Some positive linear operator approximation processes for linear integral equations

This thesis includes the examination of approximate solutions of Volterra and Fredholm integral equations obtaining by setting up some positive linear operators and the examination of exact solutions of Volterra and Fredholm integral equations. In the first chapter, the emergence of the theory of integral equations and studies on integral equations are given. In the second part, the types of integral equations and in the third part basic concepts and linear positive operators are given. Bernstein polynomials and Baskakov operators, which are linear positive operators, are introduced and the properties of these operators are given. In the fourth chapter, Adomian decomposition method and successive approximations method are introduced for exact solutions of Volterra integral equations, and Volterra integral equations are reconstructed using Bernstein polynomials and Baskakov operator to be used in the calculation of approximate solutions. At the end of the fourth chapter, the exact and approximate solutions of the Volterra integral equation are found by giving an example. The exact solution of the integral equation given in the example and its approximate solutions obtained with different n values are compared using tables and graphs. In the fifth chapter, Adomian decomposition method and successive approximation method are introduced for exact solutions of Fredholm integral equations, and Fredholm integral equations are reconstructed using Bernstein polynomials and Baskakov operator to be used in the calculation of approximate solutions. At the end of the fifth chapter, the exact and approximate solutions of the Fredholm integral equation are found by giving an example. Also, the exact solution of the integral equation and the approximate solutions obtained with different n values were compared using tables and graphs. In the conclusion part, by interpreting the tables and graphics obtained from the examples were given suggestions for future studies.