2018 Issue №4

New exact solutions are obtained for a spatially non-local wave equation with sources. The results are set out in the terms of the heat transfer theory. The non-locality of the problem is determined by the value of the fourth spatial derivative. We considered two types of volume energy sources which are alter­na­ting with respect to the temperature. For a technical source the derivative is po­sitive, since «higher» temperatures arise from the energy release. For a bio­lo­gical source the source function is negative inclined, because a biological tis­sue gives off heat in the region of «lower» temperatures. The external inf­luen­ce on a non-local medium is simulated by spatially nonuniform energy source, and we have considered five types of such sources. The analytical solutions are pre­sented in the finite form. The effects of monotonous and nonmonotonous (im­pulsive) reonomic sources are compared. The conditions for a transonic transition are indicated for the wave of perturbation in the temperature set. Resonance occurrences in the system «medium — energy source» are studied. The limits of oscillation stability/instability are determined. We found a di­men­sionless criterion including the inclination of the source function and the pa­rameter of the medium non-locality. The criterion affects the correlation «os­cillation frequency — fading parameter».

A method for construction of exact cosmological solutions of Einstein equations is developed. The method is based on representation of these equations as a second-order linear differential equation. Particularly, the method allows to construct a more general solution, parametrized by two constants, using any known solution. Also it is shown, that in a specific case new classes of exact solutions have singularities with the following property: a geodesic, which begins or ends in the singularity, has an infinite length. Such a singularity leads to an absence of event horizons (if this is a future singularity). In this case it might prove possible to construct cosmological model, satisfying every consistency requirement during a universe lifetime.