Vibrations of continuous systems are modelled in the form of a partial differential equation system. In seeking approximate analytical solutions of these systems, one common choice is to discretize the partial differential equation system and then to apply perturbation methods to the resulting ordinary differential system (the discretization-perturbation method). An alternative approach is to seek approximate solutions of the original partial differential system. In this approach, perturbation methods are applied directly to the partial differential system (the direct-perturbation method). Comparisons of these two methods have appeared in the literature for various mathematical models [1][2][3][4][5][6][7]. Some of the work has addressed the comparisons for finite mode truncations [1-4] and some others for infinite modes [4][5][6][7]. It is shown that, while both methods produce identical results for infinite number of modes, the direct-perturbation method produces more accurate results for finite mode truncations. This is because the spatial functions appearing at higher orders of approximation represent the real system better in the case of the direct-perturbation method. It is shown that [5], for finite mode truncations, the spatial functions are the converged functions obtained by using the infinite series of eigenfunctions calculated at the first level of approximation. However, for the discretization-perturbation method, the spatial functions appearing at higher orders of approximations are only approximate, reducing the accuracy of the overall system.
In all of the previous work [1-7], comparisons have been made for non-linear systems, especially for systems having quadratic and cubic non-linearities. No discussions are presented for linear systems. This may lead one to conclude that the differences in results occur due to the non-linearities. However, such a conclusion would be wrong. Differences in results arise even for linear systems. As an illustration, we present here a case of a linear parametrically excited system. Instead of treating a specific problem, the formalism given in references [2] and [5] is followed and solutions are presented for arbitrary spatial operators. The algorithms developed are then applied to a specific problem. Differences in results occur if a higher order perturbation scheme is employed and if the boundary value problems appearing at higher orders of approximations yield different solutions in the case of direct treatment.
It is to be noted that results of the discretization-perturbation method would converge to those of the direct-perturbation method if the number of modes taken into consideration