A FOUR-STEP BLOCK HYBRID BACKWARD DIFFERENTIATION FORMULAE FOR THE SOLUTION OF GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATION

Abstract

Ordinary Differential Equations (ODEs) play a crucial role in modeling various real-world phenomena across physics, engineering, and the applied sciences. Many of these equations, especially second-order ODEs, arise in fields such as mechanics, fluid dynamics, and electrical circuit analysis. Traditional numerical methods like single-step and multi-step techniques have been extensively explored for solving these equations. However, stiff and non-stiff problems often require more efficient and stable numerical schemes. Backward Differentiation Formulae (BDF) are implicit multi-step methods well known for their stability properties, making them suitable for solving stiff ODEs. Hybrid and block approaches have been introduced to enhance the accuracy, efficiency, and convergence of numerical methods. The block method enables the simultaneous solution of multiple points within a single step, improving computational efficiency, while the hybrid approach incorporates additional off-step points to increase accuracy. In this paper, the block hybrid Backward Differentiation formulae (BHBDF) for the step number k=4 was developed. For this purpose, power series was employed as the basis function for the development of schemes in a collocation and interpolation techniques at some selected grid and off- grid points which gave rise to continuous schemes and were further evaluated at those points to produce discrete schemes combined together to form block methods.  Analysis of the basic properties of the discrete schemes investigated showed consistency, zero stability and convergence of the proposed block methods. Tested problems were solved to examine the efficiency and accuracy of the proposed method. The results showed that the proposed methods with relatively small errors...