Geodesic
Implementation of the Taylor series method for solving ordinary differential equations
Numerical methods and programming, Tome 13 (2012) no. 4, pp. 497-510.

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New variable-step size variable-order algorithm and software implementations of the explicit Taylor series method for solving nonstiff ordinary differential equations with polynomial right-hand sides are proposed. The version of the method being used is based on new simple formulas for the recursive computation of the Taylor coefficients of the solutions and on new rigorous a priori local error estimates combined with conventional nonstrict a posteriori considerations. The author's Fortran 95 program is compared with three other existing Fortran programs of good quality that implement Dormand–Prince, Gragg–Bulirsch–Stoer and Taylor series explicit methods, respectively. Numerical experiments proves the competitive abilities of the program, its applicability and reliability to solve real problems of dynamics.
Keywords: polynomial ODE system; Taylor series method; dynamics. 
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     author = {L. K. Babadzanjanz and A. I. Bol'shakov},
     title = {Implementation of the {Taylor} series method for solving ordinary differential equations},
     journal = {Numerical methods and programming},
     pages = {497--510},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2012_13_4_a2/}
}
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L. K. Babadzanjanz; A. I. Bol'shakov. Implementation of the Taylor series method for solving ordinary differential equations. Numerical methods and programming, Tome 13 (2012) no. 4, pp. 497-510. http://geodesic.mathdoc.fr/item/VMP_2012_13_4_a2/