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Symmetry Analysis of Differential Equations with Mathematica® [Multiple-component retail product]

  • Formaat: Multiple-component retail product, 521 pages, kaal: 1080 g, 10 Tables, black and white; 56 Illustrations, black and white; XII, 521 p. 56 illus. With CD-ROM., 1 Item
  • Ilmumisaeg: 20-Apr-2000
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 0387985522
  • ISBN-13: 9780387985527
  • Multiple-component retail product
  • Hind: 107,20 €*
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  • Kogus:
  • Lisa ostukorvi
  • Tasuta tarne
  • Lisa soovinimekirja
  • Formaat: Multiple-component retail product, 521 pages, kaal: 1080 g, 10 Tables, black and white; 56 Illustrations, black and white; XII, 521 p. 56 illus. With CD-ROM., 1 Item
  • Ilmumisaeg: 20-Apr-2000
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 0387985522
  • ISBN-13: 9780387985527
The first book to explicitly use Mathematica so as to allow researchers and students to more easily compute and solve almost any kind of differential equation using Lie's theory. Previously time-consuming and cumbersome calculations are now much more easily and quickly performed using the Mathematica computer algebra software. The material in this book, and on the accompanying CD-ROM, will 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. The cross-platform CD-ROM contains Mathematica (version 3.0) notebooks which allow users to directly interact 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 in regard to performing algebraic computations.
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