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 MAJID^1,2,*

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

Received: 20 April 2023/Accepted: 11 July 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

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