Citation
Bluman, George W. (1968) Construction of solutions to partial differential equations by the use of transformation groups. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd04082005155822
Abstract
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A systematic approach is given for finding similarity solutions to partial differential equations by the use of transformation groups.
If a oneparameter group of transformations leaves invariant a partial differential equation and its accompanying boundary conditions, then the number of variables can be reduced by one. In order to find the group of a given partial differential equation, the "classical" and "nonclassical" methods are discussed. Initially no special boundary conditions are imposed since the invariances of the equation are used to find the general class of invariant boundary conditions.
New exact solutions to the heat equation are derived. In addition new exact solutions are found for the transition probability density function corresponding to a particular class of first order nonlinear stochastic differential equations. The equation of nonlinear heat conduction is considered from the classical point of view.
The conformal group in n "spacelike" and m "timelike" dimensions, C(n, m), which is the group leaving invariant [...], is shown to be locally isomorphic to S O (n+l, m+l) for n + m >= 3. Thus locally compact operators, besides pure rotations, leave invariant Laplace's equation in n >= 3 dimensions. These are used to find closed bounded geometries for which the number of variables in Laplace's equation can be reduced.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  boundary value problems; conformal group; symmetries and differential equations 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Applied And Computational Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  14 November 1967 
NonCaltech Author Email:  bluman (AT) math.ubc.ca 
Record Number:  CaltechETD:etd04082005155822 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd04082005155822 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  1309 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  08 Apr 2005 
Last Modified:  26 Dec 2012 02:37 
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