An Effective scheme based on quartic B-spline for the solution of Gardner equation and Harry Dym equation

In this paper, numerical solution of both the combined KdV and modified KdV equation (KdV-MKdV) known as Gardner equation and Harry Dym equation are obtained using quartic B-spline as a space approximation combined with finite difference for time discretization. We also study the stability analysis using Von-Neumann stability analysis showing that the numerical scheme is unconditionally stable. Accuracy of the method is discussed using two test problems. The numerical result shows that the presented method is a successful numerical technique for solving these problems which is showed by presenting the error at different time and space levels.