Dynamic Properties and Construction of Piecewise Smooth Periodic Solutions of Integro-Differential Equations

Abstract

A spatially distributed integro-differential equation with periodic boundary conditions is considered. It is assumed that the solution has zero mean over the spatial variable. The boundary value problem under consideration has a family of piecewise constant on the spatial variable solutions with one discontinuity point. The stability conditions for such solutions are defined. The existence of the piecewise constant solutions having more than one point of discontinuity is shown. During the numerical experiment, an algorithm based on the method of expansions in Fourier series has been used. With its help, it was expedient to calculate solutions to a boundary value problem that satisfies the condition of zero mean. We numerically study the behavior of the solutions to the boundary value problem for β 6 = 1 outside the domain of an α-stable one-parameter family of piecewise constant solutions. The presence of α-stable piecewise constant solutions with more than one discontinuity point is shown. A numerical analysis of the dynamics of the boundary value problem has been carried out.