Symmetry methods for differential equations : a beginner's guide / Peter E. Hydon

Includes bibliographical references (pages 209-210) and index

Preface Acknowledgements 1 Introduction to Symmetries 1 1.1 Symmetries of Planar Objects 1 1.2 Symmetries of the Simplest ODE 5 1.3 The Symmetry Condition for First-Order ODEs 8 1.4 Lie Symmetries Solve First-Order ODEs 11 2 Lie Symmetries of First-Order ODEs 15 2.1 The Action of Lie Symmetries on the Plane 15 2.2 Canonical Coordinates 22 2.3 How to Solve ODEs with Lie Symmetries 26 2.4 The Linearized Symmetry Condition 30 2.5 Symmetries and Standard Methods 34 2.6 The Infinitesimal Generator 38 3 How to Find Lie Point Symmetries of ODEs 43 3.1 The Symmetry Condition 43 3.2 The Determining Equations for Lie Point Symmetries 46 3.3 Linear ODEs 52 3.4 Justification of the Symmetry Condition 54 4 How to Use a One-Parameter Lie Group 58 4.1 Reduction of Order by Using Canonical Coordinates 58 4.2 Variational Symmetries 63 4.3 Invariant Solutions 68 5 Lie Symmetries with Several Parameters 74 5.1 Differential Invariants and Reduction of Order 74 5.2 The Lie Algebra of Point Symmetry Generators 79 5.3 Stepwise Integration of ODEs 89 6 Solution of ODEs with Multiparameter Lie Groups 93 6.1 The Basic Method: Exploiting Solvability 93 6.2 New Symmetries Obtained During Reduction 99 6.3 Integration of Third-Order ODEs with sl(2) 101 7 Techniques Based on First Integrals 108 7.1 First Integrals Derived from Symmetries 108 7.2 Contact Symmetries and Dynamical Symmetries 116 7.3 Integrating Factors 122 7.4 Systems of ODEs 128 8 How to Obtain Lie Point Symmetries of PDEs 136 8.1 Scalar PDEs with Two Dependent Variables 136 8.2 The Linearized Symmetry Condition for General PDEs 146 8.3 Finding Symmetries by Computer Algebra 149 9 Methods for Obtaining Exact Solutions of PDEs 155 9.1 Group-Invariant Solutions 155 9.2 New Solutions from Known Ones 162 9.3 Nonclassical Symmetries 166 10 Classification of Invariant Solutions 173 10.1 Equivalence of Invariant Solutions 173 10.2 How to Classify Symmetry Generators 176 10.3 Optimal Systems of Invariant Solutions 182 11 Discrete Symmetries 187 11.1 Some Uses of Discrete Symmetries 187 11.2 How to Obtain Discrete Symmetries from Lie Symmetries 188 11.3 Classification of Discrete Symmetries 191 11.4 Examples 195 Hints and Partial Solutions to Some Exercises 201 Bibliography 209 Index 211