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Abstract
In this work the mixed Sturm-Liouville problem of the Dirichlet-
Neumann type is considered in a circular region. The mixed problem
is reduced to the form of the Riemanian discrete problem which
in turn is reduced to a Cauchy type integral equation. This integral
equation is modified such that the unbounded solutions may be found.
The truncation of the algebraic system as well as the estimation of the resulting error is justified.
A numerical experiment is carried out to
show the usefulness of the proposed modification .
The thesis consists of four chapters, chapter one is an introductory one.
It contains the necessary mathematical background and tools on which this method is based.
In chapter two, the illustration of a typical, mixed Sturm- Liouville problem with Dirichlet-Neumann conditions.
The integral equation is reduced to an infinite system of algebraic equations. The unknowns of this system are the Fourier coefficients of the unknown
functions in the integral equation.
The expressions of the induced integrals
as well as coefficients of the algebraic system are explicitly given.
In chapter three, the application of the truncation to the algebraic system (integral equation) is justified: considering the infinite system as a limiting case of its truncated ones and using the theory of Chersky [10] it is shown that to every eigenvalue of this infinite system there corresponds a unique solution in Lp, p < 4 3 . The error due to the truncation is estimated.
In chapter four, the procedures are carried out right to the numerical results. The appearance of new solutions other than that obtained in [9], confirms the necessity of the present modification.