Symmetry Methods for Differential Equations: A Beginner's Guide
This book is a straightforward introduction to the subject of symmetry methods for solving differential equations, and is aimed at applied mathematicians, physicists, and engineers. The presentation is informal, using many worked examples to illustrate the main symmetry methods. It is written at a level suitable for postgraduates and advanced undergraduates, and is designed to enable the reader to master the main techniques quickly and easily.The book contains some methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.
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Lie Symmetries of FirstOrder ODEs
How to Find Lie Point Symmetries of ODEs
How to Use a OneParameter Lie Group
Lie Symmetries with Several Parameters
Solution of ODEs with Multiparameter Lie Groups
Techniques Based on First Integrals
How to Obtain Lie Point Symmetries of PDEs
Methods for Obtaining Exact Solutions of PDEs
Classification of Invariant Solutions
Hints and Partial Solutions to Some Exercises
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algebra of point ansatz arbitrary constants calculate canonical coordinates Chapter characteristic commutator computer algebra constraints contact transformation derivatives determining equations diffeomorphism differential equations discrete point symmetries dynamical symmetries equating powers equivalent Euler-Lagrange equation Example first-order ODE functionally independent fundamental differential invariants group of point group of symmetries heat equation Hence independent variables infinitesimal integrating factor invariant point invariant solutions invariant surface condition Lie algebra Lie point symmetries Lie symmetries linear ODE linearized symmetry condition mapped matrix nonclassical symmetries nontrivial obtain ODE of order one-parameter group one-parameter Lie group optimal system orbit original ODE packages parameter PDEs plane point transformation quadrature reduced ODE restrict attention result Riccati equation rotations satisfies scaling symmetries second-order ODE simplify solution curve solve the ODE subalgebra Suppose symmetry group symmetry methods system of ODEs tangent vector tangent vector field third-order ODE trivial symmetry variational symmetries vector field yields zero