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Symmetry Methods for Differential Equations

Symmetry Methods for Differential Equations
A Beginner's Guide

$63.99 (P)

Part of Cambridge Texts in Applied Mathematics

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

$ 63.99 (P)

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About the Authors
  • 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.

    • 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

    "[A] valuable addition to the bookshelf for both the beginner and research worker in the field." Mathematical Geology

    "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." Mathematical Reviews

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    Product details

    • Date Published: January 2000
    • format: Paperback
    • isbn: 9780521497862
    • length: 228 pages
    • dimensions: 229 x 152 x 13 mm
    • weight: 0.34kg
    • 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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