Numerical solutions of the Burgers' equation by using B-spline functions

NUMERICAL SOLUTIONS OF THE BURGERS' EQUATION BY USING B-SPLINE FUNCTIONS Ali ŞAHÎN Mathematics, MSc Thesis, 2004 ' ' Thesis Adviser: Assoc. Prof. Dr. Idris DAÛ SUMMARY Tlits thesis consists ol* eleven chapters. In the first chapter, some properties of B-spline functions are given. Quadratic, cubic and quintic B-spline functions are presented. Besides, the Burgers' equation is introduced together with initial and boundary conditions. Some of the previous numerical methods about the equation are mentioned. The quadratic B-spline collocation solution of the space splitted Burgers' equation is given in the second chapter. Cubic B-spline collocation solution of the time splitted Burgers' equation is set up in the third chapter. The space splitted Burgers' equation is solved by using cubic B-spline collocation method in the fourth chapter. Chapter fifth includes cubic B-spline collocation solution of the Burgers' equation. Solution of the Burgers' equation is obtained by using cubic B-spline collocation method in the sixth chapter. Quadratic B-spline Galerkin solution of the Burgers' equation is presented in the chapter seven. Cubic B-spline Galerkin solution of the Burgers' equation is examined in the chapter eight. In the ninth chapter, quintic B-spline Galerkin solution of the Burgers' equation is studied. The chapter ten includes quintic B-spline Galerkin solution of the time splitted Burgers' equation. Numerical solutions which are obtained from the methods compared In the last chapter. The methods which are given to solve the Burgers' equation are tested by using two standard test problems. Numerical solutions are compared with analytical solutions. L2 and Ln error norms are calculated for the reliability of the methods and also Fourier stability analyses are done for the some methods. Keywords : B-spline functions, Burgers' equation, Collocation method, Galerkin method, Finite element mehtods.