Symmetry Methods for Differential Equations: A Beginner’s Guide

Cover......Page 1
About......Page 2
Cambridge Texts in Applied Mathematics......Page 3
Symmetry Methods for Differential Equations: A Beginner's Guide......Page 4
0521497868......Page 5
Contents......Page 8
Preface......Page 10
Acknowledgements......Page 12
1.1 Symmetries of Planar Objects......Page 14
1.2 Symmetries of the Simplest ODE......Page 18
1.3 The Symmetry Condition for First-Order ODEs......Page 21
1.4 Lie Symmetries Solve First-Order ODEs......Page 24
2.1 The Action of Lie Symmetries on the Plane......Page 28
2.2 Canonical Coordinates......Page 35
2.3 How to Solve ODEs with Lie Symmetries......Page 39
2.4 The Linearized Symmetry Condition......Page 43
2.5 Symmetries and Standard Methods......Page 47
2.6 The Infinitesimal Generator......Page 51
3.1 The Symmetry Condition......Page 56
3.2 The Determining Equations for Lie Point Symmetries......Page 59
3.3 Linear ODEs......Page 65
3.4 Justification of the Symmetry Condition......Page 67
4.1 Reduction of Order by Using Canonical Coordinates......Page 71
4.2 Variational Symmetries......Page 76
4.3 Invariant Solutions......Page 81
5.1 Differential Invariants and Reduction of Order......Page 87
5.2 The Lie Algebra of Point Symmetry Generators......Page 92
5.3 Stepwise Integration of ODEs......Page 102
6.1 The Basic Method: Exploiting Solvability......Page 106
6.2 New Symmetries Obtained During Reduction......Page 112
6.3 Integration of Third-Order ODEs with sl(2)......Page 114
7.1 First Integrals Derived from Symmetries......Page 121
7.2 Contact Symmetries and Dynamical Symmetries......Page 129
7.3 Integrating Factors......Page 135
7.4 Systems of ODEs......Page 141
8.1 Scalar PDEs with Two Dependent Variables......Page 149
8.2 The Linearized Symmetry Condition for General PDEs......Page 159
8.3 Finding Symmetries by Computer Algebra......Page 162
9.1 Group-Invariant Solutions......Page 168
9.2 New Solutions from Known Ones......Page 175
9.3 Nonclassical Symmetries......Page 179
10.1 Equivalence of Invariant Solutions......Page 186
10.2 How to Classify Symmetry Generators......Page 189
10.3 Optimal Systems of Invariant Solutions......Page 195
11.1 Some Uses of Discrete Symmetries......Page 200
11.2 How to Obtain Discrete Symmetries from Lie Symmetries......Page 201
11.3 Classification of Discrete Symmetries......Page 204
11.4 Examples......Page 208
Hints and Partial Solutions to Some Exercises......Page 214
Bibliography......Page 222
Index......Page 224