Taylor series method with numerical derivatives 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. In some cases such algorithms will be much more complicated than a R-K methods, because it will require more function evaluation than well-known classical algorithms. However these evaluations can be accomplished fully parallel and the coefficients of truncated Taylor series can be calculated with matrix-vector operations. For large systems these operations suit for the parallel computers. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval and the local error of this solution at the interior points of the subinterval is less than that one at the end point. This property offers different facility for adaptive error control. This paper describes several above-mentioned algorithms and examines its consistency and stability properties. It demonstrates some numerical test results for stiff systems herewith we attempt to prove the efficiency of these new-old algorithms.