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.

Eigenvalue analysis of temperature distribution in composite walls

“…Conduction heat transfer problems in composite walls have been analysed by several investigations (Tittle, 1965;Mulholland and Cobble, 1972;O G zisik, 1980). In all of these works, temperature distribution of composite walls involving two or more layers is investigated spectrally and the results are formulated by the eigenvalues and eigenfunctions of the auxiliary spectral problem.…”

A new type of boundary value coupling for second order Sturm-Liouville systems

“…Recently, generalizations of the second order Sturm-Liouville system have been developed which, while retaining the distinctive S turm· Liouville character, are applicable to problems involving co mposite media [3,4,5,6]. The purpose of this paper is to discuss a differe nt generalization of the second order Sturm-Liouville system whic h retains the Sturm-Liouville character.…”

Analytical Modeling of Soil Solution Monitoring by Diffusion in Porous Cups

“…However, closely related functions can be defined that are orthogonal with respect to z (Tittle 1965;de Monte 2000de Monte , 2002 and which enable the constants a n to be calculated. These functions, denoted as δ n , are listed below.…”

“…The exact solution, to these equations, is developed by applying the method of separation of variables in each region and then applying the interface and boundary conditions (Tittle 1965;de Monte 2000de Monte , 2002. In order to this end, we assume the following product solutions:…”