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

Received: 12 May 2022/Accepted: 29 August 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

REFERENCES

Ali, H.A. 2009. Expansion method for solving linear delay integro-differential equation using B-spline functions. Engineering and Technology Journal 27(10): 1651-1661.

Ayad, A. 2001. The numerical solution of first order delay integro-differential equations by spline functions. International Journal of Computer Mathematics 77(1): 125-134.

Baharum, N.A., Majid, Z.A. & Senu, N. 2022. Boole’s strategy in multistep block method for Volterra integro-differential equation. Malaysian Journal of Mathematical Sciences 16(2): 237-256.

Baker, C.T.H. 2000. A perspective on the numerical treatment of Volterra equations. Journal of Computational and Applied Mathematics 125(1-2): 217-249.

Ismail, N.I.N., Majid, Z.A. & Senu, N. 2020. Hybrid multistep block method for solving neutral delay differential equations. Sains Malaysiana 49(4): 929-940.

Janodi, M.R., Majid, Z.A., Ismail, F. & Senu, N. 2020. Numerical solution of Volterra integro-differential equations by hybrid block with quadrature rules method. Malaysian Journal of Mathematical Sciences 14(2): 191-208.

Kolmanovskii, V. & Myshkis, A. 2012. Applied Theory of Functional Differential Equations. New York: Kluwer Academic Publisher.

Lambert, J.D. 1973. Computational Methods in Ordinary Differential Equations. New York: Wiley.

Majid, Z.A. & Mohamed, N.A. 2019. Fifth order multistep block method for solving Volterra integro-differential equations of second kind. Sains Malaysiana 48(3): 677-684.

Mustafa, Q.K. & Mohammed, A.M. 2018. Numerical method for solving delay integro-differential equations. Research Journal of Applied Sciences 13: 103-105.

Qin, H., Zhiyong, W., Fumin, Z. & Jinming, W. 2018. Stability analysis of additive Runge-Kutta methods for delay-integro-differential equations. International Journal of Differential Equations. 2018: Article ID. 8241784. https://doi.org/10.1155/2018/8241784

Salih, R.K., Hassan, I.H. & Atheer, J.K. 2014. An approximated solutions for n^th order linear delay integro-differential equations of convolution type using B-spline functions and Weddle method. Baghdad Science Journal 11(1): 168-177.

Salih, R.K., Hassan, I.H., Atheer, J.K. & Fuad, A.H. 2010. B-spline functions for solving n^th order linear delay integro-differential equations of convolution type. Engineering and Technology Journal 28(23): 6801-6813.

Shakourifar, M. & Dehghan, M. 2008. On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments. Computing 82(4): 241-260.

Yüzbaşı, Ş. & Karaçayır, M. 2018. A numerical approach for solving high-order linear delay Volterra integro-differential equations. International Journal of Computational Methods 15(5): 1850042.

Zaidan, L.I. 2012. Solving linear delay Volterra integro-differential equations by using Galerkin’s method with Bernstien polynomial. J. Bablyon Appl. Sci. 20: 1305-1313.

^*Corresponding author; email: am_zana@upm.edu.my