Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations

Driskell, C. N., Jr.

Gallaher, L. J.

Martin, R. H., Jr.

English

NASA Technical Reports Server

August 1968	 Brief 68-10276 
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Solution of Differential Equations by Application of

Transformation Groups 
A report has been prepared dealing with the appli-
cation of transformation groups to the solution of 
systems of ordinary differential equations and, in par-
ticular, partial differential equations. Although these 
groups are Lie groups in the usual sense, their trans-
formation properties rather than their basic structures 
are applied. 
The principal theorem, referred to as Lie's theorem, 
gives a method for finding an integrating factor for a 
system of ordinary differential equations when the 
appropriate invariance group or groups can be found. 
Lie's theorem can be extended to partial differential 
equations by considering a partial differential equa-
tion as a continuously infinite system of coupled 
ordinary differential equations. For a system of ordi-
nary differential equations the integrating factor is a 
matrix. For a partial differential equation the integrat-
ing factor is then a continuously infinite matrix. 
The proof of Lie's theorem and its use for partial 
differential equations depends on constructing an ade-
quate theory of continuously infinite matrices; this is 
done through the use of distributions or generalized 
functions. 
Chapter II of the report (Chapter lis the introduc-
tion) treats systems of ordinary differential equations. 
Lie's theorem is derived in such a way that it can be 
readily applied to the discrete approximation of a 
partial differential equation. Examples given are 
the discrete approximation to the heat flow and 
wave equations considered as initial value prob-
lems, and it is shown that the limiting form of 
the solutions obtained are those given by other 
more familiar techniques. In Chapter III, Lie's theo-
rem is derived for and applied directly to partial dif-
ferential equations, without the necessity of using a 
discrete approximation. The heat now equation is
again used as an example. While the examples given 
are linear equations, there is nothing in the method 
that restricts it to linear problems. Lie's theorem can 
in principle be applied to nonlinear partial differential 
equations, but in practice it has been difficult to find 
a nonlinear example. 
Chapters II and III are oriented towards the engi-
neer, physicist, or chemist whose prime interest is the 
solution of practical problems. The proofs or deriva-
tions here are not rigorous. 
In the Appendix, however, an effort is made to 
achieve rigor in the proofs given. It is here that the 
foundation of a theory of continuously infinite ma-
trices based on distribution theory and generalized 
functions is given. 
While the literature has not disclosed applications 
of Lie's theorem for the solution of partial differential 
equations (or even for systems of ordinary differential 
equations), it is unlikely that the material presented 
in the report is completely new. 
Note: 
Inquiries concerning this material may be directed 
to:
Technology Utilization Officer 
Marshall Space Flight Center 
Huntsville, Alabama 35812 
Reference: B68-10276 
Patent status: 
No patent action is contemplated by NASA. 
Source: R. H. Martin, Jr., C. N. Driskell, Jr., 
and L. J. Gallaher 
of Georgia Institute of Technology 
under contract to 
Marshall Space Flight Center 
(M FS- l4802) 
Category 02 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
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Solution of differential equations by application of transformation groups

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