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34
Computer Algebra Solving of First Order ODEs Using Symmetry Methods
, 1997
"... 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 nitesima ..."
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Cited by 17 (6 self)
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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.
Computation of Densities and Fluxes of Nonlinear DifferentialDifference Equations
, 2003
"... 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 ..."
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Cited by 12 (8 self)
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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
Algorithmic computation of higherorder symmetries for nonlinear evolution and lattice equations Adv
 Comput. Math
, 1999
"... A straightforward algorithm for the symbolic computation of higherorder symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the higherorder symmetries. The coeffici ..."
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Cited by 11 (6 self)
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A straightforward algorithm for the symbolic computation of higherorder symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the higherorder symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of firstorder in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semidiscrete lattice equations. With our Integrability Package, higherorder symmetries are obtained for several wellknown systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of higherorder symmetries exist. The existence of a sequence of such symmetries is a predictor for integrability.
Applying AXIOM to Partial Differential Equations
, 1995
"... 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 CartanKuranishi Theorem and ..."
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Cited by 9 (6 self)
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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 CartanKuranishi Theorem and the setting up of the determining system for the generators of classical and nonclassical Lie symmetries. Details of the implementation are described and examples of applications are given. An appendix contains tables of all exported functions.
The Symbolic Integration of Exact PDEs
, 2000
"... 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 al ..."
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Cited by 8 (5 self)
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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 nonlinear 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 overdetermined systems of partial differential 1 equations (PDEs) have t...
Symmetry group analysis of the shallow water and semigeostrophic
"... The twodimensional shallow water equations and their semigeostrophic 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 symmetr ..."
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Cited by 8 (0 self)
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The twodimensional shallow water equations and their semigeostrophic 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 semigeostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semigeostrophic equations in such a way that a welldefined 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.
Symbolic software for soliton theory
 Also: Proc. of KdV '95 Conf
, 1995
"... 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 nonlin ..."
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Cited by 7 (6 self)
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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 differencedifferential equations. Several examples illustrate the capabilities of the software. Key words: soliton theory, symbolic programs, Hirota method, Painlevé test, Lie symmetries, conserved densities.
Group analysis of variable coefficient diffusion–convection equations
 IV. Potential symmetries, 2007, in preparation
"... 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 cl ..."
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Cited by 6 (5 self)
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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 nontrivial 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
Size Reduction and Partial Decoupling of Systems of Equations
 J. Symb. Comput
, 1999
"... 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 nonlinear systems. Modifications to improve efficiency are given and examples are sho ..."
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Cited by 5 (5 self)
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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 nonlinear systems. Modifications to improve efficiency are given and examples are shown. This procedure can be used as a presimplification 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...
Symmetries and First Order ODE Patterns
 Computer Physics Communications
, 1998
"... 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 ODEsol ..."
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Cited by 4 (1 self)
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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 ODEsolver. 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/e81.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...