Sains Malaysiana 52(8)(2023): 2337-2351

http://doi.org/10.17576/jsm-2023-5208-13

 

Hybrid Multistep Block Method for Solving Neutral Volterra Integro-Differential Equation with Proportional and Mixed Delays

(Kaedah Berbilang Langkah Blok Hibrid untuk Menyelesaikan Persamaan Kamiran-Pembezaan Neutral Volterra dengan Kelengahan Berkadar dan Bercampur)

 

NUR INSHIRAH NAQIAH ISMAIL1 & ZANARIAH ABDUL MAJID1,2,*

 

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

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

 

Diserahkan: 20 April 2023/Diterima: 11 Julai 2023

 

Abstract

The neutral Volterra integro-differential equation with proportional and mixed delays (NDVIDE) is being solved by a newly proposed technique in numerical method, namely, the two-point one off-point block multistep method (1OBM3). The method is also known as a hybrid multistep block method. Subsequently, Lagrange interpolating polynomial is utilized in order to develop the hybrid block method. The foundation of the technique is taken from predictor and corrector formulae. The proposed method will solve NDVIDE in two steps simultaneously, with three predictor formulae including one off-point. The NDVIDE problems are solved via the constant step size technique. In order to solve the integral and differential parts of the problems, two alternative numerical approaches are applied. The differentiation part is approximated by deriving the divided difference formula, while the integration part is interpolated using composite Simpson’s rule. Note that the proposed method has been analysed thoroughly regarding its order, consistency, zero stability and convergence of the method. The stability region for 1OBM3 has been constructed based on the stability polynomial obtained. Consequently, numerical results are presented to demonstrate the effectiveness of the proposed method, 1OBM3.

 

Keywords: Hybrid multistep block method; mixed delay; neutral delay Volterra integro-differential equations; proportional delay

 

Abstrak

Persamaan kamiran-pembezaan neutral Volterra dengan kelengahan berkadar dan bercampur (NDVIDE) diselesaikan dengan teknik baharu yang dicadangkan dalam kaedah berangka iaitu, kaedah blok berbilang langkah dua titik dan satu luar-titik (1OBM3). Kaedah ini juga dikenali sebagai kaedah blok berbilang langkah hibrid. Interpolasi polinomial Lagrange dimanfaatkan bagi membangunkan kaedah blok hibrid. Asas kepada kaedah ini diambil daripada formula peramal-pembetul. Kaedah yang dicadangkan akan menyelesaikan NDVIDE dalam dua langkah serentak dengan tiga formula peramal termasuk satu luar-titik. Masalah NDVIDE diselesaikan melalui teknik saiz langkah malar. Untuk menyelesaikan masalah di bahagian kamiran dan pembezaan, dua pendekatan berangka alternatif digunakan. Bahagian pembezaan dianggarkan dengan memperoleh formula perbezaan terbahagi manakala bahagian kamiran di interpolasi dengan menggunakan peraturan Simpson komposit. Kaedah yang dicadangkan telah dianalisis dengan teliti dari segi peringkat, ketekalan, kestabilan sifar dan penumpuan. Kawasan kestabilan untuk 1OBM3 telah dibina berdasarkan polinomial kestabilan yang diperoleh. Keputusan berangka dibentangkan untuk menunjukkan keberkesanan kaedah 1OBM3 yang dicadangkan.

 

Kata kunci: Kaedah blok berbilang langkah hybrid; kelengahan bercampur; kelengahan berkadar; kelengahan neutral persamaan kamiran-pembezaan Volterra

 

RUJUKAN

Altun, Y. 2021a. Asymptotic behaviours of the solutions of neutral type Volterra integro-differential equations and some numerical solutions via differential transform method. MANAS Journal of Engineering 9(Special 1): 49-57.

Altun, Y. 2021b. A new result on the global exponential stability of nonlinear neutral Volterra integro-differential equation with variable lags. Mathematics in Natural Science 5(2019): 29-43.

Baharum, N.A., Majid, Z.A. & Senu, N. 2018. Solving Volterra integrodifferential equations via diagonally implicit multistep block method. International Journal of Mathematics and Mathematical Sciences 2018: 7392452.

Gurbuz, B. 2021. A numerical scheme for the solution of neutral integro-differential equations including variable delay. Mathematical Sciences 16(1-9): 13-21.

Ismail, N.I.N., Majid, Z.A. & Senu, N. 2022. Numerical solution for neutral delay differential equation of constant or proportional type using hybrid block method. Advanced in Applied Mathematics and Mechanics 14(5): 1138-1160.

Ismail, N.I.N., Majid, Z.A. & Senu, N. 2020. Solving neutral delay differential equation of pantograph type. Malaysian Journal of Mathematical Sciences (ICoAIMS2019) 14(S): 107-121.

Jator, S.N. 2010. On the hybrid method with three off-step points for initial value problems. International Journal of Mathematical Education in Science and Technology 41(1): 110-118.

Laib, H., Bellour, A. & Boulmerka, A. 2022. Taylor collocation method for high-order neutral delay Volterra integro-differential equations. Journal of Innovative Applied Mathematics and Computational Sciences 2(1): 43-77.

Lambert, J.D. 1991. Numerical Methods for Ordinary Differential Systems. New York: John Wiley & Sons.

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

Li, Y. & Li, S. 2021. Classical theory of linear multistep methods for Volterra functional differential equations. Discrete Dynamics in Nature and Society 2021: 1-15.

Mirzaee, F., Bimesl, S. & Tohidi, E. 2016. A numerical framework for solving high-order pantograph-delay Volterra integro-differential equations. Kuwait Journal of Science 43(1): 69-83.

Mohamed, N.A. & Majid, Z.A. 2016. Multistep block method for solving Volterra integro-differential equations. Malaysian Journal of Mathematical Sciences 10(2016): 33-48.

Reutskiy, S.Y. 2016. The backward substitution method for multipoint problems with linear volterra–fredholm integro-differential equations of the neutral type. Journal of Computational and Applied Mathematics 296: 724-738.

Rihan, F.A., Doha, E.H., Hassan, M.I. & Kamel, N. 2009. Numerical treatments for Volterra delay integro-differential equations. Computational Methods in Applied Mathematics 9(3): 292-318.

Sedaghat, S., Ordokhani, Y. & Dehghan, M. 2014. On spectral method for Volterra functional integro-differential equations of neutral type. Numerical Functional Analysis and Optimization 35(2): 223-239.

Vijayakumar, V. 2018, Approximate controllability results for abstract neutral integrodifferential inclusions with infinite delay in Hilbert spaces. IMA Journal of Mathematical Control and Information 5(1): 297-314.

Wen, L. & Zhou, Y. 2017. Convergence of one-leg methods for neutral delay integro-differential equations. Journal of Computational and Applied Mathematics 317: 432-446.

Wen, L. & Yu, Y. 2016. Convergence of Runge–Kutta methods for neutral delay integro-differential equations. Applied Mathematics and Computation 282(2016): 84-96.

Wu, S. & Gan, S. 2008. Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Computers & Mathematics with Applications 55(11): 2426-2443.

Yuzbasi, S. 2014. Laguerre approach for solving pantograph-type Volterra integro-differential equations. Applied Mathematics and Computation 232: 1183-1199.

Yuzbasi, S. & Karacayir, M. 2017. A numerical approach for solving high-order linear delay Volterra integro-differential equations. International Journal of Computational Methods 15(5): 1850042.

 

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

 

 

 

 

 

 

 

 

 

 

 

 

   

sebelumnya