DEVELOPMENT OF A CONTINUOUS BACKWARD DIFFERENTIATION FORMULAE FOR SOLVING FIRST-ORDER AND SECOND-ORDER INITIAL VALUE PROBLEM

Abstract

Numerical methods for solving ordinary differential equations (ODEs) are essential in modeling dynamical systems across science and engineering. While specialized methods exist for first-order and second-order ODEs, developing a unified approach that efficiently handles both classes remain an active area of research. In this paper, we present a novel two-step hybrid block method based on the backward differentiation formula (BDF), capable of approximating solutions for both first- and second-order ODEs without requiring separate derivations. The method is constructed using interpolation and collocation techniques, and its numerical analysis confirms consistency, zero-stability, and convergence. Furthermore, stability analysis via the general linear method demonstrates that the scheme is A-stable, making it suitable for stiff systems. Numerical experiments including applications to the SIR epidemic model, Riccati differential equations, nonlinear stiff chemical systems, and second-order nonlinear ODEs—validate the method’s accuracy and computational efficiency. Comparative results with existing methods in the literature highlight its superior performance in terms of error reduction and stability. This work contributes a versatile, high-precision tool for ODE solutions, bridging gaps in the adaptability of traditional BDF-based approaches.