Bernstein collocation method for numerical solutions of second order partial differential equations

In this study, Bernstein collocation method is applied to obtain approximate solutions of second order linear partial differential equation problem in general form under Neuman, Dirichlet and mixed conditions, second order linear partial differential equation problem in general form with initial and boundary conditions, singularly pertürbed one-dimensional parabolic convection-diffusion problem, two-space variable second order linear partial differential equation problem with variable coefficients and a class of nonlinear partial differential equation problem. In the presented method, equation and given conditions with the collocation points are reduced to a system of matrix equations with Bernstein coefficients putting them in the form of matrices. These matrix equations systems are solved and approximate solutions based on Bernstein polynomials are obtained. Also, the error estimation of the method, based residual error function, is presented for different types of problems. It is aimed to improve Bernstein polynomial solutions with predicted error functions and reduce errors. This study is formed of five parts. In the first part, the history of partial differential equations, other methods related to the solution of partial differential equations, which are given in the literature are mentioned briefly. The second part deals with the preliminaries, necessary definitions and also the classification of second-order partial differential equations. In the third part describes the basic matrix relations, solution method and error analysis for Bernstein collocation method for partial differential equations with linear and nonlinear variable coefficients. In the fourth chapter, numerical examples for the problems given in the thesis are given and the results are compared with other methods and shown with the help of charts and graphs. Finally, the results related with the method are discussed. Keywords: Partial Differential Equations, Bernstein Polynomials, Bernstein Collocation Method