Schwarz Algorithmic Lie Theory for Solving Ordinary Differential Equations

1. Auflage 2007
ISBN: 978-1-58488-890-1
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 448 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

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.

Zielgruppe

Mathematics researchers and graduate and senior undergraduate students.

Autoren/Hrsg.

Schwarz, Fritz

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

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