#### Oscillations, resonances and waves in a non-local medium with sources

Abstract
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».