Boundary Value Problems in Composite Media: Quasi-Orthogonal Functions

Abstract

A new theorem is stated on the construction and use of orthogonal sets for solving boundary value problems in a composite medium where the boundary function traverses more than one region. The concept of quasi-orthogonality is introduced and employed in a rigorous expansion of the boundary function into a series of the nonorthogonal eigenset that arises from separation of variables in such instances. Orthogonal sets are constructed from the nonorthogonal (quasi-orthogonal) eigenset by the use of orthogonality factors for each region derived from the orthogonality condition. Orthogonality factors for representative problems in neutron diffusion, heat conduction, scalar wave propagation, and nonrelativistic quantum mechanics are computed for rectangular, spherical, and cylindrical coordinates. The theorem allows a solution of multi-region problems by a direct expansion without resort to integral transformation or Green's functions. It is an extension of the method of separation of variables and generalized Fourier analysis to cases heretofore beyond its scope.