An efficient method for the numerical evaluation of partial derivatives of arbitrary order
ACM Transactions on Mathematical Software (TOMS)
Consistency and stability for some nonnegativity-conserving methods
Applied Numerical Mathematics
On the relational database type numerical programming
ICECT'03 Proceedings of the third international conference on Engineering computational technology
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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 this algorithms is based on the approximate calculation of higher derivatives using well-known finite-difference technique for the partial differential equations. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval integration. This property offers different facility for adaptive error control. This paper describes several explicit Taylor series algorithms with numerical derivatives and their implicit extension and examines its consistency and stability properties. The implicit extension based on a collocation term added to the explicit truncated Taylor series and the approximate solution obtained as a continuously differentiable piecewise polynomials function. Some numerical test results is presented to prove the efficiency of these new-old algorithm.