2011 YÜksek lisans tez özetleri

About The Solution of General Singular Equations

In this study, a new solution method is given for general singular equation of normal type. As an application of this method, solutions for a class of convolution type integral equation have been found.

In the section of General Parts, summaries of important studies in which solubility of some singular integral equations are presented. Initially, a short theory of Fredholm integral equations is studied, then theorems -known as Fredholm Theorems- which shows solubility of these equations are presented. Afterwards, Fredholm theorems are examined for wider and more abstract equation types and necessary parts of studies of F. Riesz - J. Schauder [17], S. Nikolskii [3] are respectively presented. The number of linearly independent solutions of the homogenous parts of the equations, which are studied in these studies, is finite and equals to the number of linearly independent solutions of the adjoint homogeneous equations. Later, a short theory of singular integral equations with Cauchy kernel, which does not satisfy the property presented above, is investigated and theorems, known as Noether Theorems, that give solubility of these equations are given. Thereafter, the study of F.V. Atkinson [4], which investigates Noether theorems for more abstract equations, is analyzed.

Z.I. Halilov [5] considered singular integral equations with Cauchy kernel as operator equations and examined abstract case of these equations. General singular equations of normal type, which is, for the first time, defined by Halilov forms the basis for this study. After Halilov indicated that, this type of equations satisfies Noether Theorems, YU. I. Cherskii [6] has reduced this type of equations to Riemann boundary value problem by using factorization and has found a solution for the equations.

The purpose of this study is to provide a new solution method for general singular equations of normal type, which is easier and more simple than Cherskii ’s method. This new method presented at Findings part enables us to find solutions for normal type general singular equations without using factorization. Finally, an application of this new method is done and solution of a class of convolution type integral equation is presented.
 [1] NIKOLSKII, S., 1943, Linear Equations in Normed Linear Space, Izv.Akad.Nauk SSSR, Ser. Mat, 7:3, 146-166 (Rusça)
[2] ATKINSON, F.V., 1951, The Normal Solubility of Linear Equations in Normed Space, Mat. Sb.(N.S.), 28(70):1, 3-14 (Rusça)
[3] HALILOV, Z.I., 1949, Linear Singular Equations in a Normed Ring, Izv.Akad.Nauk SSSR, Ser. Mat 13:2, 163-176 (Rusça)
[4] CHERSKII, YU.I., 1957, The General Singular Equation and Equations of Convolution Type, Mat.Sb .41:3(83), 277-296 (Rusça)
 [5] YOSIDA, K., 1974, Functional Analysis, Grundlehren der mathematischen Wissenschaften 123, Springer-Verlag
  

KIRIK Bahar

Danışman : Prof. Dr. Leyla ZEREN AKGÜN

Anabilim Dalı : Matematik

Mezuniyet Yılı : 2011

Tez Savunma Jürisi : Prof. Dr. Leyla ZEREN AKGÜN

Prof. Dr. Nazım SADIK

Doç. Dr. Fatma ÖZDEMİR

Yrd. Doç. Dr. Hakan Mete TAŞTAN

Yrd. Doç. Dr. Özkan DEĞER