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Invariant linearization criteria of square systems of second-order quadratically semi-linear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in…
A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the…
This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and…
This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
We present a method to obtain symmetries for second-order systems of ordinary difference equations and how to use them to reduce the order. We also introduce a technique of finding conservation laws for such systems.
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as…
A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate many novel equations. Two independent methods that can be used to derive the equations of the semigroup are…
While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by…
In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…
Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new…
We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with $n$ independent and $m$ dependent variables ($n\times m$ systems). We solve the symmetry conditions in a geometric way and…
We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear…
After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a…
Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher…
An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of…
We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution…
We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity…
We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to…