Class of Biharmonic End-Strip Problems Arising in Elasticity and Stokes Flow.

摘要

We consider boundary value problems for the biharmonic equation in the open rectangle x > 0, -1 < y < 1, with homogeneous boundary conditions on the free edges y = + or - 1, and data on the end x = 0 of a type arising both in elasticity and in Stokes flow of a viscous fluid, in which either two stresses or two displacements are prescribed. For such 'non-canonical' data, coefficients in the eigenfunction expansion can be found only from the solution of infinite sets of linear equations, for which a variety of methods of formulation have been proposed. A drawback of existing methods has been that the resulting equations are unstable with respect to the order of truncation. It is clear from an examination of the spectrum of a typical matrix that ill-conditioning is to be expected. However, a search among a wider class of possible trial functions than hitherto for use in a Galerkin method based on the actual eigenfunctions has led to the choice of a unique set, here termed optimal weighting functions, for which the resulting infinite matrix is diagonally-dominated. This ensures the existence of an inverse, which can be approximated by solving a finite subset of the equations.