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.

scholarly journals Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. P. Markakis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Closed Form Solution ◽  
First Integrals ◽  
Form Solution ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Abel Equations

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

On Stability in the Large for Systems of Ordinary Differential Equations

1961 ◽  
Vol 13 ◽  
pp. 480-492 ◽  
Author(s):  
Philip Hartman
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riemannian Manifolds ◽  
Autonomous Systems ◽  
The Stability ◽  
Image Position

Autonomous systems. This note concerns the stability of systems of (real) differential equations in the large on Euclidean space En and on certain Riemannian manifolds Mn. The results will be refinements of those of Krasovski (3), (4), (5) and of Markus and Yamabe (8) and will make clear the role of the various assumptions in the type of theorems under consideration.In this section, the main theorems are stated for autonomous systems(1)Their proofs are given in § 2, 3, 4. In § 5, 6, 7, generalizations to non-autonomous systems are made.

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