Algorithmic Lie Theory for Solving Ordinary Differential Equations: 1st Edition (Hardback) book cover

Algorithmic Lie Theory for Solving Ordinary Differential Equations

1st Edition

By Fritz Schwarz

Chapman and Hall/CRC

448 pages | 7 B/W Illus.

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Hardback: 9781584888895
pub: 2007-10-02
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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.

Reviews

"The book will serve as a valuable reference for researchers interested in ordinary differential equations, symmetry methods, and computer algebra."

Mathematical Reviews

"… The aim of this book is to discuss algorithms for solving ordinary differential equations using the Lie approach. … There are a lot of exercises, some of them with solutions. Software for computer calculations is available on a web page."

EMS Newsletter, June 2009

"… readers interested in an in-depth treatment of the computational aspects of the symmetry approach to low-order ordinary differential equations will find many informations here not available elsewhere; this includes in particular the systematic use of Janet bases for a complete classification of all possibly symmetry types."

—Werner M. Seiler, Zentralblatt Math, 2008

Table of Contents

INTRODUCTION

LINEAR DIFFERENTIAL EQUATIONS

Linear Ordinary Differential Equations

Janet's Algorithm

Properties of Janet Bases

Solving Partial Differential Equations

LIE TRANSFORMATION GROUPS

Lie Groups and Transformation Groups

Algebraic Properties of Vector Fields

Group Actions in the Plane

Classification of Lie Algebras and Lie Groups

Lie Systems

EQUIVALENCE AND INVARIANTS OF DIFFERENTIAL EQUATIONS

Linear Equations

Nonlinear First-Order Equations

Nonlinear Equations of Second and Higher Order

SYMMETRIES OF DIFFERENTIAL EQUATIONS

Transformation of Differential Equations

Symmetries of First-Order Equations

Symmetries of Second-Order Equations

Symmetries of Nonlinear Third-Order Equations

Symmetries of Linearizable Equations

TRANSFORMATION TO CANONICAL FORM

First-Order Equations

Second-Order Equations

Nonlinear Third-Order Equations

Linearizable Third-Order Equations

SOLUTION ALGORITHMS

First-Order Equations

Second-Order Equations

Nonlinear Equations of Third Order

Linearizable Third-Order Equations

CONCLUDING REMARKS

APPENDIX A: Solutions to Selected Problems

APPENDIX B: Collection of Useful Formulas

APPENDIX C: Algebra of Monomials

APPENDIX D: Loewy Decompositions of Kamke's Collection

APPENDIX E: Symmetries of Kamke's Collection

APPENDIX F: ALLTYPES Userinterface

REFERENCES

INDEX

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT007000
MATHEMATICS / Differential Equations
MAT021000
MATHEMATICS / Number Systems