Abstract
Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.
Highlights
Nonlinear partial differential equations (PDEs) have a significant role in various scientific and engineering fields
We have solved a nonlinear Korteweg-de Vries (KdV) equation using Zabusky-Kruskal’s numerical scheme, and the numerical results obtained for this experiment signify the outbreak of solitons due to perturbation of the parameters involved in the KdV equation
The novelty of this study relies on the comparative study of newly generalized integral transform, i.e., Shehu’s integral transform combined with Adomian decomposition method (ADM) together with classical finite difference method (FDM) to effectively solve some nonlinear third-order and fifth-order dispersive KdV equations
Summary
Nonlinear partial differential equations (PDEs) have a significant role in various scientific and engineering fields. We have solved a nonlinear KdV equation (with no known exact solution) using Zabusky-Kruskal’s numerical scheme, and the numerical results obtained for this experiment signify the outbreak of solitons due to perturbation of the parameters involved in the KdV equation. The novelty of this study relies on the comparative study of newly generalized integral transform, i.e., Shehu’s integral transform combined with Adomian decomposition method (ADM) together with classical FDM to effectively solve some nonlinear third-order and fifth-order dispersive KdV equations. The newly proposed methods in this work can be applied to many complicated linear and nonlinear problems since STADM does not require linearization, discretization, or perturbation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
