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Symmetry Methods for Differential Equations

Symmetry Methods for Differential Equations
A Beginner's Guide


Part of Cambridge Texts in Applied Mathematics

  • Date Published: May 2000
  • availability: Available
  • format: Paperback
  • isbn: 9780521497862

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About the Authors
  • Symmetry is the key to solving differential equations. There are many well-known techniques for obtaining exact solutions, but most of them are special cases of a few powerful symmetry methods. Furthermore, these methods can be applied to differential equations of an unfamiliar type; they do not rely on special 'tricks'. Instead, a given differential equation is forced to reveal its symmetries, which are then used to construct exact solutions. This book is a straightforward introduction to the subject, 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 methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.

    • Written in an informal style but mathematically rigorous
    • Suitable for undergraduate courses
    • Designed to enable the reader to master the main techniques quickly and easily
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    Reviews & endorsements

    'Hydon's book stands out as perhaps the best introductory level text currently available … Hydon's book is extremely well-written, and a welcome addition to the literature on Lie's methods. The author has clearly devoted a lot of effort to pedagogical details, and the exposition is designed to effortlessly bring the beginning student up to speed in basic applications of the method.' Peter Olver, ZAMM

    'I really enjoyed reading this book and I am planning on using some parts for one of my next courses.' Monatshefte für Mathematik

    '… a nice introduction to symmetry methods for ordinary and partial differential equations written with passion by a specialist … after a few pages it becomes clear that the book is written in a lucid and precise manner.' Zentralblatt MATH

    'This new book by Peter Hydon … is eminently suitable for advanced undergraduates and beginning postgraduate students … Overall I thoroughly recommend this book and believe that it will be a useful textbook for introducing students to symmetry methods for differential equations.' Journal of Nonlinear Mathematical Physics

    'Throughout the text numerous examples are worked out in detail and the exercises have been well chosen. this is the most readable text on this material I have seen and I would recommend the book for self-study (as an introduction).' MathSciNet

    'It is very suitable for, and is specifically aimed at, postgraduate courses in the field. it is the more enjoyable for being written with infectious enthusiasm and there is a good selection of examples.' Mathematical Gazette

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    Product details

    • Date Published: May 2000
    • format: Paperback
    • isbn: 9780521497862
    • length: 228 pages
    • dimensions: 230 x 153 x 26 mm
    • weight: 0.42kg
    • contains: 8 b/w illus.
    • availability: Available
  • 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.

  • Author

    Peter E. Hydon, University of Surrey

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