Similarity methods for differential equations
Abstract
We begin with a few remarks on the scope of these lecture notes. The purpose is to give an elementary introduction to similarity methods for partial differential equations for those who have had little or nor experience with these techniques. The emphasis will be on motivation and practical calculations involving several simple examples. From time-to-time we will allude to the differential-geometric structure which underlies these concepts; for, what we are really talking about is the invariance of a partial differential equation under the action of a local, Lie group of transformations. However, a deep understanding of these general concepts is not a prerequisite for being able to apply similarity methods to a given system. The first lecture is mostly an introductory lecture on the nature of self-similar solutions. The second lecture discusses dimensional analysis and the Buckingham Pi Theorem and how dimensional arguments lead in a natural way to similarity solutions. The third and fourth lectures develop the concept of invariance under a group of transformations and, given the group, show how solutions can be constructed. In the fifth and sixth lectures we show how the invariance group can be calculated. The final lecture deals with a detailed physical examplemore »
- Authors:
- Publication Date:
- Research Org.:
- California Univ., Livermore (USA). Lawrence Livermore Lab.
- OSTI Identifier:
- 5419926
- Report Number(s):
- UCID-19316
ON: DE82011225
- DOE Contract Number:
- W-7405-ENG-48
- Resource Type:
- Technical Report
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFERENTIAL EQUATIONS; ANALYTICAL SOLUTION; BOUNDARY CONDITIONS; INVARIANCE PRINCIPLES; LECTURES; LIE GROUPS; SPACE-TIME; DOCUMENT TYPES; EQUATIONS; SYMMETRY GROUPS; 658000* - Mathematical Physics- (-1987)
Citation Formats
Logan, D. Similarity methods for differential equations. United States: N. p., 1982.
Web.
Logan, D. Similarity methods for differential equations. United States.
Logan, D. Fri .
"Similarity methods for differential equations". United States.
@article{osti_5419926,
title = {Similarity methods for differential equations},
author = {Logan, D.},
abstractNote = {We begin with a few remarks on the scope of these
lecture notes. The purpose is to give an elementary introduction to
similarity methods for partial differential equations for those who have
had little or nor experience with these techniques. The emphasis will
be on motivation and practical calculations involving several simple
examples. From time-to-time we will allude to the differential-geometric
structure which underlies these concepts; for, what we are really
talking about is the invariance of a partial differential equation under
the action of a local, Lie group of transformations. However, a deep
understanding of these general concepts is not a prerequisite for being
able to apply similarity methods to a given system. The first lecture is
mostly an introductory lecture on the nature of self-similar solutions.
The second lecture discusses dimensional analysis and the Buckingham Pi
Theorem and how dimensional arguments lead in a natural way to
similarity solutions. The third and fourth lectures develop the concept
of invariance under a group of transformations and, given the group,
show how solutions can be constructed. In the fifth and sixth lectures
we show how the invariance group can be calculated. The final lecture
deals with a detailed physical example taken from the area of detonation
physics.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {1982},
month = {1}
}