Symmetry Analysis of Differential Equations

193 Pages·2015·1.32 MB·English

by  Daniel J. Arrigo

Symmetry Analysis of
Differential Equations

Symmetry Analysis of
Differential Equations
An Introduction
Daniel J. Arrigo
DepartmentofMathematics
UniversityofCentralArkansas
Conway,AR
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LibraryofCongressCataloging-in-PublicationData:
Arrigo,DanielJ.(DanielJoseph),1960-
Symmetryanalysisofdifferentialequations:anintroduction/DanielJ.Arrigo,
DepartmentofMathematics,UniversityofCentral
Arkansas,Conway,AR.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-72140-7(cloth)
1. Liegroups–Textbooks.2. Liegroups–Studyandteaching(Higher)3. Lie
groups–Studyandteaching(Graduate)4. Differential
equations,Partial–Textbooks. I.Title.
QA387.A772014
515′.353–dc23
2014007305
PrintedintheUnitedStatesofAmerica.
10987654
TothelateBillAmes,
myteacher,mymentor,myfriend.

Contents
Preface xi
Acknowledgments xiii
1 An Introduction 1
1.1 WhatIsaSymmetry? 1
1.2 LieGroups, 4
1.3 InvarianceofDifferentialEquations, 6
1.4 SomeOrdinaryDifferentialEquations, 8
Exercises, 12
2 Ordinary Differential Equations 15
2.1 InfinitesimalTransformations, 19
2.2 Lie’sInvarianceCondition, 23
Exercises, 27
2.3 StandardIntegrationTechniques, 28
2.3.1 LinearEquations, 28
2.3.2 BernoulliEquation, 30
2.3.3 HomogeneousEquations, 31
2.3.4 ExactEquations, 32
2.3.5 RiccatiEquations, 35
Exercises, 37
2.4 InfinitesimalOperatorandHigherOrderEquations, 38
2.4.1 TheInfinitesimalOperator, 38
2.4.2 TheExtendedOperator, 39
2.4.3 ExtensiontoHigherOrders, 40
2.4.4 First-OrderInfinitesimals(revisited), 40
2.4.5 Second-OrderInfinitesimals, 41
2.4.6 TheInvarianceofSecond-OrderEquations, 42
2.4.7 Equationsofarbitraryorder, 43
2.5 Second-OrderEquations, 43
Exercises, 55
vii
viii Contents
2.6 HigherOrderEquations, 56
Exercises, 61
2.7 ODESystems, 61
2.7.1 FirstOrderSystems, 61
2.7.2 HigherOrderSystems, 67
Exercises, 71
3 Partial Differential Equations 73
3.1 First-OrderEquations, 73
3.1.1 WhatDoWeDowiththeSymmetriesof
PDEs? 77
3.1.2 DirectReductions, 80
3.1.3 TheInvariantSurfaceCondition, 83
Exercises, 84
3.2 Second-OrderPDEs, 84
3.2.1 HeatEquation, 84
3.2.2 Laplace’sEquation, 91
3.2.3 Burgers’EquationandaRelative, 94
3.2.4 HeatEquationwithaSource, 100
Exercises, 107
3.3 HigherOrderPDEs, 109
Exercises, 115
3.4 SystemsofPDEs, 115
3.4.1 First-OrderSystems, 115
3.4.2 Second-OrderSystems, 120
Exercises, 124
3.5 HigherDimensionalPDEs, 126
Exercises, 132
4 Nonclassical Symmetries and Compatibility 133
4.1 NonclassicalSymmetries, 133
4.1.1 InvarianceoftheInvariantSurface
Condition, 135
4.1.2 TheNonclassicalMethod, 137
4.2 NonclassicalSymmetryAnalysisandCompatibility, 146
4.3 BeyondSymmetriesAnalysis–General
Compatibility, 147
4.3.1 CompatibilitywithFirst-OrderPDEs–Charpit’s
Method, 149