Gerd Baumann
Symmetry Analysis of Differential Equations
Chapter 1
Chapter 2
2.1 Groups and Lie Groups
2.1.1 Groups
2.1.2 Isomorphism
2.1.3 Lie Groups
2.2 Lie Algebras
2.2.1 Representation of a Lie Algebra
2.2.2 Properties of Lie Algebras
Chapter 3
3.1. Ordinary and Partial Derivatives
3.2 Tangent Vector
3.3 The Total Derivative
3.4 Prolongations
3.5 The Fréchet Derivative
3.6 The Euler Derivative
3.6.1 The Problem of Variations
3.6.2 Euler's Equation
3.6.3 Euler Operator
3.6.4 Algorithm used in the Calculus of Variations
3.6.5 Euler Operator for q Dependent Variables
3.6.6 Euler Operator for q+p-Dimensions
3.7 Prolongation of Vector Fields
Chapter 4
4.1 Introduction
4.2 Symmetry Transformations of Functions
4.2.1 Symmetries
4.2.2 Infinitesimal Transformations
4.2.3 Group Invariants
4.2.4 Tangent Vector
4.2.5 Prolongation of Transformations
4.3 Symmetry Transformations of Differential Equations
4.3.1 Definition of a Symmetry Group
4.3.2 Main Properties of Symmetry Groups
4.3.3 Calculation of the Infinitesimal Symmetries
4.3.4 Canonical Variables
4.4 Analysis of Ordinary Differential Equations
4.4.1 First-Order Equations
4.4.1.1 The Skeleton of an Ordinary Differential Equation
4.4.1.2 Integrating Factor
4.4.1.3 Infinitesimals of First-Order Ordinary Differential Equations
4.4.2 Second-Order Ordinary Differential Equations
4.4.2.1 Integration by Group Classification
4.4.2.2 The Integrating Factor Method
4.4.2.2 Method of Canonical Variables
4.4.3 Higher-Order Ordinary Differential Equations
4.4.3.1 Integrating Factor Method
Chapter 5
5.1 Introduction
5.2 Lie's Theory Used in MathLie
5.3 Invariance Based on Fréchet Derivatives
5.4 Application of the Theory
5.4.1 Calculation of Prolongations
5.4.2 Derivation of Determining Equations
5.4.3 Interactive Solution of Determining Equations
5.4.4 Data Basis of Symmetries
5.5 Similarity Reduction of Partial Differential Equations
5.6 Working Examples
5.6.1 The Diffusion Equation
5.6.2 The Earthworm's New Year Problem
5.6.3 Single Flux Line in Superconductors
5.6.4 The Korteweg-de Vries equation and its Generalizations
5.6.5 Stokes' Solution of the Creeping Flow
5.6.6 Two-Dimensional Boundary-Layer Flows: Group Classification
5.6.6.1 The Blasius Solution
5.6.6.2 Falkner Skan Solution
5.6.6.3 Exponential Mainstream Velocity
5.6.6.4 Group Classification
5.6.7 The Plane Jet
5.6.8 Drop Formation
5.6.9 The Rayleigh Particle
5.6.10 Molecular Beam Epitaxy
5.6.10.1 Surface Diffusion with Nonlinearity
5.6.10.2 Desorption with Nonlinearity
5.6.11 The First Atomic Explosion
Chapter 6
6.1 Introduction
6.2 Mathematical Background of the Non-Classical Method
6.3 Applications of the Non-Classical Method
6.3.1 The Heat Equation
6.3.2 The Boussinesq Equation
6.3.3 The Fokker-Planck Equation
Chapter 7
7.1 Introduction
7.2 Basics of Potential Symmetries
7.3 Calculation of Potential Symmetries
7.4 Applications of Potential Symmetries
7.4.1 A Nonlinear Reaction Diffusion Equation
7.4.2 Cylindrical Korteweg-de Vries Equation
7.4.3 The Burgers Equation
Chapter 8
8.1 Introduction
8.2 Approximations
8.3 One-Parameter Approximation Group
8.4 Approximate Group Generator
8.5 The Determining Equations and an Algorithm of Calculation
8.6 Examples
8.6.1 Isentropic Liquid
8.6.2 Perturbed Korteweg de Vies Equation
Chapter 9
9.1
9.2 Elements of Generalized Symmetries
9.3 Algorithm for Calculation of Generalized Symmetries
9.4 Examples
9.4.1 Diffusion Equation
9.4.2 Potential Burgers Equation
9.4.3 Generalized Korteweg de Vries Equations
9.4.4 Coupled System of Wave Equations
9.5 Second-Order ODEs and the Euler-Lagrange Equation
9.5.1 Generalized Symmetries and Second-Order ODEs
9.5.2 Conservation laws
9.6 Algorithm for Conservation Laws of Second-Order ODEs
9.7 Examples for Second-Order ODEs
9.7.1 The Hénon-Heiles Model
9.7.2 Two-Dimensional Quartic Oscillators
9.7.3 Two Ions in a Trap
Chapter 10
10.1 Introduction
10.2 General Canonical Form of PDEs
10.2.1 Application of the General Canonical Form Algorithm
10.3 Solution of Linear PDEs
10.3.1 Integration of Monomials
10.3.2 Integrating ODEs and Pseudo-ODEs
10.3.3 Integrating Exact PDEs
10.3.4 Potential Representation
10.4 Simplification of Equations
10.4.1 Direct Separation
10.4.2 Indirect Separation
10.4.3 Reducing the Number of Dependent Variables
10.5 Example
10.5.1 Liouville Type Equation of Quantum Gravity Theory
Chapter 11 Appendix
A Marius Sophus Lie: A Mathematician's Life
B List of Key Symbols Used in Mathematica
C Installing MathLie
Gerd Baumann
Created by Mathematica (September 15, 2003)