Boundary shape distortion involving dirichlet conditions with application to temperature distribution

Summary

This paper presents a variant of the method of separation of variables which enables the determination of the solution of an elliptic differential equation having Dirichlet conditions along an arbitrary curve D forming part of the boundary. The coefficients in the eigenfunction expansions representing the general solution are determined by comparison with a special series representation of the Dirichlet condition along D. This representation is obtained by means of the Gram-Schmidt orthogonalization process which uses as its basic non-orthogonal set of functions a special set derived directly from the eigenfunction expansion. A simple numerical example concerning the temperature distribution in a semi-infinite parallel slab with a skew end face on which there is a sinusoidal temperature varation illustrates the application of the method. It is shown that the rate of convergence is good and that the asymptotic solution is estimated rapidly and accurately by this method.