Symmetry analysis of differential equations: an introduction
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley
2015
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 176 S. graph. Darst. |
| ISBN: | 9781118721407 1118721403 |
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| adam_text | A SELF-CONTAINED INTRODUCTION TO THE METHODS AND
TECHNIQUES OF SYMMETRY ANALYSIS USED TO SOLVE ODEs AND PDEs
s
Symmetry Analysis of Differential Equations: An Introduction presents an accessible
approach to the uses of symmetry methods in solving both ordinary differential equations
(ODEs) and partial differential equations (PDEs). Providing comprehensive coverage,
the book fills a gap in the literature by discussing elementary symmetry concepts and
invariance, including methods for reducing the complexity of ODEs and PDEs in an effort
to solve the associated problems.
Thoroughly class-tested, the author presents classical methods in a systematic, logical,
and well-balanced manner. As the book progresses, the chapters graduate from
elementary symmetries and the invariance of algebraic equations, to ODEs and PDEs,
followed by coverage of the nonclassical method and compatibility. Symmetry Analysis
of Differential Equations: An Introduction also features:
* Detailed, step-by-step examples to guide readers through the methods of symmetry
analysis
• End-of-chapter exercises, varying from elementary to advanced, with select solutions
to aid in the calculation of the presented algorithmic methods
Symmetry Analysis of Differential Equations: An Introduction is an ideal textbook for
upper-undergraduate and graduate-level courses in symmetry methods and applied
mathematics. The book is also a useful reference for professionals in science, physics,
and engineering, as well as anyone wishing to learn about the use of symmetry methods
in solving differential equations.
DANIEL J. ARRIGO, PhD, is Professor in the Department of Mathematics atthe University
of Central Arkansas. The author of over 30 journal articles, his research interests include
the construction of exact solutions of PDEs; symmetry analysis of nonlinear PDEs; and
solutions to physically important equations, such as nonlinear heat equations and
governing equations modeling of granular materials and nonlinear elasticity. In 2008,
Dr. Arrigo received the Oklahoma-Arkansas Section of the Mathematical Association of
America s Award for Distinguished Teaching of College or University Mathematics.
Contents
Preface xi
Acknowledgments xlll
1 An Introduction 1
1.1 What Is a Symmetry? 1
1.2 Lie Groups, 4
1.3 Invariance of Differentia I liquations, 6
1.4 Some Ordinary Differential Equations. X
Exercises, 12
2 Ordinary Differential Equations 15
2.1 Infinitesimal Transformations. 19
2.2 Lie’s Invariance Condition. 23
Exercises, 27
2.3 Standard Integration Techniques, 28
2.3.1 Linear Equations, 28
2.3.2 Bernoulli Equation. 30
2.3.3 Homogeneous liquations. 31
2.3.4 Exact Equations. 32
2.3.5 Riccati Equations. 35
Exercises, 37
2.4 Infinitesimal Operator and Higher Order liquations. 38
2.4.1 The Infinitesimal Operator. 38
2.4.2 The Extended Operator. 39
2.4.3 Extension to Higher Orders. 40
2.4.4 First-Order Infinitesimals (revisited). 40
2.4.5 Second-Order Infinitesimals. 41
2.4.6 The Invariance of Second-Order Equations. 42
2.4.7 Equations of arbitrary order. 43
2.5 Second-Order Equations. 43
Exercises, 55
VII
Vlii Contents
2.6 Higher Order Equations, 56
Exercises, 61
2.7 ODE Systems, 61
2.7.1 First Order Systems, 61
2.7.2 Higher Order Systems, 67
Exercises, 71
3 Partial Differential Equations 73
3.1 First-Order Equations, 73
3.1.1 What Do We Do with the Symmetries of
PDEs? 77
3.1.2 Direct Reductions, 80
3.1.3 The Invariant Surface Condition, 83
Exercises, 84
3.2 Second-Order PDEs, 84
3.2.1 Heat Equation, 84
3.2.2 Laplace’s Equation, 91
3.2.3 Burgers’ Equation and a Relative, 94
3.2.4 Heat Equation with a Source, 100
Exercises, 107
3.3 Higher Order PDEs, 109
Exercises, 115
3.4 Systems of PDEs, 115
3.4.1 First-Order Systems, 115
3.4.2 Second-Order Systems, 120
Exercises, 124
3.5 Higher Dimensional PDEs, 126
Exercises, 132
4 Nonclassical Symmetries and Compatibility 133
4.1 Nonclassical Symmetries, 133
4.1.1 Invariance of the Invariant Surface
Condition, 135
4.1.2 The Nonclassical Method, 137
4.2 Nonclassical Symmetry Analysis and Compatibility, 146
4.3 Beyond Symmetries Analysis-General
Compatibility, 147
4.3.1 Compatibility with First-Order PDEs-Charpit’s
Method, 149
Contents ix
4.3.2 Compatibility of Systems, 157
4.3.3 Compatibility of the Nonlinear Heat
Equation, 159
Exercises, 160
4.4 Concluding Remarks, 161
Solutions 163
References 171
Index 175
|
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| language | English |
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| spelling | Arrigo, Daniel J. 1960- Verfasser (DE-588)1067544461 aut Symmetry analysis of differential equations an introduction Daniel J. Arrigo Hoboken, NJ Wiley 2015 XIII, 176 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Lie groups / Textbooks Lie groups / Study and teaching (Higher) Lie groups / Study and teaching (Graduate) Differential equations, Partial / Textbooks Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Lie-Gruppe (DE-588)4035695-4 s Differentialgleichung (DE-588)4012249-9 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027396706&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027396706&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arrigo, Daniel J. 1960- Symmetry analysis of differential equations an introduction Lie groups / Textbooks Lie groups / Study and teaching (Higher) Lie groups / Study and teaching (Graduate) Differential equations, Partial / Textbooks Lie-Gruppe (DE-588)4035695-4 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4012249-9 (DE-588)4151278-9 |
| title | Symmetry analysis of differential equations an introduction |
| title_auth | Symmetry analysis of differential equations an introduction |
| title_exact_search | Symmetry analysis of differential equations an introduction |
| title_full | Symmetry analysis of differential equations an introduction Daniel J. Arrigo |
| title_fullStr | Symmetry analysis of differential equations an introduction Daniel J. Arrigo |
| title_full_unstemmed | Symmetry analysis of differential equations an introduction Daniel J. Arrigo |
| title_short | Symmetry analysis of differential equations |
| title_sort | symmetry analysis of differential equations an introduction |
| title_sub | an introduction |
| topic | Lie groups / Textbooks Lie groups / Study and teaching (Higher) Lie groups / Study and teaching (Graduate) Differential equations, Partial / Textbooks Lie-Gruppe (DE-588)4035695-4 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Lie groups / Textbooks Lie groups / Study and teaching (Higher) Lie groups / Study and teaching (Graduate) Differential equations, Partial / Textbooks Lie-Gruppe Differentialgleichung Einführung |
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