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Determination of approximate symmetries of differential equations. (English) Zbl 1092.34018

Winternitz, P. (ed.) et al., Group theory and numerical analysis. Selected papers based on the presentation at the workshop, Montréal, Canada, May 26–31, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3565-3/pbk). CRM Proceedings and Lecture Notes 39, 249-266 (2005).
The authors make a compelling case for the method of symboliconumerical computation described in their paper by a consideration of the Lie point symmetries of the equation \[ y'' + ayy' +by ^ 3= 0, \tag{1} \] where \(a \) and \(b \) are specific numbers almost connected by the known relation for which the equation admits eight Lie point symmetries and is thereby linearisable by means of a point transformation. For other values of \(a \) and \(b,\) the equation possesses just the two Lie point symmetries of invariance under time translation and rescaling. (It is still linearisable, but the transformation is nonlocal.) The analysis of the approximate symmetries of (1) in the sense of Baikov et al would have been an interesting addition to the paper.
The authors argue that their approach is useful in the context of equations with numerical coefficients which may be approximate. Perhaps there exists a ‘nearby’ equation with greater symmetry and this equation could be used to provide a starting point for the solution of the equation at hand since the solution is an analytic function of the parameters in the equation if they occur analytically.
The argument against the use of fully symbolic codes through the replacement of numerical coefficients by parameters is not completely convincing since the equations treated as examples are of few terms. In practice, say with systems of equations arising in the models of biological systems, the number of parameters can be large and this leads to computational difficulties as the authors state.
The approach is interesting and makes a useful contribution to the practice of symmetry methods. It is a pity that the paper is marred by an extraordinary facility in the splitting of infinitives and the misuse of the word ‘exact’.
For the entire collection see [Zbl 1074.65002].

MSC:

34C14 Symmetries, invariants of ordinary differential equations
68W30 Symbolic computation and algebraic computation

Software:

ConLaw; ApplySym; SYMMGRP
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