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Validated Solution of Ordinary Differential Equations with the Taylor Model
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2004 International Mathematica Symposium
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Banff, Canada
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In this paper, we describe a high-precision validation method for the solution of initial value problems (IVP's) of ordinary differential equations with the Taylor model. Validated integration of ODEs with the Taylor model is based on local modelling of the solution with a high-order Taylor polynomial and an error polynomial. It is proposed to use symbolic algebra to get Taylor model polynomials. During the past few years there have been combininations of symbolic-algebraic methods and validated methods into so-called 'hybrid' methods. Numerical values of these polynomials are calculated with high precision interval arithmetic. It is shown in this paper that the precision of the solution obtained progressively degrades as a result of round-off errors. Additionally, in the case of solving a system of ODEs, the so-called 'wrapping' effect arises.
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Taylor model, ODE, ordinary differential equations, validation method, initial value problems, Taylor polynomial, hybrid methods
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