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

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.

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. R. Seadawy ◽  
A. Sayed

The modeling of unidirectional propagation of long water waves in dispersive media is presented. The Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations are derived from water waves models. New traveling solutions of the KdV and BBM equations are obtained by implementing the extended direct algebraic and extended sech-tanh methods. The stability of the obtained traveling solutions is analyzed and discussed.


2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Norhashidah Hj. Mohd. Ali

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.


2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 341-352 ◽  
Author(s):  
Akbar Ali ◽  
Norhashidah Ali ◽  
Abdul-Majid Wazwaz

In this article, the modified simple equation (MSE) method is introduced to examine the closed form wave solutions of the fractional non-linear Cahn-Allen equation and of the fractional generalized reaction Duffing equation. The fractional derivatives are delineated in the sense of Jumarie?s modified Riemann-Liouville derivative. A fractional complex transformation is used to transform the fractional-order PDE into integer order ODE. The reduced equations are then examined by using the MSE method and some new and further general solutions of these equations are successfully established. The approach of this method is simple, standard and the obtained solutions are highly encouraging. It is also powerful, reliable and effective.


Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 219-226 ◽  
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dianchen Lu

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.


Author(s):  
Hülya Durur

In this study, the Lonngren-wave equation, which is physically semiconductor, is taken into consideration. Traveling wave solutions of this equation are presented with generalized exponential rational function method, which is one of the mathematically powerful analytical methods. These solutions are produced in bright (non-topological) soliton and complex trigonometric-type traveling wave solutions. Three-dimensional (3D), 2D and contour graphics are presented with the help of a ready-made package program with special values given to constants in these solutions. The effect of the change in wave velocity on the traveling wave solution showing energy transport is presented with the help of simulation. It is argued that velocity is one of the important factors in wave diffraction. In the results and discussion section, the advantages and disadvantages of the method are discussed.


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