The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

An eigenfunction expansion associated with a two-parameter system of differential equations. III

1983 ◽  
Vol 93 (3-4) ◽  
pp. 189-195 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Differential Equations ◽  
Uniform Convergence ◽  
Ordinary Differential Equations ◽  
Eigenfunction Expansion ◽  
Second Order ◽  
System Of Differential Equations ◽  
Parameter System ◽  
Two Parameter

SynopsisWe continue with the work of earlier papers concerning the use of partial dilferential equations to prove the uniform convergence of the eigenfunction expansion associated witha left definite two-parameter system of ordinary differential equations of the second order.

An eigenfunction expansion associated with a two-parameter system of differential equations. I

1981 ◽  
Vol 89 (1-2) ◽  
pp. 143-155 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Uniform Convergence ◽  
Ordinary Differential Equations ◽  
Eigenfunction Expansion ◽  
Second Order ◽  
System Of Differential Equations ◽  
Parameter System ◽  
Partial Differential ◽  
Two Parameter

SynopsisTechniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

An eigenfunction expansion associated with a two-parameter system of differential equations. II

1982 ◽  
Vol 92 (1-2) ◽  
pp. 87-93 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Uniform Convergence ◽  
Ordinary Differential Equations ◽  
Eigenfunction Expansion ◽  
Second Order ◽  
System Of Differential Equations ◽  
Parameter System ◽  
Partial Differential ◽  
Two Parameter

SynopsisWe continue with the work of an earlier paper concerning the use of partial differential equations to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

scholarly journals Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complete Classification ◽  
First Integrals ◽  
Second Order ◽  
Point Transformations ◽  
Maximal Algebra ◽  
The Relationship ◽  
Linear Odes

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

scholarly journals An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Dunqian Cao
Keyword(s):  
Differential Equations ◽  
Difference Scheme ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Spline Function ◽  
Boundary Value ◽  
Second Order ◽  
Smooth Functions ◽  
Exponential Spline ◽  
The Relationship

In this paper, we mainly study an exponential spline function space, construct a basis with local supports, and present the relationship between the function value and the first and the second derivative at the nodes. Using these relations, we construct an exponential spline-based difference scheme for solving a class of boundary value problems of second-order ordinary differential equations (ODEs) and analyze the error and the convergence of this method. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of 1/240h6 was achieved with smooth functions.

Sign in / Sign up

Export Citation Format

Share Document