Although the application of Lie methods to differential equations is algorithmic, it usually involves a lot of cumbersome and tedious calculations. Today, many powerful Computer Algebra Systems (either commercial or open source) are available, and the needed algebraic manipulations can be rapidly done almost automatically. In fact, many specific packages are currently available in the literature. This Chapter describes two computer programs. The first one, written in the open source Computer Algebra System Reduce, is able to perform almost automatically much of the computations needed for analyzing differential equations from a Lie group standpoint. This program, ReLie, is able to determine point, contact, variational and conditional symmetries, both exact and approximate. The user is requested to input a minimal set of data for defining the problem, and the program can do almost all the work automatically. The second program, SymboLie, is written in Wolfram Mathematica™. It is devoted to find families of optimal Lie subalgebrasOptimal Lie subalgebras of a finite-dimensional Lie algebra. The goal is achieved by classifying the Lie subalgebras of a finite-dimensional Lie algebra by means of the inner automorphisms that are obtained by exponentiating the adjoint groups. Both programs are freely available.

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Computer Algebra Programs

  • Francesco Oliveri

摘要

Although the application of Lie methods to differential equations is algorithmic, it usually involves a lot of cumbersome and tedious calculations. Today, many powerful Computer Algebra Systems (either commercial or open source) are available, and the needed algebraic manipulations can be rapidly done almost automatically. In fact, many specific packages are currently available in the literature. This Chapter describes two computer programs. The first one, written in the open source Computer Algebra System Reduce, is able to perform almost automatically much of the computations needed for analyzing differential equations from a Lie group standpoint. This program, ReLie, is able to determine point, contact, variational and conditional symmetries, both exact and approximate. The user is requested to input a minimal set of data for defining the problem, and the program can do almost all the work automatically. The second program, SymboLie, is written in Wolfram Mathematica™. It is devoted to find families of optimal Lie subalgebrasOptimal Lie subalgebras of a finite-dimensional Lie algebra. The goal is achieved by classifying the Lie subalgebras of a finite-dimensional Lie algebra by means of the inner automorphisms that are obtained by exponentiating the adjoint groups. Both programs are freely available.