The operational matrix properties of the class of special polynomials and their applications to delay integro differential equations

The aim of this thesis study is to obtain the approximate solutions of the delay functional integro differential equations arising in science and engineering by using matrix collocation methods based on the classes of special polynomials. These methods are made up of the matrix relations of the polynomials, collocation points and a fundamental matrix equation. In the method, equations and given conditions with the collocation points are reduced to an algebraic system with the special polynomial coefficients by putting them in the matrix forms; then, by solving this system, the approximate solutions of the problem are obtained. Also, the error analysis related to residual functions is performed and some illustrative examples are presented. The obtained results are demonstrated by tables and graphics; the usability and efficiency of the methods are interpreted. The study consists of five parts. In the first part, the general development of the functional integro differential equations and the family of special polynomials and orthogonal polynomials solution methods, historical informations and fundamental properties are given. Finally, in this chapter also the aim of the thesis work are given. In the second part, basic concepts about these equations, different methods used for their numerical solutions, resource summaries about the family of special polynomials and error analysis are available. In the third part, special polynomials collocation method, matrix relations, method of solution and error analysis are explained. In the fourth section, numerical examples are given for each problem type and the results are shown with the help of tables and graphs. Finally, in the fifth section, conclusions and suggestions are included.