A Maple V R.3/4 computer algebra package, ODEtools, 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 explicit determination of the coefficients of the in nitesimal symmetry generator; the construction of the most general invariant 1st order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results.

We present an AXIOM environment called JET for geometric computations with partial differential equations within the framework of the jet bundle formalism. This comprises especially the completion of a given differential equation to an involutive one according to the Cartan-Kuranishi Theorem and the setting up of the determining system for the generators of classical and non-classical Lie symmetries. Details of the implementation are described and examples of applications are given. An appendix contains tables of all exported functions.

this paper we focus on one aspect of the integrability of DDEs, namely the computation of conserved densities and associated fluxes (of a given form) via direct methods which could be implemented in computer algebra systems line Maple, Mathematica, and muPAD

program tests for the existence of solitons for nonlinear PDEs. It explicitly constructs solitons using Hirota’s bilinear method. In the second program, the Painlevé integrability test for ODEs and PDEs is implemented. The third program provides an algorithm to compute conserved densities for nonlinear evolution equations. The fourth software package aids in the computation of Lie symmetries of systems of differential and difference-differential equations. Several examples illustrate the capabilities of the software. Key words: soliton theory, symbolic programs, Hirota method, Painlevé test, Lie symmetries, conserved densities.

An algorithm is described which decides if a given polynomial differential expression \Delta of multivariate functions is exact, i.e. whether there exists a first integral P such that D x P = \Delta for any one x of a set of n variables and to provide the integral P . A generalization is given to allow integration in the case that the exactness is prevented by terms which contain only functions of less than n independent variables. 1 Motivation The common way to deal with problems that involve the solution of non-linear differential equations is to try different ansatze which are either geometrically motivated or just chosen to simplify computations. Typical examples are the investigation of infinitesimal symmetries, the search for classes of integrating factors and related first integrals/conservation laws or the search for a variational principle equivalent to a given system of equations. In all these cases over-determined systems of partial differential 1 equations (PDEs) have t...

The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an ‘infinite linear cascade’. 1.

A method is presented that reduces the number of terms of systems of linear equations (algebraic, ODEs or PDEs). As a byproduct these systems become partially decoupled. A variation of this method is applicable to non-linear systems. Modifications to improve efficiency are given and examples are shown. This procedure can be used as a pre-simplification step of a computation of the radical of a differential ideal (pseudo differential Grobner basis). Algorithms for applying integrability conditions to a system of differential equations in a systematic way in order to generate simplified differential equations are implemented in a number of programs ([3, 4, 7, 8, 10, 11, 12] and more in [6]). Such calculations result in the radical or a (pseudo) differential Grobner Basis of the differential ideal generated by the original system. A common problem of these algorithms, and consequently their implementations, is an explosive expression swell. Optimizations like Buchbergers 2 nd criterion...

This is the second part of the series of papers on symmetry properties of a class of variable coefficient (1+1)-dimensional nonlinear diffusion–convection equations of general form f(x)ut = (g(x)A(u)ux)x +h(x)B(u)ux. At first, we review the results of [12] on equivalence transformations and group classification of the class under consideration. Investigation of non-trivial limits of parameterized subclasses of equations from the given class, which generate contractions of the corresponding maximal Lie invariance algebras, leads to the natural notion of contractions of systems of differential equations. After a brief discussion on contractions of symmetries, equations and solutions in general case, such types of contractions are studied for diffusion–convection equations. A detailed symmetry analysis of an interesting equation from the class under consideration is performed. Exact solutions of some subclasses of the considered class are also given. 1

A scheme for determining symmetries for certain families of rst order ODEs, without solving any dierential 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 rst order ODE examples of Kamke's book [1] is shown. (Revised Version. To appear in Computer Physics Communications) 1 Department of Mathematics, University of British Columbia, Vancouver, Canada. 2 Symbolic Computation Group, Department of Theoretical Physics, State University of Rio de Janeiro, Brasil. 3 Department of Computer Science, Faculty of Mathematics, University of Waterloo, Ontario, Canada. Available as http://dft.if.uerj.br/preprint/e8-1.tex; also as http://lie.uwaterloo.ca/odetools/ode iv.tex PROGRAM SUMMARY Title of the software package: Extension to the Maple ODEtools package Catalogue number: (sup...

We present an informal overview of a number of approaches to differential equations which are popular in computer algebra. This includes symmetry and completion theory, local analysis, differential ideal and Galois theory, dynamical systems and numerical analysis. A large bibliography is provided. 1 Introduction Differential equations represent one of the largest fields within mathematics. Besides being an interesting subject of their own right one can hardly overestimate their importance for applications. They appear in natural and engineering sciences and increasingly often in economics and social sciences. Whenever a continuous process is modeled mathematically, chances are high that differential equations are used. Thus it is not surprising that differential equations also play an important role in computer algebra and most general purpose computer algebra systems provide some kind of solve command. Many casual users believe that designing and improving such procedures is a centra...