A necessary and sufficient condition is established
for the existence of an invertible mapping of a system of
nonlinear differential equations to a system of linear
differential equations based on a group analysis of differential
equations. It is shown how to construct the mapping, when it
exists, from the invariance group of the nonlinear system.
It is demonstrated that the hodograph transformation, the
Legendre transformation and Lie's transformation of the
Monge-Ampere equation are obtained from this theorem. The
equation (ux)Puxx-uyy=0 is studied and it is determined
for what values of p this equation is transformable to a linear equation by an invertible mapping.
Many of the known non-invertible mappings of nonlinear equations to linear equations are shown to be related to invariance groups of equations associated with the given nonlinear equations. A number of such; examples are given, including Burgers' equation uxx +uuz-ut=0 a nonlinear
diffusion equation (u⁻²ux ) x -ut =0, equations of wave propagation
{Vy-wx=0, Vy-avw-bv-cw=0}, equations of a fluid flow {wy+vx=0,
wx -v⁻¹wP=0} and the Liouville equation uxy=eu.
As another application of group analysis, it is shown how conservation laws associated with the Korteweg-deVries equation, the cubic Schrodinger equation, the sine-Gordon equation and Hamilton's field equation are related to the
invariance groups of the respective equations.
All relevant background information is in the thesis, including an appendix on the known algorithm for computing the invariance group of a given system of differential equations. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/23079 |
| Date | January 1981 |
| Creators | Kumei, Sukeyuki |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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