An efficient method for the numerical evaluation of partial derivatives of arbitrary order
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
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
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
Implicit extension of Taylor series method for initial value problems
ICCMSE '03 Proceedings of the international conference on Computational methods in sciences and engineering
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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 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.