Symmetry Analysis of Differential Equations with Mathematica®

by Gerd Baumann

• ISBN13: 9780387985527
• ISBN10: 0387985522
• Format: Nonspecific Binding
• Copyright: 2013-11-21
• Publisher: Springer Nature

Summary

This is the first book which explicitly uses Mathematica (computer algebra system) to allow researchers and students to more easily compute and solve almost any kind of differential equation using Lie's theory. Heretofore time-consuming and cumbersome calculations if done by hand, are much more easily and quickly performed via the Mathematica computer algebra software. The material in this book, and on the accompanying CD-ROM, should be of interest to a broad group of scientists, mathematicians and engineers involved in dealing with symmetry analysis of differential equations. Each section of the book starts with a theoretical discussion of the material, then shows the application in connection with Mathematica. This book contains a large number of working examples relating to these applications of Lie's theory. The cross-platform CD-ROM contains Mathematica (version 3.0) notebooks which provide users with the capability of directly interacting with the code presented within the book. In addition, the author's proprietary "MathLie" software is included, so users can readily learn to use this powerful tool to perform algebraic computations.

Author Biography

Gerd Baumann is a senior assistant professor in the Mathematical Physics Division at the University of Ulm, Germany.

Table of Contents

Preface	vii
Introduction	1	(5)
Elements of Symmetry Analysis	6	(31)
Groups and Lie Groups	6	(15)
Groups	6	(8)
Isomorphism	14	(1)
Lie Groups	14	(7)
Lie Algebras	21	(16)
Representation of a Lie Algebra	26	(3)
Properties of Lie Algebras	29	(8)
Derivatives	37	(59)
Ordinary and Partial Derivatives	37	(8)
Tangent Vector	45	(5)
The Total Derivative	50	(2)
Prolongations	52	(2)
The Frechet Derivative	54	(5)
The Euler Derivative	59	(15)
The Problem of Variation	59	(4)
Euler's Equation	63	(2)
Euler Operator	65	(1)
Algorithm Used in the Calculus of Variations	65	(4)
Euler Operator for q Dependent Variables	69	(2)
Euler Operator for q + p Dimensions	71	(3)
Prolongation of Vector Fields	74	(22)
Symmetries of Ordinary Differential Equations	96	(120)
Introduction	96	(2)
Symmetry Transformations of Functions	98	(25)
Symmetries	98	(5)
Infinitesimal Transformations	103	(4)
Group Invariants	107	(5)
Tangent Vector	112	(5)
Prolongation of Transformations	117	(6)
Symmetry Transformations of Differential Equations	123	(25)
Definition of a Symmetry Group	123	(1)
Main Properties of Symmetry Groups	124	(1)
Calculation of the Infinitesimal Symmetries	125	(14)
Canonical Variables	139	(9)
Analysis of Ordinary Differential Equations	148	(68)
First-Order Equations	148	(26)
Second-Order Ordinary Differential Equations	174	(27)
Higher-Order Ordinary Differential Equations	201	(15)
Point Symmetries of Partial Differential Equations	216	(149)
Introduction	216	(1)
Lie's Theory Used in MathLie	217	(3)
Invariance Based on Frechet Derivatives	220	(2)
Application of the Theory	222	(35)
Calculation of Prolongations	223	(6)
Derivation of Determining Equations	229	(6)
Interactive Solution of Determining Equations	235	(8)
Data Basis of Symmetries	243	(14)
Similarity Reduction of Partial Differential Equations	257	(25)
Working Examples	282	(83)
The Diffusion Equation	282	(1)
The Earthworm's New Year Problem	282	(7)
Single Flux Line in Superconductors	289	(7)
The Korteweg-de Vries Equation and its Generalizations	296	(8)
Stokes' Solution of the Creeping Flow	304	(7)
Two-Dimensional Boundary-Layer Flows: Group Classification	311	(12)
The Plane Jet	323	(7)
Drop Formation	330	(10)
The Rayleigh Particle	340	(6)
Molecular Beam Epitaxy	346	(9)
The First Atomic Explosion	355	(10)
Non-Classical Symmetries of Partial Differential Equations	365	(27)
Introduction	365	(1)
Mathematical Background of the Non-classical Method	366	(4)
Applications of the Non-classical Method	370	(22)
The Heat Equation	370	(7)
The Boussinesq Equation	377	(6)
The Fokker-Planck Equation	383	(9)
Potential Symmetries of Partial Differential Equations	392	(12)
Introduction	392	(1)
Basics of Potential Symmetries	393	(1)
Calculation of Potential Symmetries	394	(4)
Applications of Potential Symmetries	398	(6)
A Non-linear Reaction Diffusion Equation	398	(1)
Cylindrical Korteweg-de Vries Equation	399	(3)
The Burgers Equation	402	(2)
Approximate Symmetries of Partial Differential Equations	404	(20)
Introduction	404	(1)
Approximations	405	(1)
One-Parameter Approximation Group	405	(2)
Approximate Group Generator	407	(1)
The Determining Equations and an Algorithm of Calculation	408	(2)
Examples	410	(14)
Isentropic Liquid	410	(9)
Perturbed Korteweg-de Vies Equation	419	(5)
Generalized Symmetries	424	(33)
Introduction	424	(1)
Elements of Generalized Symmetries	425	(2)
Algorithm for Calculation of Generalized Symmetries	427	(1)
Examples	428	(5)
Diffusion Equation	428	(2)
Potential Burgers Equation	430	(1)
Generalized Korteweg-de Vries Equations	431	(1)
Coupled System of Wave Equations	432	(1)
Second-Order ODEs and the Euler-Lagrange Equation	433	(4)
Generalized Symmetries and Second-Order ODEs	434	(2)
Conservation Laws	436	(1)
Algorithm for Conservation Laws of Second-Order ODEs	437	(1)
Examples for Second-Order ODEs	438	(19)
The Henon-Heiles Model	438	(8)
Two-Dimensional Quartic Oscillators	446	(6)
Two Ions in a Trap	452	(5)
Solution of Coupled Linear Partial Differential Equations	457	(26)
Introduction	457	(1)
General Canonical Form of PDEs	458	(13)
Application of the General Canonical Form Algorithm	462	(9)
Solution of Linear PDEs	471	(4)
Integration of Monomials	472	(1)
Integrating ODEs and Pseudo-ODEs	473	(1)
Integrating Exact PDEs	473	(1)
Potential Representation	474	(1)
Simplification of Equations	475	(4)
Direct Separation	475	(1)
Indirect Separation	476	(1)
Reducing the Number of Dependent Variables	477	(2)
Example	479	(4)
Liouville-Type Equation of Quantum Gravity Theory	480	(3)
Appendix	483	(10)
Marius Sophus Lie: A Mathematician's Life	483	(4)
List of Key Symbols Used in Mathematica	487	(1)
Installing MathLie	488	(5)
References	493	(10)
Index for MathLie and Mathematica Functions	503	(2)
Subject Index	505