Symmetries and Differential Equations

Contents......Page 6
Preface......Page 10
Introduction......Page 12
1.2.1 Assumptions Behind Dimensional Analysis......Page 16
1.2.2 Conclusions from Dimensional Analysis......Page 18
1.2.3 Proof of the Buckingham Pi-Theorem......Page 19
1.2.4 Examples......Page 22
1.3 Application of Dimensional Analysis to PDEs......Page 27
1.3.1 Examples......Page 28
1.4 Generalization of Dimensional Analysis: Invariance of PDEs Under Scalings of Variables......Page 36
1.5 Discussion......Page 42
2.1 Introduction......Page 44
2.2.2 Examples of Groups......Page 45
2.2.4 One-Parameter Lie Group of Transformations......Page 47
2.2.5 Examples of One-Parameter Lie Groups of Transformations......Page 48
2.3 Infinitesimal Transformations......Page 49
2.3.1 First Fundamental Theorem of Lie......Page 50
2.3.2 Examples Illustrating Lie’s First Fundamental Theorem......Page 52
2.3.3 Infinitesimal Generators......Page 53
2.3.4 Invariant Functions......Page 57
2.3.5 Canonical Coordinates......Page 58
2.3.6 Examples of Sets of Canonical Coordinates......Page 60
2.4 Point Transformations and Extended Transformations (Prolongations)......Page 63
2.4.1 Extended Group of Point Transformation: One Dependent and One Independent Variable......Page 64
2.4.2 Extended Infinitesimal Transformations: One Dependent and One Independent Variable......Page 71
2.4.3 Extended Transformations: One Dependent and n Independent Variables......Page 73
2.4.4 Extended Infinitesimal Transformations: One Dependent and n Independent Variables......Page 76
2.4.5 Extended Transformations and Extended Infinitesimal Transformations: m Dependent and n Independent Variables......Page 79
2.5 Multiparameter Lie Groups of Transformations and Lie Algebras......Page 83
2.5.1 r-Parameter Lie Groups of Transformations......Page 84
2.5.2 Lie Algebras......Page 88
2.5.3 Examples of Lie Algebras......Page 90
2.5.4 Solvable Lie Algebras......Page 93
2.6.1 Invariant Surfaces, Invariant Curves, Invariant Points......Page 96
2.6.2 Mappings of Curves......Page 100
2.6.3 Examples of Mappings of Curves......Page 101
2.6.4 Mappings of Surfaces......Page 102
2.7.1 Point Transformations......Page 103
2.7.2 Contact and Higher-Order Transformations......Page 105
2.7.3 Examples of Local Transformations......Page 106
2.8 Discussion......Page 108
3.1 Introduction......Page 112
3.1.1 Elementary Examples......Page 113
3.2 First-Order ODEs......Page 117
3.2.1 Canonical Coordinates......Page 118
3.2.2 Integrating Factors......Page 120
3.2.3 Mappings of Solution Curves......Page 121
3.2.4 Determining Equation for Symmetries of a First-Order ODE......Page 123
3.2.5 Determination of First-Order ODEs Invariant Under a Given Group......Page 125
3.3 Invariance of Second- and Higher-Order ODEs Under Point Symmetries......Page 132
3.3.1 Reduction of Order Through Canonical Coordinates......Page 133
3.3.2 Reduction of Order Through Differential Invariants......Page 135
3.3.3 Examples of Reduction of Order......Page 137
3.3.4 Determining Equations for Point Symmetries of an nth-Order ODE......Page 143
3.3.5 Determination of nth-Order ODEs Invariant Under a Given Group......Page 148
3.4.1 Invariance of a Second-Order ODE Under a Two-Parameter Lie Group......Page 152
3.4.2 Invariance of an nth-Order ODE Under a Two-Parameter Lie Group......Page 156
3.4.3 Invariance of an nth-Order ODE Under an r-Parameter Lie Group with a Solvable Lie Algebra......Page 161
3.4.4 Invariance of an Overdetermined System of ODEs Under an r-Parameter Lie Group with a Solvable Lie Algebra......Page 170
3.5 Contact Symmetries and Higher-Order Symmetries......Page 176
3.5.1 Determining Equations for Contact Symmetries and Higher-Order Symmetries......Page 178
3.5.2 Examples of Contact Symmetries and Higher-Order Symmetries......Page 180
3.5.3 Reduction of Order Using Point Symmetries in Characteristic Form......Page 186
3.5.4 Reduction of Order Using Contact Symmetries and Higher-Order Symmetries......Page 190
3.6 First Integrals and Reduction of Order Through Integrating Factors......Page 196
3.6.1 First-Order ODEs......Page 198
3.6.2 Determining Equations for Integrating Factors of Second-Order ODEs......Page 202
3.6.3 First Integrals of Second-Order ODEs......Page 207
3.6.4 Determining Equations for Integrating Factors of Third- and Higher-Order ODEs......Page 219
3.6.5 Examples of First Integrals of Third- and Higher-Order ODEs......Page 232
3.7 Fundamental Connections Between Integrating Factors and Symmetries......Page 243
3.7.1 Adjoint-Symmetries......Page 244
3.7.2 Adjoint Invariance Conditions and Integrating Factors......Page 247
3.7.3 Examples of Finding Adjoint-Symmetries and Integrating Factors......Page 249
3.7.4 Noether’s Theorem, Variational Symmetries, and Integrating Factors......Page 256
3.7.5 Comparison of Calculations of Symmetries, Adjoint-Symmetries, and Integrating Factors......Page 262
3.8 Direct Construction of First Integrals Through Symmetries and Adjoint-Symmetries......Page 266
3.8.1 First Integrals from Symmetry and Adjoint-Symmetry Pairs......Page 267
3.8.2 First Integrals from a Wronskian Formula Using Symmetries or Adjoint-Symmetries......Page 273
3.8.3 First Integrals for Self-Adjoint ODEs......Page 281
3.9 Applications to Boundary Value Problems......Page 286
3.10 Invariant Solutions......Page 290
3.10.1 Invariant Solutions for First-Order ODEs: Separatrices and Envelopes......Page 295
3.11 Discussion......Page 301
4.1.1 Invariance of a PDE......Page 308
4.1.2 Elementary Examples......Page 310
4.2.1 Invariant Solutions......Page 314
4.2.2 Determining Equations for Symmetries of a kth-Order PDE......Page 316
4.2.3 Examples......Page 321
4.3 Invariance for a System of PDEs......Page 341
4.3.1 Invariant Solutions......Page 342
4.3.2 Determining Equations for Symmetries of a System of PDEs......Page 344
4.3.3 Examples......Page 346
4.4 Applications to Boundary Value Problems......Page 362
4.4.1 Formulation of Invariance of a Boundary Value Problem for a Scalar PDE......Page 364
4.4.2 Incomplete Invariance for a Linear Scalar PDE......Page 380
4.4.3 Incomplete Invariance for a Linear System of PDEs......Page 390
4.5 Discussion......Page 398
References......Page 402
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