Malaysian Journal of Mathematical Sciences, September 2026, Vol. 20, No. 3
Block Backward Differentiation Formula with a Singly Diagonally Implicit Structure for Numerical Simulation of Dynamics Models of Diabetes Mellitus
Sunwandi, N. S., Aksah, S. J., Rasid, N. A., and Ijam, H. M.
Corresponding Email: hazizahijam@uitm.edu.my
Received date: 3 September 2025
Accepted date: 10 March 2026
Abstract:
This paper presents a Singly Diagonally Implicit Block Backward Differentiation Formula of order two for the numerical solution of stiff ordinary differential equations arising in diabetes mellitus models. The proposed method is formulated as a two–step block scheme that computes approximations at two consecutive time levels simultaneously. Its singly diagonally implicit coefficient structure enables sequential solution of the block equations, significantly reducing computational cost per time step while preserving strong stability properties for stiff systems. The performance of the proposed method is investigated using several diabetes models, including models with available analytical solutions. Numerical experiments demonstrate improved accuracy compared with the conventional block backward differentiation formula method and stable, non-oscillatory behavior across a range of stiff parameter regimes. A stability analysis based on the linear test equation confirms that the proposed method is both A–stable and L–stable, ensuring effective damping of rapidly decaying components. These results show that the proposed method is suitable for the efficient and stable numerical simulation of stiff diabetes models.
Keywords: block backward differentiation formula; dynamics models of diabetes mellitus; singly diagonally implicit.
