Using of fast expansions in the construction of two-dimensional exact solutions of the Poisson equation

Using the fast expansion method, we obtained several exact solutions of the boundary value problem for the Poisson equation in a rectangular domain. We have given graphs of exact solutions corresponding to different boundary conditions and different types of the Poisson equation free term. We showed the influence of the type of boundary conditions and the right-hand side on the type of exact solution. We obtained a solution to the problem of membrane deflections under the action of variable load. We have given graphs of the stress components, from the analysis of which it follows that the greatest stress is in the middle of the rectangular membrane long sides.