E.S. Cheb-Terrab

Computer algebra solving of second order ODEs using symmetry methods

Abstract A Maple V R.3/4 computer algebra package, ODEtools , for the analytical solving of first order ODEs using Lie group symmetry methods is presented. The set of commands includes a first order ODE solver and mutines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant first order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results.

Abel ODEs: Equivalence and integrable classes

A classification, according to invariant theory, of non-constant invariant Abel ODEs known as solvable and found in the literature is presented. A set of new integrable classes depending on one or no parameters, derived from the analysis of the works by Abel (1881), Liouville (1886) and Appell (1889), is also shown. Computer algebra routines were developed to solve ODEs members of these classes by solving their related equivalence problem. The resulting library permits a systematic solving of Abel type ODEs in the Maple symbolic computing environment.

A computational approach for the analytical solving of partial differential equations

A strategy for the analytical solving of partial differential equations and a first implementation of it as the PDEtools software package of commands, using the Maple V R.3 symbolic computing system, are presented. This implementation includes a PDE-solver, a command for changing variables and some other related tool-commands.

Integrating Factors for Second-order ODEs

A systematic algorithm for building integrating factors of the form ?(x,y), ?(x,y?) or ?(y,y?) for second-order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the ?(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamkes book is shown.

Symmetries and first order ODE patterns

A scheme for determining symmetries for certain families of first order ODEs, without solving any differential equations, and based mainly in matching an ODE to patterns of invariant ODE families, is presented. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A statistics of the performance of this approach in solving the first order ODE examples of Kamkes book (E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen (Chelsea, New York, 1959)) is shown.

Poincaré sections of Hamiltonian systems

Abstract A set of Maplev R.3 software routines, for plotting 2 D 3 D projections of Poincare surfaces-of-section of Hamiltonian dynamical systems, is presented. The package consists of a plotting-command plus a set of facility-commands for a quick setup of the Hamilton equations of motion, initial conditions for numerical experiments, and for the zooming of plots.

Vlasov-Maxwell equations for complicated plasma geometry

Abstract The explicit form of Vlasovs equation for complicated plasma geometries (toroidal and spherical types) is obtained using the computer algebra package Maple . The use of the result in particular plasma physics problems is discussed. In addition, some remarks towards a closed analytical solution of the set of Vlasov-Maxwell equations are given.

Maple procedures for partial and functional derivatives

Abstract A package with new Maple V commands for the evaluation of partial derivatives, functional derivatives and integrals containing the Dirac delta function is presented. Some examples of the use of these commands in physics are shown.

ANALYTICAL SOLVING OF PARTIAL DIFFERENTIAL EQUATIONS USING SYMBOLIC COMPUTING

This work presents a brief discussion and a plan towards the analytical solving of Partial Dierential Equations (PDEs) using symbolic computing, as well as a implementation of part of this plan as the PDEtools software-package of commands.