Similarity solutions of nonlinear partial differential equations invariant to a family of affine groups

Conference · Sat Aug 01 04:00:00 EDT 1987

OSTI ID:5221870

Dresner, L

Problems of technological interest can very often be described by partial differential equations (PDEs) with one dependent and two independent variables (call them c, z, and t, respectively). Many such PDEs are invariant to one-parameter families of one-parameter affine groups. Similarity solutions are solutions of the PDE that are invariant to one group of the family. The great utility of similarity solutions is that they may be calculated by solving an ODE rather than a PDE and are thus much more easily accessible than other solutions. The form of the principal ODE depends, of course, on the form of the PDE, but it can be proved quite generally that the principal ODE is itself invarient to the one-parameter affine group or associated group. because of the invariance of the principal ODE to the associated group, the dependence on the boundary and initial conditions of certain special values of the function y(x), e.g., y(O), y(infinity), y(O), ets., may be predicted a priori without solving the principal ODE. The nonlinear PDE of heat transport in superfluid He-II, is used as an illustration of these ideas in this review.

• Research Organization: Oak Ridge National Lab., TN (USA)
• DOE Contract Number: AC05-84OR21400
• OSTI ID: 5221870
• Report Number(s): CONF-870860-1; ON: DE88003404
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990210* -- Supercomputers-- (1987-1989)
ANALYTICAL SOLUTION
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
ENERGY TRANSFER
EQUATIONS
EVEN-EVEN NUCLEI
FLUIDS
HEAT TRANSFER
HELIUM 4
HELIUM II
HELIUM ISOTOPES
INVARIANCE PRINCIPLES
ISOTOPES
LIGHT NUCLEI
NONLINEAR PROBLEMS
NUCLEI
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM FLUIDS
STABLE ISOTOPES