Applications of Lie's Theory of Ordinary and Partial Differential Equations: 1st Edition (Paperback) book cover

Applications of Lie's Theory of Ordinary and Partial Differential Equations

1st Edition

By L Dresner

CRC Press

240 pages | 16 B/W Illus.

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Paperback: 9780750305310
pub: 1998-01-01
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Description

Lie's group theory of differential equations unifies the many ad hoc methods known for solving differential equations and provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations and is not restricted to linear equations. Applications of Lie's Theory of Ordinary and Partial Differential Equations provides a concise, simple introduction to the application of Lie's theory to the solution of differential equations. The author emphasizes clarity and immediacy of understanding rather than encyclopedic completeness, rigor, and generality. This enables readers to quickly grasp the essentials and start applying the methods to find solutions. The book includes worked examples and problems from a wide range of scientific and engineering fields.

Table of Contents

Conventions Used in This Book

One-Parameter Groups

Groups of transformations

Infinitesimal transformations

Group invariants

Invariant curves and families of curves

Transformation of derivatives: the extended group

Transformation of derivatives (continued)

Invariant differential equations of the first order

First-Order Ordinary Differential Equations

Lie's integrating factor

The converse of Lie's theorem

Invariant integral curves

Singular solutions

Change of variables

Tabulation of differential equations

Notes to chapter two

Second-Order Ordinary Differential Equations

Invariant differential equations of the second order

Lie's reduction theorem

Stretching groups

Streching groups (continued)

Stretching groups (continued)

Other groups

Equations invariant to two groups

Two-parameter groups

Noether's theorem

Noether's theorem (continued)

Similarity Solutions of Partial Differential Equations

One-parameter families of stretching groups

Similarity solutions

The associated group

The asymptotic behavior of similarity solutions

Proof of the ordering theorem

Functions invariant to an entire family of stretching groups

A second example

Further use of the associated group

More wave propagation problems

Wave propagation problems (continued)

Shocks

Traveling-Wave Solutions

One-parameter families of translation groups

The diffusion equation with source

Determination of the propagation velocity a

Determination of the propagation volocity: role of the initial condition

The approach to traveling waves

The approach to traveling waves (continued)

A final example

Concluding remarks

Notes of chapter five

Approximate Methods

Introduction

Superfluid diffusion equation with a slowly varying face temperature

Ordinary diffusion equation with a nonconstant diffusion coefficient

Check on the accuracy of the approximate formula

Epilogue

Appendix 1: Linear, First-Order Partial Differential Equations

Appendix II: Riemann's Method of Characteristics

Appendix III: The Calculus of Variations and the Euler-Lagrange Equation

Appendix IV: Computation of Invariants and First Differential Invariants from the Transformation Equations

Solutions to the Problems

References

Symbols and Their Definitions

Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics