A Variational approximation to the solution of the Burgers' equation

This thesis consists of five chapters. In chapter 1, the history of the Burgers' equation was presented. Chapter 2 was devoted to the presentation of the solutions of the systems of equations and the method of discretization in time and the least squares method and general concepts and theorems concerning with the subject. In Chapter 3, by considering the relation between Burgers' equation and heat equation the exact solution of the Burgers' equation was given. Chapter 4 is the original part of this thesis. In this chapter, Burgers' equation which is one dimensional nonlinear partial differential equation was converted to p nonlinear ordinary differential equations by using the method of discretization in time. Each of them was solved by the least squares method which is a variational method. Chapter 5 is devoted to the numerical results of Chapter 4. For e > 0.01 at different time steps, the obtained numerical solutions were compared with the exact visolutions by giving tables and graphs. It was seen that both of them were in excellent agreement. While the exact solution of the equation could not been computed for e < 0.01 at small time steps, it was shown that the mathematical structure of the problem for the obtained numerical solutions for e = 0.00001 at small time steps did not decay. KEYWORDS: Burgers' Equation, The Method of Discretization in Time, The Least Squares Method, The Newton Method for Solving Nonlinear Systems of Equ ations. vn