Algorithmic Lie Theory for Solving Ordinary Differential Equations

Description

Despite the fact that Sophus Lie's theory was virtually the only systematic method for solving nonlinear ordinary differential equations (ODEs), it was rarely used for practical problems because of the massive amount of calculations involved. But with the advent of computer algebra programs, it became possible to apply Lie theory to concrete problems. Taking this approach, Algorithmic Lie Theory for Solving Ordinary Differential Equations serves as a valuable introduction for solving differential equations using Lie's theory and related results.After an introductory chapter, the book provides the mathematical foundation of linear differential equations, covering Loewy's theory and Janet bases. The following chapters present results from the theory of continuous groups of a 2-D manifold and discuss the close relation between Lie's symmetry analysis and the equivalence problem. The core chapters of the book identify the symmetry classes to which quasilinear equations of order two or three belong and transform these equations to canonical form. The final chapters solve the canonical equations and produce the general solutions whenever possible as well as provide concluding remarks. The appendices contain solutions to selected exercises, useful formulae, properties of ideals of monomials, Loewy decompositions, symmetries for equations from Kamke's collection, and a brief description of the software system ALLTYPES for solving concrete algebraic problems.

The focus of Schwarz's text is on algorithmic methods for finding explicit solutions to ordinary differential equations. The book concentrates on three approaches: the solution of linear ordinary differential equations by factorization, mapping ordinary differential equations into a solved canonical form, and reduction of order through the use of Lie symmetries. The goal is to present a comprehensive survey, and so the book omits many proofs and much background material, but does include extensive historical remarks and references. A highlight is the inclusion of many worked examples and computational exercises, mostly based on Kamke's famous tables of solved differential equations [E. Kamke, Differentialgleichungen, Neunte Auflage, Teubner, Stuttgart, 1977; MR0466672]. The book will serve as a valuable reference for researchers interested in ordinary differential equations, symmetry methods, and computer algebra. In more detail:
Following a brief introductory chapter, the first section of Chapter 2 develops the factorization method for linear ordinary differential equations. The key complication is that, unlike, say, polynomials, linear differential operators do not admit unique factorizations; an elementary example is due to Landau:

(D−1x)2=(D−1x(1+ax))(D−1+2axx(1+ax)),

valid for any constant a. To circumvent this complication, Schwarz concentrates on a decomposition introduced by A. Loewy in 1906, combined with the Galois theory for linear differential equations due to Picard and Vessiot, to design effective algorithms.
The second part of Chapter 2 launches into the analysis of linear homogeneous partial differential equations using an algebraic method developed by Janet in the 1920's. Janet bases are, interestingly, a direct precursor of Buchberger's celebrated Gröbner basis method in computational algebra [D. A. Cox, J. B. Little and D. B. O'Shea, Ideals, varieties, and algorithms, Second edition, Springer, New York, 1997; MR1417938]. The main applications come later, including determination of infinitesimal symmetries of differential equations and computation of invariants.
Chapter 3 provides a "traditional'' introduction to Lie transformation groups and their infinitesimal generators—the corresponding Lie algebra. Lie's classification of group actions in the plane is discussed in detail. The final section applies Janet bases to analyze Lie systems—first-order linear partial differential equations defining Lie algebras of vector fields.
Chapter 4 deals with solution techniques based on transforming the differential equation to a canonical form. For linear ordinary differential equations, the Laguerre-Forsyth normal forms are developed. Extensions to nonlinear equations rely on a direct analysis of the equivalence problem and the associated differential invariants. In particular, conditions for the linearizability of the differential equation are based on the vanishing of certain invariants. There is no mention of the more powerful (but more difficult) methods for solving equivalence problems developed by Cartan, cf. [P. J. Olver, Equivalence, invariants, and symmetry, Cambridge Univ. Press, Cambridge, 1995; MR1337276].
Chapter 5 develops the standard Lie infinitesimal algorithms for computing symmetries of differential equations. Only ordinary differential equations are discussed, leaving out a broad range of important applications of Lie methods to solving and analyzing partial differential equations [B. J. Cantwell, Introduction to symmetry analysis, Cambridge Univ. Press, Cambridge, 2002; MR1929892; P. E. Hydon, Symmetry methods for differential equations, Cambridge Univ. Press, Cambridge, 2000; MR1741548; P. J. Olver, Applications of Lie groups to differential equations, Second edition, Springer, New York, 1993; MR1240056; P. J. Olver, op. cit.; MR1337276]. A detailed classification of first-, second- and third-order equations according to their symmetry groups is included. Chapter 6 applies symmetries for transforming a differential equation to canonical form. Chapter 7 applies symmetry techniques to develop solution algorithms based on reducing the order of an equation with symmetries. Finally, Chapter 8 combines everything into a general flowchart for solving scalar ordinary differential equations with symmetries.
There are 6 appendices: solutions to selected problems, collections of useful formulas, basics on the algebra of monomials, a list of Loewy decompositions and then the Lie symmetries of the differential equations in Kamke's list, and, finally, a brief summary of Schwarz's computer algebra system ALLTYPES for handling the often intricate calculations.

Reviewed by Peter J. Olver