Development of Two – Step Simpson – Type Hybrid Block Method for the Solution of Differential Equations - Suleiman, Fahad; Mohammed, Umaru; Abdullahi, Habibah (FUDMA Journal of Sciences, 2026)
This paper presents the development of a Two-Step Simpson-Type Second Derivative Hybrid Block Method (TSSTHBM) for the numerical solution of stiff and non-stiff systems of first-order ordinary differential equations. The method is derived using polynomial interpolation and collocation techniques, incorporating second derivative information and four off-grid (intra-step) points to enhance accuracy and stability. The resulting scheme is implemented in block form, allowing simultaneous computation of solution values without the need for starting procedures. The basic properties of the method, including order, consistency, zero-stability, and convergence, are established, confirming the reliability of the scheme. Numerical experiments are carried out on standard test problems, including nonlinear systems, linear stiff systems, Riccati equations, and a nonlinear biosorption model. The results demonstrate that the proposed method provides improved accuracy and better stability when compared with existing methods in the literature, even with relatively larger step sizes. Therefore, the TSSTHBM is an efficient and reliable method for solving stiff differential equations arising in scientific and engineering applications.