This book is a straightforward introduction to symmetry methods, 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 designed to enable postgraduates and advanced undergraduates to master the main techniques quickly and easily.
This book is a straightforward introduction to symmetry methods, 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 designed to enable postgraduates and advanced undergraduates to master the main techniques quickly and easily.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
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.
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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