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Excerpt from A Finite Difference Scheme for the Neumann and the Dirichlet Problem A specific feature of the finite difference scheme here proposed is that in it the boundary condition and the differential equation are treated simultaneously. It thus differs essentially from the schemes for the Neumann problem referred to by Forsythe and Nasow (in Section 2o.lo of their book) and the scheme of H. Keller (to appear), in which a special construction is employed to set up a finite difference boundary Condition. Our scheme is close to those in which the bound ary condition is derived as a natural one from a variational principle for the finite difference solution and which therefore has a symmetric coefficient matrix. Such a scheme, indicated for Laplace's equation by Forsythe and Nasow (section was developed for systems of equations by G. White (to appear). Our scheme will also result from a variational principle, but from the variational principle of the original differential equation problem, simply by using the Ritz method employing piecewise linear approximation functions. Thus a rather uniform treatment of boundary condition and differential equation will result. The coefficients of the substitute boundary condition will be given as the areas of sections of triangles cut out by the boundary; they are therefore not sensitive to variations of the direction of the boundary. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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