On solution of differential equations by matrix method based on Freud polynomials

In this study, a matrix collocation method based on Freud Polynomials, which we propose to find the exact or approximate solutions of an important class of differential equations, starting from first and second order linear differential equations, to first and second order Pantograph differential equations, is given. The approximation function for the analytical solution is an estimated series with the base as Freud polynomials and the coefficients as unknown values. The algebraic equation system is reached by substituting the colocation points in the matrix equation obtained by writing each term in the differential equation in matrix form. By solving the system obtained by taking into account the conditions given by the equation, the coefficients of the Freud polynomials in the terms of the series forming the approximation function are found. After the proposed solution method is given in detail, various examples are given to show that this theory is suitable for solving the above-mentioned equations effectively and efficiently. In the layout of the thesis, first, the types of differential equations that we will consider throughout this thesis and their general usage areas and some solution techniques used for the solution of these equations are mentioned. In the second part, Hungarian mathematician Gaze Freud, who revealed Freud Polynomials, his studies and the general structure of Freud polynomials are given. In the third chapter, linear generalized pantograph differential equations with first and second order variable coefficients, the Freud matrix collocation method, which we proposed, for the first time, for these equations, are mentioned. Finally, various examples given were solved with this method and the results obtained were explained in detail with the help of tables and graphics.