B-Spline Galerkin Finite Element Numerical Solutions For (1+1)-Dimensional Nonlinear Evolution Equation

With the development of nonlinear science, obtaining the solutions of nonlinear evolution equations has become a hot question to a great number of science workers who study physics,mechanics,geoscience,life sciences,applied mathematics and engineering technique. At present, although a number of methods are proposed and developed to look for the exact solutions of nonlinear evolution equations, unfortunately, in most cases, we can only obtain their approximate solutions by numerical methods. In this dissertation, we study the B-spline Galerkin finite element methods, and apply it to solve the Burgers'equation and the KdVB equation.Besides, the B-spline is improved here to triumphantly obtain the solutions of KdVB which values are not nil on the ends of the interval. This dissertation mainly discusses the following contents:1. The (1+1)-dimensional nonlinear Burgers'equation under the condition of the homogeneous boundaries is solved by a quintic B-spline Galerkin finite element method. The validity and accuracy of the method are demonstrated.2. A cubic B-spline Galerkin finite element method is presented for solving the (1+1)-dimensional nonlinear KdVB equation under the condition of the unhomogeneous boundaries. The numerical examples have shown that the method is capable of simulating the original problems.