Nonlinear partial differential equations invariant to a one-parameter family of stretching groups
Conference
·
OSTI ID:10155819
Nonlinear partial differential equations (PDEs) in one dependent and two independent variables (call them c, z, and t) occur in many technological applications. Typical PDEs and the contexts in which they arise are the following: c{sub t} = (c{sup n}){sub zz}, which occurs in plasma physics, hydrology, gas flow in porous media, and applied superconductivity; cc{sub t} = c{sub zz}, which describes the expulsion of fluid from a long, slender, heated pipe; c{sub t} = (c{sub z} {sup 13}){sub z}, which describes heat transport in turbulent superfluid He-II; and c{sub tt} = (c{sub zz/2}) {integral} {sub o}{sup 1}c{sub z}{sup 2} dz, which describes the motion of a shock-loaded elastic membrane. All of these equations are invariant to a one-parameter family of one-parameter stretching groups of the form c{prime} = {lambda}{sup a}c, t{prime} = {lambda}{sup {beta}}t, z{prime} = {lambda}z, 0 < {lambda} < {infinity} where {lambda} is the group parameter that labels the individual transformations of a group and {alpha} and {beta} are the parameters that label groups of the family. The parameters {alpha} and {beta} are connected by a linear relation Ma + N{beta} = L where M, N, and L are numbers determined by the structure of the PDE. Similarity solutions are of the PDE that are invariant to one group of the family, say, that for which {alpha} = {alpha}* and {beta} = {beta}*. Such solutions have the form c = t{sup {alpha}*/{beta}*} y(z/t{sup 1/{beta}*}) where y is a function of the single variable x = z/t{sup 1{beta}*}. When substituted into the PDE yields an ordinary differential equation for the function y(x).
- Research Organization:
- Oak Ridge National Lab., TN (United States)
- Sponsoring Organization:
- USDOE, Washington, DC (United States)
- DOE Contract Number:
- AC05-84OR21400
- OSTI ID:
- 10155819
- Report Number(s):
- CONF-940719--2; ON: DE94012753
- Country of Publication:
- United States
- Language:
- English
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