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# Construction of solutions to partial differential equations by the use of transformation groups

## 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. doi:10.7907/X1E8-1E61. https://resolver.caltech.edu/CaltechETD:etd-04082005-155822

## Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

A systematic approach is given for finding similarity solutions to partial differential equations by the use of transformation groups.

If a one-parameter 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 "non-classical" 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 "space-like" and m "time-like" 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.)) boundary value problems; conformal group; symmetries and differential equations California Institute of Technology Engineering and Applied Science Applied And Computational Mathematics Public (worldwide access) Lagerstrom, Paco A. (advisor)Cole, Julian D. (advisor) Unknown, Unknown 14 November 1967 bluman (AT) math.ubc.ca CaltechETD:etd-04082005-155822 https://resolver.caltech.edu/CaltechETD:etd-04082005-155822 10.7907/X1E8-1E61 No commercial reproduction, distribution, display or performance rights in this work are provided. 1309 CaltechTHESIS Imported from ETD-db 08 Apr 2005 21 Dec 2019 02:11

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