Some applications of finite element method with B-spline functions

Partial differential equations can be examined in two classes as linear and nonlinear partial differential equations. Depending on the application areas, the varieties are increasing in themselves. In this master thesis, the numerical solution will be investigated for some one-dimensional nonlinear partial differential equations depending on time and position. As the model problem, KdV Burgers, Gardner, Advection-Diffusion, MRLW equations will be chosen. while researching solutions, Crank-Nicolson method will be used for time parsing and Galerkin method will be used for location parsing. These numerical methods will be applied to model problems. In the first part, some basic concepts that will be needed in the following chapters are explained. Firstly, solitary waves, soliton, error norms, finite difference method, finite element method, galerkin method and collocation method are introduced. After explaining the concept of spline function, cubic B-spline and expanded cubic B-spline functions are defined. At the end of this chapter, literature review has been made for KDVB Equation, Gardner Equation, Advection-Diffusion Equation, MRLW Equation, whose numerical solution will be investigated in the following chapters. In other chapters, numerical solutions of KDV Burger, Gardner, Advection - Diffusion, Modified Regularized Long Wave (MRLW) equations by expanded cubic B-spline method, respectively, have been examined. Two test problems for each were used to compare the proposed method with the exact results. In the last section, the results obtained about the application of the expanded cubic B-spline method to these equations are compared.