Solutions of some partial differential equations by operator splitting B-spline collocation method

This thesis consists of five chapters. In the introduction chapter, the structure of Burgers', modified Burgers', regularized long wave (RLW) and coupled Burgers' equations of which numerical solutions are obtained as well as various physical phenomena expressed with partial differential equations, a brief summary of the finite element and operator splitting methods used in the thesis are presented. In the second chapter, the fundamental definitions and concepts related to finite element and operator splitting methods to be used in the thesis are given. The collocation method used in conjunction with the finite elements method, Spline functions, B-spline functions and cubic B-spline functions are presented in detail. In addition, a short literature search of splitting methods, operator splitting methods and Lie-Trotter splitting scheme, Strang (S_{Δt}) splitting scheme, local splitting error of Lie-Trotter and Strang splitting methods and Ext4 and Ext6 methods obtained using Strang method via extrapolation technique are explained. In the third chapter, nonlinear Burgers', modified Burgers' (mBE) and regularized long wave (RLW) equations are splitted via operator splitting into two equations, one linear and the other nonlinear equation. Then S_{Δt}, Ext4 ve Ext6 splitting methods with cubic B-spline collocation finite element method are applied to the sub equations obtained for each equation. Three test problems for the Burgers 'equation , one for the modified Burgers' equation and three for the RLW equation are considered. The numerical results calculated with S_{Δt}, Ext4 and Ext6 are compared with the exact results and / or the L₂, L_{∞} and ‖e₁‖ error norms of the studies available in the literature. In addition, the stability analysis of the schemes obtained by the finite element and operator splitting method for each equation is examined by von Neumann method. In the fourth chapter, nonlinear coupled viscous Burgers' equation was splitted into two sub-equations, each linear and non-linear, resulting in a total of four sub-equations. Then S_{Δt}, Ext4 and Ext6 methods with cubic B-spline collocation finite element method are applied to the sub equations obtained. For the present equation, three test problems are considered and the calculated numerical results are compared with the exact solution and the error norms L₂ and L_{∞} of the other studies in the literature. Again, in this section, stability analyses of the resulting finite element schemes are performed. Finally, in the fifth and the last chapter of the thesis, a general evaluation is made for the equations of which the numerical solutions are calculated using the finite element cubic B-spline collocation method with S_{Δt}, Ext4 and Ext6 methods in the third and fourth sections of the thesis.