Numerical solutions of modified equal width wave equation with operator splitting cubic trigonometric B-spline collocation finite element method

This thesis consists of four chapters. In the introduction, general information regarding nonlinear partial differential equations is provided. Additionally, some partial differential equations that model various wave phenomena are discussed, along with the modified equal width (MEW) wave equation, whose numerical solutions are computed in this thesis. A brief overview of the operator splitting method, another technique used in this thesis, is given. The literature review on the MEW equation, whose numerical solutions are calculated, is also included in the introduction. Furthermore, the conservation constants I_1, I_2 and I_3 satisfied by the MEW equation and the commonly used L_2 and L_∞ error norms in the literature are presented. In the second chapter, topics related to the finite difference methods, weighted residual methods, collocation methods, finite element methods, spline functions, trigonometric B-spline functions, and cubic trigonometric B-spline functions, which are used in the numerical solutions of the MEW equation, are discussed. Then, the solution schemes of the Lie-Trotter and Strang splitting methods, which are types of operator splitting methods, are presented. In the third chapter, the operator splitting method is applied to the modified equal width wave equation, splitting it into two sub-equations: one linear and the other nonlinear. Subsequently, each sub-equation is solved using cubic trigonometric B-spline collocation finite element methods. The stability analysis of the Lie-Trotter and Strang splitting schemes is examined using the Fourier Von Neumann stability analysis method. Following this, the numerical results computed using the methods employed in this thesis are compared with the results obtained in different studies in the literature. In the fourth and final chapter, a general evaluation of the numerical results is provided.