Solution of boundary value problems with moving boundaries by using an approximate method for constructing solutions of integro – differential equations

The problem of oscillations of objects with moving boundaries, formulated as a differential equation with boundary and initial conditions, is a non-classical generalization of a problem of hyperbolic type. To facilitate the construction of a solution to this problem and justify the choice of a solution form, equivalent integro-differential equations are constructed with symmetric and time-dependent kernels and integration limits varying in time. The method for constructing solutions of integro-differential equations is based on the direct integration of differential equations in combination with the standard replacement of the desired function with a new variable. The method is extended to a wider class of model boundary value problems that take into account the bending stiffness of an oscillating object, the resistance of the environment, and the rigidity of the substrate. Particular attention is paid to the consideration of the most common in practice case when external disturbances act at the boundaries. The solution is made in dimensionless variables accurate to second-order values of smallness with respect to small parameters characterizing the speed of the border.