Similarity methods for differential equations

Technical Report · Fri Jan 01 04:00:00 EST 1982

OSTI ID:5419926

Logan, D

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.

• Research Organization: California Univ., Livermore (USA). Lawrence Livermore Lab.
• DOE Contract Number: W-7405-ENG-48
• OSTI ID: 5419926
• Report Number(s): UCID-19316; ON: DE82011225
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
DOCUMENT TYPES
EQUATIONS
INVARIANCE PRINCIPLES
LECTURES
LIE GROUPS
SPACE-TIME
SYMMETRY GROUPS