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Symmetry Analysis of Differential Equations with Mathematica®

  • Book
  • © 2000

Overview

Authors:
  1. Gerd Baumann

    1. Department of Mathematical Physics, University of Ulm, Ulm, Germany

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  • The author shows how to use Mathematica to quickly perform computations that would be both time consuming and cumbersome if done by hand
  • Includes a CD-ROM containing Mathematica 3.0 notebooks for interacting directly with the code provided in the book
  • Also includes the authors proprietary "MathLie" software, which aids in performing algebraic computations
  • Each section of the book describes the theoretical foundations of the material, then shows the application in connection with Mathematica
  • The book contains many working examples relating to the applications of Lie's theory

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Gerd Baumann
    Pages 1-5

  3. Elements of Symmetry Analysis

    • Gerd Baumann
    Pages 6-36

  4. Derivatives

    • Gerd Baumann
    Pages 37-95

  5. Symmetries of Ordinary Differential Equations

    • Gerd Baumann
    Pages 96-215

  6. Point Symmetries of Partial Differential Equations

    • Gerd Baumann
    Pages 216-364

  7. Non-classical Symmetries of Partial Differential Equations

    • Gerd Baumann
    Pages 365-391

  8. Potential Symmetries of Partial Differential Equations

    • Gerd Baumann
    Pages 392-403

  9. Approximate Symmetries of Partial Differential Equations

    • Gerd Baumann
    Pages 404-423

  10. Generalized Symmetries

    • Gerd Baumann
    Pages 424-456

  11. Solution of Coupled Linear Partial Differential Equations

    • Gerd Baumann
    Pages 457-482

  12. Appendix

    • Gerd Baumann
    Pages 483-492

  13. Back Matter

    Pages 493-521
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Keywords

About this book

The purpose of this book is to provide the reader with a comprehensive introduction to the applications of symmetry analysis to ordinary and partial differential equations. The theoretical background of physics is illustrated by modem methods of computer algebra. The presentation of the material in the book is based on Mathematica 3.0 note­ books. The entire printed version of this book is available on the accompanying CD. The text is presented in such a way that the reader can interact with the calculations and experiment with the models and methods. Also contained on the CD is a package called MathLie-in honor of Sophus Lie---carrying out the calculations automatically. The application of symmetry analysis to problems from physics, mathematics, and en­ gineering is demonstrated by many examples. The study of symmetries of differential equations is an old subject. Thanks to Sophus Lie we today have available to us important information on the behavior of differential equations. Symmetries can be used to find exact solutions. Symmetries can be applied to verify and to develop numerical schemes. They can provide conservation laws for differential equations. The theory presented here is based on Lie, containing improve­ ments and generalizations made by later mathematicians who rediscovered and used Lie's work. The presentation of Lie's theory in connection with Mathematica is novel and vitalizes an old theory. The extensive symbolic calculations necessary under Lie's theory are supported by MathLie, a package written in Mathematica.

Reviews

“The book is essentially an extensive user manual for the Mathematica program … . The book concentrates very much on computational aspects and should be useful for engineers and scientists who want to learn fast the basic computational recipes for a symmetry analysis. … The strength of the book are the close integration of computer algebra and the large number of examples often stemming from real applications.” (Werner M.Seiler, zbMATH 0946.35002, 2021)

Authors and Affiliations

  • Department of Mathematical Physics, University of Ulm, Ulm, Germany

    Gerd Baumann

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