Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity

Mostafa M. A. Khater; S. H. Alfalqi; J. F. Alzaidi; Samir A. Salama; Fuzhang Wang; Mostafa M. A. Khater; S. H. Alfalqi; J. F. Alzaidi; Samir A. Salama; Fuzhang Wang

doi:10.3934/math.2022004

AIMS Mathematics

2022, Volume 7, Issue 1: 54-81. doi: 10.3934/math.2022004

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Research article

Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity

1.
Department of Mathematics, Faculty of Science, Jiangsu University, 212013 Zhenjiang, China
2.
Department of Mathematics, Obour High Institute For Engineering and Technology, 11828 Cairo, Egypt
3.
Department of Mathematics, Faculty of Science and Arts, Mahayil Asir King Khalid University, Abha, Saudi Arabia
4.
Division of Biochemistry, Department of Pharmacology, College of Pharmacy, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
5.
School of Mathematical and Statistics, Xuzhou University of Technology, Xuzhou 221018, Jiangsu, China
6.
Nanchang Institute of Technology, Nanchang 330044, China
7.
College of Computer Science and Technology, Huaibei Normal University, 235000 Huaibei, China

Received: 27 June 2021 Accepted: 16 August 2021 Published: 29 September 2021
MSC : 35C08, 34A25, 49M05

This paper applies two computational techniques for constructing novel solitary wave solutions of the ill-posed Boussinesq dynamic wave (IPB) equation. Jacques Hadamard has formulated this model for studying the dynamic behavior of waves in shallow water under gravity. Extended simple equation (ESE) method and novel Riccati expansion (NRE) method have been applied to the investigated model's converted nonlinear ordinary differential equation through the wave transformation. As a result of this research, many solitary wave solutions have been obtained and represented in different figures in two-dimensional, three-dimensional, and density plots. The explanation of the methods used shows their dynamics and effectiveness in dealing with certain nonlinear evolution equations.
Citation: Mostafa M. A. Khater, S. H. Alfalqi, J. F. Alzaidi, Samir A. Salama, Fuzhang Wang. Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity[J]. AIMS Mathematics, 2022, 7(1): 54-81. doi: 10.3934/math.2022004

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Abstract

This paper applies two computational techniques for constructing novel solitary wave solutions of the ill-posed Boussinesq dynamic wave (IPB) equation. Jacques Hadamard has formulated this model for studying the dynamic behavior of waves in shallow water under gravity. Extended simple equation (ESE) method and novel Riccati expansion (NRE) method have been applied to the investigated model's converted nonlinear ordinary differential equation through the wave transformation. As a result of this research, many solitary wave solutions have been obtained and represented in different figures in two-dimensional, three-dimensional, and density plots. The explanation of the methods used shows their dynamics and effectiveness in dealing with certain nonlinear evolution equations.

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