GENERALIZED THIRD DERIVATIVE METHOD FOR SOLVING STIFF INITIAL VALUE PROBLEMS

ABSTRACT

Boundary Value Method (BVM) is derived as a unifying approach where some conditions are imposed at the initial points and the remaining ones at the final points. The unifying approach used, makes it feasible to obtain methods which overcomes the well-known Dahlquist barriers and Daniel- Moore conjecture for an A-stable Linear Multistep Formulae (LMF). To date, A-stability is a severe requirement to the integration of stiff Ordinary Differential Equations (ODEs). In this study, a family of Third Derivative LMF called Generalized Third Derivative Method (GTDM) is Constructed using collocation and interpolation technique. The new schemes, ETLMFs is implemented as a Boundary Value Method (BVM) for the numerical integration of Stiff initial value problems (IVPs). This class of methods is of order 2k + 2 and has a regular stability region over the conventional Third Derivative LMF. The coefficients of the methods is derived through Taylor’s series expansion and the method of undetermined coefficient. The Condition Number of the arising Topliz matrices of GTDMs is bounded. Hence, the GTDMs is 0v,k−v-stable and Av,k−v-stable for all step-length k. Numerical results on some stiff problems are presented to show accuracy of the method and are compared with some existing ones in the literature.