Müntz-Legendre collocation method for approximate solutions of systems of delay differential equations

In this thesis, a matrix-collocation method is applied to obtain the approximate solutions of ordinary differential equations, delay differential equations, systems of ordinary differential equations, systems of proportional delay differential equations and systems of delay differential equations which take an important place in physiscs, chemists, biology and engineering by using Müntz-Legendre polynomials. In the method, the functions and the coefficients in the problem are represented in the matrix form and the problem is reduced into an algebraic equations system. Then this algebraic system is solved by a code written in MATLAB in order to find approximate solutions. Besides by using residual function, an error analysis has been done to improve the approximate solutions. This study consists of five sections. In the first section, general information about the origin of differential equations and usage of those and the aim of this study is given. In the second section, general information, a short summary of literature, the methods which are used previously, fundamental definitions about the delay differential equations, Müntz-Legendre polynomials, other special polynomials and error analysis are introduced. In the third section, matrix-collocation method is given for each kind of problem in details such as matrix relations, solution method and error analysis. In the fourth section, numerical examples are given for each kind of problem and the results are shown in tables and figures. On the other hand actual absolute error, estimated absolute error and corrected (improved) errors are compared in tables. Finally, in the fifth section, a conclusion and suggestions are given. The results which are obtained in this study, show that the method works truely and effectively on the problems.