Numerical Solution of the Korteweg De Vries Equation by Finite Difference and Adomian Decomposition Method

Abstract

The Korteweg de Vries (KDV) equation which is a non-linear PDE plays an important role in studying the propagation of low amplitude water waves in shallow water bodies, the solution to this equation leads to solitary waves or solitons. In this paper, we present the analytic solution and use the explicit and implicit finite difference schemes and the Adomian decomposition method to obtain approximate solutions to the KDV equation. As the behavior of the solitons generated from the KDV depends on the nature of the initial wave, this work aims to study two possible scenarios (hyperbolic tangent initial condition and a sinusoidal initial condition) and obtained solution analytically, numerically with the aforementioned methods. Comparison between the four different solutions is done with the aid of tables and diagrams. We observed that valid analytical solutions for the KDV equation are restricted to time values close to the initial time and that the Adomian decomposition method is a wonderful tool for solving the KDV equation and other non-linear PDEs.