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Jaafar, Badrul Amin, Mohd Zawawi, Iskandar Shah. (1405). An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme. سامانه مدیریت نشریات علمی, 14(2), 734-753. doi: 10.22034/cmde.2025.65372.3001
Badrul Amin Jaafar; Iskandar Shah Mohd Zawawi. "An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme". سامانه مدیریت نشریات علمی, 14, 2, 1405, 734-753. doi: 10.22034/cmde.2025.65372.3001
Jaafar, Badrul Amin, Mohd Zawawi, Iskandar Shah. (1405). 'An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme', سامانه مدیریت نشریات علمی, 14(2), pp. 734-753. doi: 10.22034/cmde.2025.65372.3001
Jaafar, Badrul Amin, Mohd Zawawi, Iskandar Shah. An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme. سامانه مدیریت نشریات علمی, 1405; 14(2): 734-753. doi: 10.22034/cmde.2025.65372.3001

An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme

Computational Methods for Differential Equations
مقاله 16، دوره 14، شماره 2، تیر 2026، صفحه 734-753    اصل مقاله (669.25 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.65372.3001
نویسندگان
Badrul Amin Jaafar1؛ Iskandar Shah Mohd Zawawi* 2
11. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia.\\ 2. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Sarawak, Kampus Samarahan 2, 94300 Kota Samarahan, Sarawak, Malaysia.
2Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia.
چکیده
In this study, a two-point composite block method based on the backward differentiation formula (CBBDF) is introduced to solve stiff ordinary differential equations. The CBBDF method incorporates an additional intermediate point among the interpolating points, developed in two stages: the first stage employs the Euler's method as a fundamental building block, while the second stage utilizes CBBDF of order three. A key distinction of the method with the classical block method is the introduction of an independent parameter $\gamma$, which eliminates the need for an external startup calculation, while maintaining the accuracy and stability of numerical solutions. The theoretical analysis verifies that the proposed method is convergent and $A$-stable. It fulfills the essential properties of consistency and zero-stability, and it lies within the $A$-stability region. To demonstrate the effectiveness of the proposed approach, several stiff initial value problems of linear and non-linear are solved. For validation, the results are compared with existing literature. While approximating the solution at multiple points simultaneously, the composite block method offers the ability to use larger step sizes for solution approximation. The CBBDF method shows promising results, achieving a reliable degree of accuracy as indicated by its maximum error and average error measurements.
کلیدواژه‌ها
Backward differentiation formula؛ Block method؛ Composite scheme؛ Linear multi-sub-step method؛ Stiff differential equations
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