Taylor Series Method with Numerical Derivatives for Numerical Solution of ODE Initial Value Problems

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. Currently this algorithm is not applied frequently. This is because when one solves systems of ordinary differential equations, calculating the higher order derivatives formally is an over-elaborate task, this is true even if one uses the computer algebraic systems such as MATHEMATICA or MAPLEV. The other reason is that only the explicit versions of this algorithm are known. 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. These algorithms have several advantageous properties over the widely used classical methods. 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. We remark that for the explicit Taylor series methods is possible to give its implicit extension. Using the fact, that the approximate solution is a continuous function, (in the case of implicit version it is a continuously differentiable function), adaptive examination and control of some qualitative properties of algorithms will be more simple than the case when the approximate solution are given only at discrete grid points. This paper describes several above-mentioned algorithms and examines its consistency and stability properties. It demonstrates some numerical test results for systems of equations herewith we attempt to prove the efficiency of these new-old algorithms.