Numerical solution of cauchy type singular integral equations of the first kind by fourier-chebyshev series expansion

In this study, the numerical solutions of singular integral equations and the system of singular integral equations by using Fourier-Chebyshev series are considered.The study comprises of six parts. In the study, it is aimed to give numerical solutions transforming Cauchy type singular integral equations into a system of linear equation by using Fourier-Chebyshev series.In the first chapter, the properties and orthogonality of Chebyshev polynomials are introduced.In the second chapter, Fourier series representation of a function having 1st, 2nd, 3rd and 4th kinds are given depending on Chebyshev polynomials are given.In the third chapter, the solutions of Cauchy type characteristic singular integral equations are presented for four cases: unbounded at the limit points of ; bounded at the limit points of ; bounded at the limit point of , unbounded at the other point and bounded at the right limit point and unbounded at the left limit point.In the fourth chapter, numerical solutions are presented for four cases mentioned above by transforming first kind of singular integral equations to the system of linear equation using Fourier-Chebyshev series expansion.In the fifth chapter, the numerical solution of the first kind of Cauchy type singular integral equation system is carried out by reducing the system of linear equation for four cases mentioned above via of Fourier-Chebyshev series. In the last chapter, numerical solutions of the examples are given for the equations presented in the fourth and fifth chapters.Key Words: Characteristic Equations, Chebyshev Polynomials, Fourier Series Expansion, Singular Integral Equations.