Analytic Approximation for the Dirichlet Problem

The 1964 Benster translation of the Kantorovich and Krylov text "Approximate Methods of Higher Analysis" details a scheme for approximating the solution of boundary value problems for homogeneous elliptic equations on plane regions. One example (Laplace's equation on a square) is laboriously provided, the calculations having been done "by hand."
In this video, the recipe given by Kantorovich and Krylov is implemented in Maple, and applied to the text's example and to three others. The ease with which the requisite calculations can be executed is the message "between the lines." The overt message is that the method works, and gives reasonable accuracy for solutions to problems that would otherwise have to be obtained strictly numerically.
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