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Sains Malaysiana 54(12)(2025): 2937-2952
http://doi.org/10.17576/jsm-2025-5412-12
A
Single-Step Approach with Seven Hybrid Points for the Solution of Stiff
Differential Equations
(Pendekatan Langkah Tunggal dengan Tujuh Titik Hibrid untuk Penyelesaian Persamaan Pembezaan Kaku)
IBRAHIM MOHAMMED DIBAL1,*, YEAK SU
HOE2 & SUZARINA AHMED SUKRI2
1Department of General Studies,
School of General and Remedial Studies, Federal Polytechnic
Damaturu, 1006 Yobe State, Nigeria
2Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia,
81310 Skudai, Johor Bahru, Johor, Malaysia
Received: 29 March 2025/Accepted: 25 December 2025
Abstract
This research presents a novel single-step hybrid
block method with seven intra-step points that achieves ninth-order accuracy,
providing an accurate and computationally efficient approach for solving
first-order stiff differential equations. The method is designed to solve
first-order stiff differential equations with high efficiency and precision
while maintaining a constant step size throughout the computation. To further
improve accuracy, Lagrange polynomial interpolation
techniques are employed to approximate function values at selected points
within each step. The fundamental properties of the proposed scheme are
rigorously analysed to establish its mathematical validity. These analyses
confirm that the method satisfies the essential conditions of consistency,
stability, and convergence, thereby ensuring its reliability for applications. The
proposed method performs effectively when applied to stiff and oscillatory
differential equations. Comprehensive numerical experiments are conducted, and
the results consistently demonstrate the robustness and effectiveness of the
proposed method across various test problems. Furthermore, the findings
indicate that the method often outperforms several existing numerical
techniques in terms of both accuracy and computational efficiency.
Keywords: Block
hybrid method; intra-step points; stiff equations; zero stability
Abstrak
Penyelidikan ini memberikan satu kaedah blok hibrid satu langkah yang baharu dengan tujuh titik intra-langkah yang mencapai ketepatan tertib kesembilan, sekali gus menyediakan pendekatan yang tepat dan cekap dari segi pengiraan untuk menyelesaikan persamaan pembezaan biasa tegar tertib pertama. Kaedah ini direka dengan teliti untuk menyelesaikan persamaan pembezaan tertib pertama dengan kecekapan dan ketepatan yang tinggi sambil mengekalkan saiz langkah yang malar sepanjang pengiraan. Bagi meningkatkan lagi ketepatan, teknik interpolasi digunakan untuk menganggar nilai fungsi pada titik terpilih dalam setiap langkah. Sifat asas skim yang dicadangkan dianalisis secara teliti bagi membuktikan kesahihan matematiknya. Analisis ini mengesahkan bahawa kaedah tersebut memenuhi syarat penting iaitu kekonsistenan, kestabilan dan penumpuan, sekali gus menjamin kebolehpercayaannya untuk aplikasi praktikal. Selain itu, kaedah ini menunjukkan kebolehsuaian yang tinggi, menjadikannya sesuai untuk pelbagai jenis permasalahan. Secara khusus, kaedah ini berprestasi dengan berkesan apabila digunakan pada persamaan pembezaan tegar dan berayun. Uji kaji berangka yang menyeluruh telah dijalankan dan keputusannya secara tekal menunjukkan keteguhan dan keberkesanan kaedah yang dicadangkan merentasi pelbagai masalah ujian. Tambahan pula, keputusan kajian menunjukkan bahawa kaedah ini sering mengatasi beberapa teknik berangka sedia ada dari segi ketepatan dan kecekapan pengiraan.
Kata kunci: Kaedah hibrid blok; kestabilan sifar; persamaan tegar; titik intra
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*Corresponding
author; email: mohammeddibal5@gmail.com
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