73 research outputs found
Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations
In this paper we propose some linearizability tests of partial difference
equations on a quad-graph given by one point, two points and generalized
Hopf-Cole transformations. We apply the so obtained tests to a set of
nontrivial examples
Complexity reduction of C-algorithm
The C-Algorithm introduced in [Chouikha2007] is designed to determine
isochronous centers for Lienard-type differential systems, in the general real
analytic case. However, it has a large complexity that prevents computations,
even in the quartic polynomial case.
The main result of this paper is an efficient algorithmic implementation of
C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of
it is proposed in the rational case. It is called RCA (Rational C-Algorithm)
and is widely used in [BardetBoussaadaChouikhaStrelcyn2010] and
[BoussaadaChouikhaStrelcyn2010] to find many new examples of isochronous
centers for the Li\'enard type equation
Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches
In this review article we discuss four recent methods for computing
Maurer-Cartan structure equations of symmetry groups of differential equations.
Examples include solution of the contact equivalence problem for linear
hyperbolic equations and finding a contact transformation between the
generalized Hunter-Saxton equation and the Euler-Poisson equation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Algorithmic approach to linearization of scalar ordinary differential equation
Аналитическая теория дифференциальных уравнени
Algorithms for Mappings and Symmetries of Differential Equations
Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points
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