Symmetry Methods for Differential Equations: A Beginner's Guide / Edition 1by Peter E. Hydon, Hydon Peter E.
Pub. Date: 10/28/2005
Publisher: Cambridge University Press
A good working knowledge of symmetry methods is very valuable for those working with mathematical models. This book is a straightforward introduction to the subject for applied mathematicians, physicists, and engineers. The informal presentation uses many worked examples to illustrate the major symmetry methods. Written at a level suitable for postgraduates and
A good working knowledge of symmetry methods is very valuable for those working with mathematical models. This book is a straightforward introduction to the subject for applied mathematicians, physicists, and engineers. The informal presentation uses many worked examples to illustrate the major symmetry methods. Written at a level suitable for postgraduates and advanced undergraduates, the text will enable readers to master the main techniques quickly and easily. The book contains some methods not previously published in a text, including those methods for obtaining discrete symmetries and integrating factors.
Table of Contents
1. Introduction to symmetries; 1.1. Symmetries of planar objects; 1.2. Symmetries of the simplest ODE; 1.3. The symmetry condition for first-order ODEs; 1.4. Lie symmetries solve first-order ODEs; 2. Lie symmetries of first order ODEs; 2.1. The action of Lie symmetries on the plane; 2.2. Canonical coordinates; 2.3. How to solve ODEs with Lie symmetries; 2.4. The linearized symmetry condition; 2.5. Symmetries and standard methods; 2.6. The infinitesimal generator; 3. How to find Lie point symmetries of ODEs; 3.1 The symmetry condition. 3.2. The determining equations for Lie point symmetries; 3.3. Linear ODEs; 3.4. Justification of the symmetry condition; 4. How to use a one-parameter Lie group; 4.1. Reduction of order using canonical coordinates; 4.2. Variational symmetries; 4.3. Invariant solutions; 5. Lie symmetries with several parameters; 5.1. Differential invariants and reduction of order; 5.2. The Lie algebra of point symmetry generators; 5.3. Stepwise integration of ODEs; 6. Solution of ODEs with multi-parameter Lie groups; 6.1 The basic method: exploiting solvability; 6.2. New symmetries obtained during reduction; 6.3. Integration of third-order ODEs with sl(2); 7. Techniques based on first integrals; 7.1. First integrals derived from symmetries; 7.2. Contact symmetries and dynamical symmetries; 7.3. Integrating factors; 7.4. Systems of ODEs; 8. How to obtain Lie point symmetries of PDEs; 8.1. Scalar PDEs with two dependent variables; 8.2. The linearized symmetry condition for general PDEs; 8.3. Finding symmetries by computer algebra; 9. Methods for obtaining exact solutions of PDEs; 9.1. Group-invariant solutions; 9.2. New solutions from known ones; 9.3. Nonclassical symmetries; 10. Classification of invariant solutions; 10.1. Equivalence of invariant solutions; 10.2. How to classify symmetry generators; 10.3. Optimal systems of invariant solutions; 11. Discrete symmetries; 11.1. Some uses of discrete symmetries; 11.2. How to obtain discrete symmetries from Lie symmetries; 11.3. Classification of discrete symmetries; 11.4. Examples.
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