scholarly journals Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 219-226 ◽  
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Fourth Order ◽  
Traveling Wave ◽  
Water Wave ◽  
Dynamical Equation ◽  
Traveling Wave Solutions ◽  
Simple Equation ◽  
Equation Method ◽  
Localized Structures ◽  
Wave Solutions ◽  
Waves Interaction

Abstract The aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

scholarly journals Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations

2021 ◽  
Vol 67 (6 Nov-Dec) ◽  
Author(s):  
Gizel Bakicierler ◽  
Suliman Alfaqeih ◽  
Emine Misirli
Keyword(s):  
Traveling Wave ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Simple Equation ◽  
Wave Solutions ◽  
Non Linear ◽  
Package Program ◽  
Modified Simple Equation Method

Recently, non-linear fractional partial differential equations are used to model many phenomena in applied sciences and engineering. In this study, the modified simple equation scheme is implemented to obtain some new traveling wave solutions of the non-linear conformable time-fractional approximate long water wave equation and the non-linear conformable coupled time-fractional Boussinesq-Burger equation, which are used in the expression of shallow-water waves. The time- fractional derivatives are described in terms of conformable fractional derivative sense. Consequently, new exact traveling wave solutions of both equations are achieved. The graphics and correctness of the wave solutions are obtained with the Mathematica package program.

scholarly journals The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Norhashidah Hj. Mohd. Ali
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Simple Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Simple Equation Method ◽  
Bbm Equation

The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established that the MSE method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics.

Detection of new multi-wave solutions in an unbounded domain

2019 ◽  
Vol 33 (34) ◽  
pp. 1950425 ◽  
Author(s):  
Mohamed R. Ali ◽  
Wen-Xiu Ma
Keyword(s):  
Wave Equation ◽  
Evolution Equation ◽  
Unbounded Domain ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Reduction Process ◽  
Three Dimensions ◽  
Integrating Factor ◽  
Wave Solutions

We deduce new explicit traveling wave solutions for Zoomeron evolution equation and (3[Formula: see text]+[Formula: see text]1)-dimensional shallow water wave equation. The reduction process using Lie vectors leads in some cases to ordinary differential equations (ODEs) that having no quadrature. The integrating factor property has been used to derive several new solutions for these nonsolvable ODEs. These solutions have been illustrated with three dimensions plots. Comparison with other works are presented.

scholarly journals Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 742 ◽  
Author(s):  
Tao Liu
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Equation Method ◽  
Singular Solutions ◽  
Trial Equation Method ◽  
Wave Solutions ◽  
Fifth Order ◽  
Fkdv Equation

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

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