Sains Malaysiana 49(4)(2020): 929-940

http://dx.doi.org/10.17576/jsm-2020-4904-22

Hybrid Multistep Block Method for Solving Neutral Delay Differential Equations

(Kaedah Blok Berbilang Langkah HibridBagi Menyelesaikan PersamaanPembezaan Lengah Neutral)

NUR INSHIRAH NAQIAH ISMAIL¹, ZANARIAH ABDUL MAJID^1,2* & NORAZAK SENU¹

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

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

Received: 26 September 2019/Accepted: 13 January 2020

ABSTRACT

The initial-value problem for first order single linear neutral delay differential equations (NDDEs) of constant and pantograph delay types have been solved by using hybrid multistep block method. The method has been derived by applying Taylor series interpolation polynomial and implementing the predictor-corrector formulas in PE(CE)m  mode where m  is the number of iterations for the proposed method. Both types of NDDEs will be solved at two-point simultaneously including the off-step point with constant step-size. In order to find the solution for NDDEs, the delay solutions of the unknown function will be interpolated using Lagrange interpolation polynomial and the derivative of the delay solutions will be obtained by applying divided difference formula. The order, consistency and convergence of the proposed method have been discussed in detail in the methods section. The properties of stability region for NDDEs have also been analysed. Numerical results presented have concluded that the proposed method is comparable with the existing method and is assumed to be reliable for solving first order NDDEs with constant and pantograph delay.

Keywords: Constant delay; multistep block method; neutral delay differential equations; off-step point; pantograph delay

ABSTRAK

Masalah nilai permulaan untuk terbitan pertama tunggal linear Persamaan Pembezaan Lengah Neutral (PPLN) bagi jenis kelengahan malar dan pantograf telah diselesaikan dengan menggunakan kaedah blok berbilang langkah hibrid. Kaedah ini diperoleh dengan menggunakan polinomial penyuaian siri Taylor dan melaksanakan rumusan peramal pembetul dalam mod PE(CE)m dengan m adalah bilangan pengulangan bagi kaedah yang dicadangkan. Kedua-dua jenis PPLN akan diselesaikan pada dua titik serentak termasuk titik luar langkah dengan saiz langkah yang malar. Bagi mencari penyelesaian untuk PPLN, nilai kelengahan bagi fungsi yang tidak diketahui akan diperoleh melalui penggunaan polinomial penyuaian Lagrange dan pembezaan penyelesaian kelengahan akan diperoleh dengan menggunakan formula perbezaan pembahagian. Penentuan peringkat, tahap ketekalan dan penumpuan bagi kaedah yang dicadangkan telah dibincangkan secara terperinci dalam bahagian metod. Ciri-ciri kawasan kestabilan untuk PPLN juga telah dianalisis. Keputusan berangka yang dibentangkan telah menyimpulkan bahawa kaedah yang dicadangkan adalah setanding dengan kaedah yang telah sedia ada dan dianggap dapat menyelesaikan peringkat pertama PPLN dengan kelengahan malar dan pantograf.

Kata kunci: Kaedah blok berbilang langkah; ketundaan malar; ketundaan pantograf; persamaan pembezaan lengah neutral; titik luar langkah

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*Corresponding author; email: am_ zana@upm.edu.my