Solution of model boundary value problems on oscillations of mechanical systems with moving boundaries by the duhamel method

Abstract

The Kantorovich – Galerkin method in conjunction with the Duhamel method is considered as applied to solving problems describing the oscillations of visco–elastic objects with conditions on moving boundaries. The mathematical formulation of the problem includes an inhomogeneous partial differential equation with respect to the desired displacement function, inhomogeneous boundary and initial conditions. By introducing a new function into the problem, the boundary and initial conditions are reduced to homogeneous. The solution is made in dimensionless variables with an accuracy of the second order of smallness with respect to small parameters characterizing the velocity of the boundary and viscoelasticity. Using the Kantorovich – Galerkin method and the Duhamel method, an approximate solution of the problem of forced longitudinal oscillations of a viscoelastic rod of variable length is found.