Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis

Abstract

In this study, the fractional derivative and finite difference operators are analyzed. The time fractional KdV equation with initial condition is considered. Discretized equation is obtained with the help of finite difference operators and used Caputo formula. The inherent truncation errors in the method are defined and analyzed. Stability analysis is explored to demonstrate the accuracy of the method. While doing this analysis, considering conservation law, with the help of using the definition discovered by Lax-Wendroff, von Neumann stability analysis is applied. The numerical solutions of time fractional KdV equation are obtained by using finite difference method. The comparison between obtained numerical solutions and exact solution from existing literature is made. This comparison is highlighted with the graphs as well. Results are presented in tables using the Mathematica software package wherever it is needed.