One-dimensional generalized Rosenau-KdV equation and its numerical solution

In this study, conservative Crank-Nicolson implicit finite difference schemes based on finite difference methods are given to obtain approximate solutions of nonlinear generalized Rosenau-KdV itself and space split generalized coupled Rosena-KdV equations with initial and boundary conditions. This thesis is planned as five chapters. In the first chapter, the generalized Rosenau-KdV equation itself as well as some of its main studies available in the literature are briefly mentioned. Also, in this chapter two problems to be taken into consideration throughout the thesis are introduced. In the second chapter, finite difference approximation formulas for derivatives obtained with the help of Taylor series expansion and notations to be used in the thesis are presented. In the third chapter, two numerical schemes based on the conservative Crank-Nicolson implicit finite difference method are given to obtain approximate solutions of the generalized Rosenau-KdV equation. Stability analysis of only one method is also performed by the von Neumann method. Besides, in order to check the accuracy and reliability of the presented numerical schemes, two test problems are considered and some error norms calculated using the same parameters with conservation of energy and accuracy orders of the schemes are presented in the tables. Moreover, some graphs are given to show how good the computed numerical results represent the correct physical behavior of the problems. In the fourth chapter, similar studies given in the third chapter are applied to the space split generalized coupled Rosenau-KdV equation. In the fifth chapter that is the last chapter of the thesis, the computed results in the third and fourth chapters are compared, and a brief conclusion and suggestion for future studies are made.