Matrix properties of Dickson polynomials and aplications to pantograph type functional equations

In this study, a hybrid matrix-collocation method based on Dickson polynomials together with Taylor polynomials is developed to solve pantograph type functional differential equations with mixed delays under the inital conditions. The used method reduces the coefficients of solution to the matrix form and hence the approximate solution of the problem is reached. Besides, the parameter- in Dickson polynomials is used for obtaining the optimum solutions. An error estimation related with the residual function and the mean-value theorem is implemented and also some illustrative examples are presented. It is observed that the proposed method is easy to be applied. The thesis consists of five chapters. In the first chapter, the usage areas and the emergence of pantograph type functional differential equations are examined. In the second chapter, general information is given; resource descriptions, definition of Dickson polynomials, recurrence relations and graphics are explained. In the third chapter, the matrix-collocation methods are explained by using the basic matrix relations for equations in every section of chapter. Then, an error analysis based on residual functional for the solution method is performed. In the fourth chapter, numerical examples are given for each section. The results are supported by tables and figures. In tables, the obtained solutions and exact solutions together with absolute errors are compared. Finally, in the fifth chapter, conclusions and recommendations are given. The findings show that the method works correctly and effectively on the problems.