Related papers: The Geometry of Fourth Order Differential Operator
Four coframes of invariant 1-forms are explicitly constructed using the Inductive Cartan equivalence method with rank zero corresponding to four distinct branches. These coframes are employed to characterize non-linearizable fourth-order…
Linearization problem of ordinary differential equations by a new set of tangent transformations is considered in the paper. This set of transformations allows one to extend the set of transformations applied for the linearization problem.…
We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication…
Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the exampes the…
The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'"=f(x,u,u',u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method…
We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also…
The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits the maximal ten-dimensional contact symmetry Lie algebra. The…
The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of \'Elie Cartan. This solver returns a target equation equivalent to the equation to be…
This is the first part of a series of papers. The whole series aims to develop the tools for the study of all almost Hermitian symmetric structures in a unified way. In particular, methods for the construction of invariant operators, their…
Let y''' = f(x, y, y', y'') be a 3rd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates.
The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits a five-dimensional point symmetry Lie algebra. The invariant…
In the first part of this series of papers we developed the invariant differentiation with respect to a Cartan connection, we described this procedure in the terms of the underlying principal connections, and we discussed applications of…
This paper is a continuation of Part I where the general setup was developed. Here we discuss the general equivalence problem for geometric structures and provide criteria for the equivalence, local and global, of transitive structures.…
The aim of the present paper is to propose an algorithm for a new ODE--solver which should improve the abilities of current solvers to handle second order differential equations. The paper provides also a theoretical result revealing the…
We extend the difference analogue of Cartan's second main theorem for the case of slowly moving periodic hyperplanes, and introduce two different natural ways to find a difference analogue of the truncated second main theorem. As…
We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving point transformations $x\longmapsto \varphi(x)$, $y\longmapsto \psi(x,y)$ by using Moser's method of…
Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and…
It is well known that in general relativity theory two spacetimes whose metrics are related by a coordinate transformation are physically equivalent. However, given two line elements, it is virtually impossible to implement the most general…
The aim of the paper is firstly to study domains of definitions in terms of boundary conditions of minimal and maximal operators, as well as selfadjoint extensions of a minimal operator associated with the fourth-order differential operator…
We apply Cartan's method of equivalence to find a B\"acklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.