Taylor series method for dynamical systems with control: Convergence and error estimates |
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Authors: | L K Babadzhanjanz D R Sarkissian |
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Institution: | (1) Department of Applied Mathematics and Processes of Management, St. Petersburg State University, Russia;(2) Department of Mathematics and Statistics, Mississippi State University, MS 39765, USA |
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Abstract: | To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is
very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard
Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize.
We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with
respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the
estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori
step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems
of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and
nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision,
while the aggregative Taylor series method achieves the best computational time.
The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of
ODEs describing the optimal flight of a spacecraft from the Earth to the Moon.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical
Systems and Optimization, 2005. |
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Keywords: | |
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