A Three-Step Numerical Approximant Based on Block Hybrid Backward Differentiation Formula for Stiff System of Ordinary Differential Equations

Abstract

— As long as the field of Engineering, Science and Technology exists, the place of Mathematical modelling that involves stiff systems cannot be overemphasized. Models involving stiff system may result in ordinary differential equations (ODEs) or sometimes as system of ordinary differential equations which must be solved by experts working in that field.However, solving these models using analytical approach may sometimes be challenging or even near impossible. Therefore, it puts a great measure of importance on research into numerical algorithmsfor solution of this class of ordinary differential equations.Premised on the above mentioned, we have formulated, in this paper, a class of backward differentiation formula (BDF) which is a three-step numerical approximant for stiff systems of ODEs. The method was obtained through continuous collocation approach with Legendre polynomial as basis function. We incorporated three off-grid points at interpolation in order that we may retain theBDF’s single function evaluation characteristic. Analyzing basic properties of numerical methods led us to see that the method was consistent, having a uniform order six, zero-stable and in turn, convergent. The method's region of absolute stability was determined using the general linear method, which was plotted and shown to be stable over a vast area. The approach enumerates the solution of stiff of systems ODEs block by block using some discrete schemes that are secured from the corresponding continuous scheme. The method was tested using numerical experiments, and the results, when compared to exact or analytical answers as well as some methods published in the literature, proved that the method is efficient and accurate