Symmetry analysis of differential equations with Mathematica:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer [u.a.]
2000
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 521 S. Ill., graph. Darst. CD-ROM (12 cm) |
| ISBN: | 0387985522 |
Staff View
MARC
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| 245 | 1 | 0 | |a Symmetry analysis of differential equations with Mathematica |c Gerd Baumann |
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| 650 | 4 | |a Datenverarbeitung | |
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Record in the Search Index
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| adam_text | Contents
Preface vii
Chapter 1 Introduction 1
Chapter 2 Elements of Symmetry Analysis 6
2.1 Groups and Lie Groups 6
2.1.1 Groups 6
2.1.2 Isomorphism 14
2.1.3 Lie Groups 14
2.2 Lie Algebras 21
2.2.1 Representation of a Lie Algebra 26
2.2.2 Properties of Lie Algebras 29
Chapter 3 Derivatives 37
3.1 Ordinary and Partial Derivatives 37
3.2 Tangent Vector 45
3.3 The Total Derivative 50
3.4 Prolongations 52
3.5 The Frechet Derivative 54
3.6 The Euler Derivative 59
3.6.1 The Problem of Variation 59
3.6.2 Euler s Equation 63
3.6.3 Euler Operator 65
3.6.4 Algorithm Used in the Calculus of Variations 65
3.6.5 Euler Operator for q Dependent Variables 69
x Contents
3.6.6 Euler Operator for q + p Dimensions 71
3.7 Prolongation of Vector Fields 74
Chapter 4 Symmetries of Ordinary Differential Equations 96
4.1 Introduction 96
4.2 Symmetry Transformations of Functions 98
4.2.1 Symmetries 98
4.2.2 Infinitesimal Transformations 103
4.2.3 Group Invariants 107
4.2.4 Tangent Vector 112
4.2.5 Prolongation of Transformations 777
4.3 Symmetry Transformations of Differential Equations 123
4.3.1 Definition of a Symmetry Group 123
4.3.2 Main Properties of Symmetry Groups 124
4.3.3 Calculation of the Infinitesimal Symmetries 725
4.3.4 Canonical Variables 139
4.4 Analysis of Ordinary Differential Equations 148
4.4.1 First Order Equations 148
4.4.2 Second Order Ordinary Differential Equations 774
4.4.3 Higher Order Ordinary Differential Equations 207
Chapter 5 Point Symmetries of Partial Differential Equations 216
5.1 Introduction 276
5.2 Lie s Theory Used in MathLie 277
5.3 Invariance Based on Frechet Derivatives 220
5.4 Application of the Theory 222
5.4.1 Calculation of Prolongations 223
5.4.2 Derivation of Determining Equations 229
5.4.3 Interactive Solution of Determining Equations 235
5.4.4 Data Basis of Symmetries 243
5.5 Similarity Reduction of Partial Differential Equations 257
5.6 Working Examples 282
5.6.1 The Diffusion Equation 282
5.6.2 The Earthworm s New Year Problem 282
5.6.3 Single Flux Line in Superconductors 289
5.6.4 The Korteweg de Vries Equation and its Generalizations 296
5.6.5 Stokes Solution of the Creeping Flow 304
5.6.6 Two Dimensional Boundary Layer Flows: Group
Classification 311
5.6.7 The Plane Jet 323
5.6.8 Drop Formation 330
5.6.9 The Rayleigh Particle 340
5.6.10 Molecular Beam Epitaxy 346
5.6.11 The First Atomic Explosion 355
Contents xi
Chapter 6 Non Classical Symmetries of Partial Differential Equations 365
6.1 Introduction 365
6.2 Mathematical Background of the Non classical Method 366
6.3 Applications of the Non classical Method 370
6.3.1 The Heat Equation 370
6.3.2 The Boussinesq Equation 377
6.3.3 The Fokker Planck Equation 383
Chapter 7 Potential Symmetries of Partial Differential Equations 392
7.1 Introduction 392
7.2 Basics of Potential Symmetries 393
7.3 Calculation of Potential Symmetries 394
1A Applications of Potential Symmetries 398
7.4.1 A Non linear Reaction Diffusion Equation 398
7.4.2 Cylindrical Korteweg de Vries Equation 399
7.4.3 The Burgers Equation 402
Chapter 8 Approximate Symmetries of Partial Differential Equations 404
8.1 Introduction 404
8.2 Approximations 405
8.3 One Parameter Approximation Group 405
8.4 Approximate Group Generator 407
8.5 The Determining Equations and an Algorithm of Calculation 408
8.6 Examples 410
8.6.1 Isentropic Liquid 410
8.6.2 Perturbed Korteweg de Vies Equation 419
Chapter 9 Generalized Symmetries 424
9.1 Introduction 424
9.2 Elements of Generalized Symmetries 425
9.3 Algorithm for Calculation of Generalized Symmetries 427
9.4 Examples 428
9 A.I Diffusion Equation 428
9.4.2 Potential Burgers Equation 430
9 A3 Generalized Korteweg de Vries Equations 431
9.4.4 Coupled System of Wave Equations 432
9.5 Second Order ODEs and the Euler Lagrange Equation 433
9.5.1 Generalized Symmetries and Second Order ODEs 434
9.5.2 Conservation Laws 436
9.6 Algorithm for Conservation Laws of Second Order ODEs 437
9.7 Examples for Second Order ODEs 438
9.7.1 The Henon Heiles Model 438
9.7.2 Two Dimensional Quartic Oscillators 446
9.7.3 Two Ions in a Trap 452
xii Contents
Chapter 10 Solution of Coupled Linear Partial Differential Equations 457
10.1 Introduction 457
10.2 General Canonical Form of PDEs 458
10.2.1 Application of the General Canonical Form Algorithm 462
10.3 Solution of Linear PDEs 471
10.3.1 Integration of Monomials 472
10.3.2 Integrating ODEs and Pseudo ODEs 473
10.3.3 Integrating Exact PDEs 473
10.3.4 Potential Representation 474
10.4 Simplification of Equations 475
10.4.1 Direct Separation 475
10.4.2 Indirect Separation 476
10.4.3 Reducing the Number of Dependent Variables 477
10.5 Example 479
10.5.1 Liouville Type Equation of Quantum Gravity Theory 480
Chapter 11 Appendix 483
A Marius Sophus Lie: A Mathematician s Life 483
B List of Key Symbols Used in Mathematica 487
C Installing MathLie 488
References 493
Index for MathLie and Mathematica Functions 503
Subject Index 505
|
| any_adam_object | 1 |
| author | Baumann, Gerd 1956- |
| author_GND | (DE-588)142960241 |
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| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
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| id | DE-604.BV012487389 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:28:27Z |
| institution | BVB |
| isbn | 0387985522 |
| language | English |
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| physical | XII, 521 S. Ill., graph. Darst. CD-ROM (12 cm) |
| publishDate | 2000 |
| publishDateSearch | 2000 |
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| publisher | Springer [u.a.] |
| record_format | marc |
| spelling | Baumann, Gerd 1956- Verfasser (DE-588)142960241 aut Symmetry analysis of differential equations with Mathematica Gerd Baumann New York [u.a.] Springer [u.a.] 2000 XII, 521 S. Ill., graph. Darst. CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Differentiaalvergelijkingen gtt EQUAÇÕES DIFERENCIAIS ORDINÁRIAS (ANÁLISE MATEMÁTICA) larpcal Lie-groepen gtt Mathematica (computerprogramma) gtt Numerieke methoden gtt Symmetrie gtt TEORIA QUALITATIVA larpcal Datenverarbeitung Differential equations Numerical solutions Data processing Mathematica (Computer program language) Symmetry (Physics) Data processing Symmetrie (DE-588)4058724-1 gnd rswk-swf Mathematica Programm (DE-588)4268208-3 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mathematica Programm (DE-588)4268208-3 s DE-604 Differentialgleichung (DE-588)4012249-9 s Symmetrie (DE-588)4058724-1 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008475996&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Baumann, Gerd 1956- Symmetry analysis of differential equations with Mathematica Differentiaalvergelijkingen gtt EQUAÇÕES DIFERENCIAIS ORDINÁRIAS (ANÁLISE MATEMÁTICA) larpcal Lie-groepen gtt Mathematica (computerprogramma) gtt Numerieke methoden gtt Symmetrie gtt TEORIA QUALITATIVA larpcal Datenverarbeitung Differential equations Numerical solutions Data processing Mathematica (Computer program language) Symmetry (Physics) Data processing Symmetrie (DE-588)4058724-1 gnd Mathematica Programm (DE-588)4268208-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4058724-1 (DE-588)4268208-3 (DE-588)4012249-9 |
| title | Symmetry analysis of differential equations with Mathematica |
| title_auth | Symmetry analysis of differential equations with Mathematica |
| title_exact_search | Symmetry analysis of differential equations with Mathematica |
| title_full | Symmetry analysis of differential equations with Mathematica Gerd Baumann |
| title_fullStr | Symmetry analysis of differential equations with Mathematica Gerd Baumann |
| title_full_unstemmed | Symmetry analysis of differential equations with Mathematica Gerd Baumann |
| title_short | Symmetry analysis of differential equations with Mathematica |
| title_sort | symmetry analysis of differential equations with mathematica |
| topic | Differentiaalvergelijkingen gtt EQUAÇÕES DIFERENCIAIS ORDINÁRIAS (ANÁLISE MATEMÁTICA) larpcal Lie-groepen gtt Mathematica (computerprogramma) gtt Numerieke methoden gtt Symmetrie gtt TEORIA QUALITATIVA larpcal Datenverarbeitung Differential equations Numerical solutions Data processing Mathematica (Computer program language) Symmetry (Physics) Data processing Symmetrie (DE-588)4058724-1 gnd Mathematica Programm (DE-588)4268208-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Differentiaalvergelijkingen EQUAÇÕES DIFERENCIAIS ORDINÁRIAS (ANÁLISE MATEMÁTICA) Lie-groepen Mathematica (computerprogramma) Numerieke methoden Symmetrie TEORIA QUALITATIVA Datenverarbeitung Differential equations Numerical solutions Data processing Mathematica (Computer program language) Symmetry (Physics) Data processing Mathematica Programm Differentialgleichung |
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