Super class of implicit extended backward differentiation formulae for the numerical integration of stiff initial value problems

An implicit Superclass of non-block Extended Backward Differentiation Formulae (SEBDF) for the numerical integration of first-order stiff system of Ordinary Differential Equations (ODEs) in Initial Value Problems (IVPs) with optimal stability properties is presented. The stability and convergence properties of the SEBDF schemes show that the methods are consistent, zero stable and convergent. The plotted Region of Absolute Stability (RAS)  of the methods using boundary locus shows that the methods are A-stable of order up to order 5 and A(α)-stable of order up to 9. The algorithm is described whereby the required approximate solution is predicted using classical explicit Euler’s method and conventional Backward Differentiation Formula (BDF) schemes of order k and then corrected using a Super class of Extended Backward Differentiation Formula (SEBDF) schemes of higher orders k+1. The SEBDF schemes are implemented using a Modified Newton iteration algorithm iterated to convergence in which some selected systems of first-order stiff IVPs are solved, and the numerical results obtained for the proposed methods are often better than the existing BDF and SBDF methods for solving stiff IVPs.