The solution of non-self-adjoint and non-separable initial-boundary-value problems is treated by using a Green function approach. Specifically, the analysis of the long-time response of a finite, isotropic, homogeneous, linearly elastic cantilever plate, in a state of plane strain to an antisymmetric and dynamically applied distributed surface load, has been shown to constitute such a non-self-adjoint and non-separable problem. The Green function of the problem is determined, based on the method given by J. Miklowitz for solving non-separable waveguide problems, by using a double Laplace transform and an entirety condition on the solution. The Green functions for two near-field and far-field domains are obtained. Hence, the response of the plate to any antisymmetric dynamically applied distributed surface load is determined in the form of an integral equation for the two domains, with the respective Green functions as the kernels of integration. It is concluded that the "elementary" theories and the "engineering methods" in which a "dynamic load factor" is applied to the static solution tend to underestimate the maximum values of the plate deflections in the vicinity of the free end of the cantilever plate.

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Journal Journal of Sound and Vibration
Afagh, F, & Leipholz, H.H.E. (1989). Long-time dynamic response of linearly elastic non-self-adjoint and non-separable systems: A green function approach. Journal of Sound and Vibration, 128(3), 471–486. doi:10.1016/0022-460X(89)90786-4