To Approximate Solution of Ordinary Differential Equations, I

2013 ◽  
pp. 165-181 ◽  
Author(s):  
Tamaz S. Vashakmadze
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations

scholarly journals On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

2021 ◽  
Vol 29 (1) ◽  
pp. 63-72
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Projective Geometry ◽  
Autonomous Systems ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Duality Principle ◽  
Integrals Of Motion ◽  
Quadratic Integral

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

Approximate solution of random ordinary differential equations

1978 ◽  
Vol 10 (1) ◽  
pp. 172-184 ◽  
Author(s):  
William E. Boyce
Keyword(s):  
Approximate Solution ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Approximate Methods ◽  
Expository Paper

This is a largely expository paper on approximate methods of solving random ordinary differential equations, with an emphasis on direct numerical methods. Two methods are discussed in some detail and several others are mentioned briefly.

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