A Sixth Order Implicit Hybrid Backward Differentiation Formula (HBDF) for Block Solution of Ordinary Differential Equations

Abstract

The Hybrid Backward Differentiation Formula (HBDF) for case K=5 was reformulated into continuous form using the idea of multistep collocation. Multistep Collocation is a continuous finite difference (CFD) approximation method by the idea of interpolation and collocation. The hybrid 5-step Backward Differentiation Formula (BDF) and additional methods of order (6,6,6,6,6,)𝑇𝑇 were obtained from the same continuous scheme and assembled into a block matrix equation which was applied to provide the solutions of IVPs over non-overlapping intervals.The continous form was im-mediately employed as block methods for direct solution of Ordinary Differential Equation (𝑦𝑦′=𝑓𝑓(𝑥𝑥,𝑦𝑦)). Some benefits of this study are, the proposed block methods will be self starting and does not call for special predictor to estimate 𝑦𝑦’ in the integrators and all the discrete methods obtained will be evaluated from a single continuous formula and its derivatives at various grids and off grid points. These study results help to speed up computation, also the requirement of a starting value and the overlap of solution model which are normally associated with conventional Linear Multistep Methods were elimi-nated by this approach. In conclusion, a convergence analysis of the derived hybrid schemes to establish their effectiveness and reliability was presented. Numerical example carried out on stiff problem further substantiates their performance.