Publication Date

5-31-1968

Abstract

In this dissertation we consider a class of integral equations of the general form

Ψ(z)=g(z) + ∫yx Y(z’)K(|z-z’|) Ψ(z’)dz’

Where K has an integral representation of the type

K(u)= ∫0∞ k(s’)e-a(s’)uds’

With this integral equation, we associate the pseudo transport problem

(sgn s) ∂/∂z N(z,s) + a(s)N(z,s) = k(s)Y(z) ∫-∞∞ N(z,s’)ds’,

Y<=z<=x, |s| < ∞

with certain boundary conditions. (Equation (2) is a generalization or the equation for particle transport in a slab geometry). Equivalence between the problems (1) and (2) is established. Thus any results concerning the problem (2) produce corresponding results concerning the problem (1).

Functional equations are derived relating the solutions (one fo1 11 > o and one for s < O) of (2). The functions involved in these equations are solutions to initial value problems, and may be thought of as generalizations of the reflection and transmission functions which arise in transport theory.

These results are applied to the eigenvalue problem associated with (1 ). Under certain symmetry conditions a new positive, symmetric, L^2 kernel, S(x,a,B), is generated. The condition then for unity to be the smallest eigenvalue for the integral operator associated with (1) is that the parameter x be so determined that unity is the smallest positive eigenvalue for the integral operator defined by the kernel S(x,a,B). A particular example is considered and some numerical results obtained.

Although much of the analysis was motivated by transport phenomena, all derivations are mathematically rigorous and do not depend on the purely physical arguments of transport theory.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

George Milton Wing

Second Committee Member

Bernard Epstein

Third Committee Member

Illegible

Language

English

Document Type

Dissertation

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