithm for the approximate solutions were constructed and analyzed using the 2PBM method with Newton-Cotes quadrature rules. This paper focused on constant and pantograph delay types, and the previous values are used to interpolate the delay solutions. Moreover, the studies also carried out on the stability analysis of the proposed method. Some numerical results are tested to validate the competency of the multistep block method with quadrature rule approach.

Keywords: Multistep block; Newton-Cotes rule; Volterra delay integro-differential equation

Abstrak

Persamaan pembezaan lengah kamilan bagi jenis Volterra telah diselesaikan menggunakan kaedah blok berbilang langkah dua titik (2PBM) untuk langkah yang malar. Kaedah blok peringkat tiga yang dicadangkan telah dirumus menggunakan pengembangan Taylor dan akan menganggar penyelesaian berangka secara serentak pada dua titik. Kaedah 2PBM dibangunkan dengan menggabungkan formula peramal dan pembetul dalam mod PECE. Kaedah peramal adalah tak tersirat manakala kaedah pembetul adalah tersirat. Algoritma penyelesaian anggaran dibina dan dianalisis menggunakan kaedah 2PBM dengan peraturan kuadratur Newton-Cotes. Kertas ini memberi tumpuan kepada jenis kelengahan malar dan pantograf serta nilai sebelumnya digunakan untuk menginterpolasi penyelesaian kelengahan. Selain itu, kajian juga dijalankan ke atas analisis kestabilan bagi kaedah yang dicadangkan. Beberapa keputusan berangka diuji untuk mengesahkan kecekapan kaedah blok berbilang langkah dengan pendekatan peraturan kuadratur.

Kata kunci: Blok berbilang langkah; peraturan Newton-Cotes; persamaan pembezaaan lengah-kamilan Volterra

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^*Pengarang untuk surat-menyurat; email: am_zana@upm.edu.my