Comparative Study of Some Numerical and Semi-analytical Methods for Some 1D and 2D Dispersive KdV-type Equations

Abey Sherif Keli̇l

doi:10.54974/fcmathsci.1002281

Abstract

This paper aims to investigate an approximate-analytical and numerical solutions for some 1D and 2D dispersive homogeneous and non-homogeneous KdV equations by employing two reliable methods namely reduced differential transform method (RDTM) and a classical finite-difference method. RDTM provides an analytical approximate solution in the form of a convergent series. The classical finite-difference method (FDM) to solve dispersive KdV equations is employed by primarily checking Von Neumann’s stability criterion. The performance of the mentioned methods for the considered experiments are compared by computing absolute and relative errors at some spatial nodes at a given time; and to the best of our knowledge, the comparison between these two methods for the considered experiments is novel. Knowledge acquired will enable us to build methods for other related PDEs such as KdV-Burgers, stochastic KdV and fractional KdV-type equations.

Highlights

Nonlinear partial differential equations are obtained when problems in numerous domains in science and engineering are modeled
The aim of this paper is to provide a comparative study for solving 1D homogeneous and 2D non-homogeneous dispersive Korteweg-de Vries (KdV) type equations using reduced differential transform method (RDTM) and classical finite-difference method (FDM) for the first time in literature
Numerical Experiment We investigate two numerical experiments, as stated below: (i) One-dimensional homogeneous nonlinear dispersive KdV equation [23]

Summary

Introduction

Nonlinear partial differential equations are obtained when problems in numerous domains in science and engineering are modeled. The authors in [3] investigated classical and multisymplectic schemes for linearized KdV equations using some numerical methods and dispersion analysis was studied. The aim of this paper is to provide a comparative study for solving 1D homogeneous and 2D non-homogeneous dispersive KdV type equations using RDTM and classical FDM for the first time in literature. This section compares numerical results with the approximate series solutions obtained via RDTM. Reduced differential transform method (RDTM) is very powerful method to obtain analytical approximate solutions to linear and nonlinear partial differential equations [18] and for systems of differential equations [12].

Application of RDTM to the 1D-dispersive

Application of RDTM to the 2D-Linearized KdV Equation

Solution of Numerical Experiment 1 Using Finite Difference Scheme

Solution of Numerical Experiment 2 Using Classical FDM

Discussion and Conclusions