Oscillatory processes under impulsive perturbations

The problem of oscillatory processes under impulsive perturbations at fixed moments in time is considered. The mathematical model consists of a classical partial differential equation describing oscillatory processes, supplemented by boundary and initial conditions, along with additional conditions characterizing the impulsive effects at the specified time instances.Using the method of separation of variables, the solution is constructed in the form of a Fourier series involving the eigenfunctions of Laplace operator. Conditions under which the resulting solution is classical are discussed. A special case is examined in which the domain is a rectangle, and in this case, an explicit analytic solution is obtained.The results obtained in this paper may be especially useful in the analysis of mathematical models for various applied problems involving short-term effects, as well as in the further development of the theory of impulsive differential equations, including those involving partial derivatives.

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