Abundant Traveling Wave and Numerical Solutions of Weakly Dispersive Long Waves Model

Wu Li,Dianchen Lu,Mostafa M A Khater,Lanre Akinyemi

doi:10.3390/sym13061085

Abstract

In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical).

Highlights

Published: 17 June 2021Recently, the phenomenon of shallow water waves has attracted the attention of many researchers in different fields
The well-known Navier–Stokes equation explains that the conservation of mass means that the vertical velocity scale of the fluid is smaller than the horizontal velocity scale when the horizontal length scale is much larger than the vertical length scale [5,6,7]
Distinct schemes have been derived such as the well-known ΨΨ -expansion methods, the auxiliary equation method, exponential expansion method, Kudryashov method, sech-tanh expansion method, direct algebraic equation method, Adomian decomposition method, iteration method, Khater methods, B-spline schemes and so on [14,15,16,17,18,19,20]

Summary

Introduction

The phenomenon of shallow water waves has attracted the attention of many researchers in different fields. Many nonlinear evolution equations have been formulated to demonstrate the waves’ dynamic behavior through shallow water waves This phenomenon has many applications in engineering and science, such as plasma physics, cosmology, fluid dynamics, electromagnetic theory, acoustics, electrochemistry astrophysics, and so on [8,9,10,11,12,13]. Distinct schemes have been derived such as the well-known ΨΨ -expansion methods, the auxiliary equation method, exponential expansion method, Kudryashov method, sech-tanh expansion method, direct algebraic equation method, Adomian decomposition method, iteration method, Khater methods, B-spline schemes and so on [14,15,16,17,18,19,20] These techniques have been applied to several nonlinear evolution equations to construct the solutions.

Accuracy of Computational Solutions

MK Method’s Solutions

Semi-Analytical Solutions

Interpretation of Results

Conclusions