Sеlf-similar solutions of hydrodynamic equations of non-local physics

Abstract

For linear partial differential equations there are various techniques for reducing the partial differential equations (PDE) to the ordinary differential equations (ODE) or at least to equations in a smaller number of independent variables. These include various integral transforms and eigenfunction expansions. Usually such techniques are not applicable in dealing with nonlinear partial differential equations. From this point of view the presented approach is much more interesting. It identifies equations for which the solution depends on certain groupings of the independent variables rather than depending on each of the independent variables separately. The name of these solutions, self-similar, comes from the fact that the spatial distribution of the characteristics of motion remains similar to itself at all times during the motion. Roughly speaking, the idea is to look whether the solution of a problem u(x, y) can be collapsed in a function u(x, y) = U(y / f (x)). The function f (x) may be found by substitution in the PDE, in order to obtain an ODE for U . Self-similar solutions of the non-local equations describing the explosion with the spherical symmetry are investigated for the case of the astrophysical applications. Namely, in the quasi-stationary Hubble regime only the implicit dependence on time for the unknown values exists. It means that for the intermediate (Hubble) regime the complicated PDE set can be transformed in the set ODE. This possibility can be realized also in the case if the self-similar solutions exist. The mentioned self-similar solutions are found for Hubble regime.

Keywords

• non-local hydrodynamics, self-similar solutions, explosion with the spherical symmetry