Practical considerations in the numerical solution of theN-body problem by Taylor series methods

1973 ◽  
Vol 8 (3) ◽  
pp. 357-370 ◽  
Author(s):  
W. Black
Keyword(s):  
Numerical Solution ◽  
Taylor Series ◽  
Body Problem ◽  
Taylor Series Methods

scholarly journals Projected explicit and implicit Taylor series methods for DAEs

2021 ◽  
Author(s):  
Diana Estévez Schwarz ◽  
René Lamour
Keyword(s):  
Taylor Series ◽  
Multibody Systems ◽  
Optimization Problem ◽  
Automatic Differentiation ◽  
Taylor Coefficients ◽  
Integration Schemes ◽  
Consistent Initial Values ◽  
Taylor Series Methods ◽  
Systems Simulation ◽  
Derivative Array

AbstractThe recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.

scholarly journals A Numerical Method of Integration by Means of Taylor-Steffensen Series and Its Possible Use in the Study of the Motions of Comets and Minor Planets

1972 ◽  
Vol 45 ◽  
pp. 83-85
Author(s):  
V. F. Myachin ◽  
O. A. Sizova
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Taylor Series ◽  
Body Problem ◽  
Variable Number ◽  
Taylor Formula ◽  
Integration Step ◽  
Minor Planets ◽  
Simultaneous Integration ◽  
Variable Step

The Taylor formula is used directly in a method of numerical integration of the n-body problem of celestial mechanics; the derivatives in the expansion of the coordinates are calculated successively at each integration step according to the generalized Steffensen rule. The proposed method is the most precise of all numerical methods based on the predetermined part of the Taylor series. The method is used with a variable number of derivatives at each integration step and also with a variable step. The cumulative error in the coordinates increases more slowly in our method than in any other. We can apply the method to the study of the motion of a comet or minor planet, taking into account the perturbations by eight major planets; the method allows for the simultaneous integration of a great number of objects of zero mass.

scholarly journals Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Jianhua Hou ◽  
Beibo Qin ◽  
Changqing Yang
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Taylor Series ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Hybrid Function ◽  
Nonlinear Fredholm Integral Equations

A numerical method to solve nonlinear Fredholm integral equations of second kind is presented in this work. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Taylor series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of algebraic equations. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

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