1st Edition
Applications of Lie's Theory of Ordinary and Partial Differential Equations
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.
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
Biography
L Dresner
