Implicit extension of taylor series method for initial value problems

Abstract

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher derivatives using well-known technique for the partial differential equations. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval. This property offers different facility for adaptive error control. This paper describes several explicit Taylor series with implicit extension algorithms and examines its consistency and stability properties. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which conduces to the implicit R-K algorithms [12]. It demonstrates some numerical test results for stiff systems herewith we attempt to prove the efficiency of these new-old algorithms.