scholarly journals Constructions of the soliton solutions to the good Boussinesq equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammed Bakheet Almatrafi ◽  
Abdulghani Ragaa Alharbi ◽  
Cemil Tunç
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Solution Stability ◽  
Exact Results ◽  
The Stability ◽  
Stability And Accuracy

Abstract The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the $L_{2}$ L 2  error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics

2020 ◽  
Vol 34 (29) ◽  
pp. 2050282
Author(s):  
Asıf Yokuş ◽  
Doğan Kaya
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Traveling Wave Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Traveling Wave Solution ◽  
Von Neumann ◽  
Wave Solutions ◽  
De Vries ◽  
The Stability

The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The [Formula: see text] and [Formula: see text] norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.

Multiple wave solutions for nonlinear burgers equations using the multiple exp-function method

2021 ◽  
pp. 2150149
Author(s):  
Khaled A. Gepreel ◽  
E. M. E. Zayed
Keyword(s):  
Traveling Wave ◽  
Software Package ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Multiple Wave ◽  
Wave Solutions ◽  
The One

In this paper, we use the multiple exp-function method to explicity present traveling wave solutions, double-traveling wave (DTW) solutions and triple-traveling wave solutions (TWs) which include one-soliton, double-soliton and triple-soliton solutions for nonlinear partial differential equations (NPDEs) via, the (2+1)-dimensional and (3+1)-dimensional nonlinear Burgers PDEs in mathematical physics. In this work, we build some series of straightforward and new solutions successfully with the help of a computerized symbol computational software package like Maple or Mathematica. We will make some drawings in some cases with specific values for the relevant parameters for each obtained solutions such as the one-traveling wave solutions, double-traveling wave solutions and TWs. This method is efficient and powerful in solving a wide class of NPDEs.

Symbolic computations: Dispersive soliton solutions for (3+1)-dimensional Boussinesq and Kadomtsev–Petviashvili dynamical equations and its applications

2019 ◽  
Vol 33 (29) ◽  
pp. 1950342 ◽  
Author(s):  
Aly R. Seadawy ◽  
Kalim U. Tariq ◽  
Jian-Guo Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Equation Method ◽  
Dynamical Equations ◽  
Wave Solutions ◽  
Kp Equations

In this paper, the auxiliary expansion equation method is applied to compute the analytical wave solutions for (3[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq and Kadomtsev–Petviashvili (KP) equations. A simple transformation is carried out to reduce the set of nonlinear partial differential equations (NPDEs) into ODEs. These obtained results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance.

scholarly journals Exact Traveling Wave Solutions of the Gardner Equation by the Improved tanΘϑ-Expansion Method and the Wave Ansatz Method

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hatıra Günerhan
Keyword(s):  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Tool ◽  
Ansatz Method ◽  
Gardner Equation ◽  
Wave Solutions ◽  
Analytical Approaches

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.

scholarly journals Stability Analysis and Assortment of Exact Traveling Wave Solutions for the (2+1)-Dimensional Boiti-Leon-Pempinelli System

2021 ◽  
pp. 2393-2400
Author(s):  
Mizal H. Alobaidi ◽  
Wafaa M. Taha ◽  
Ali H. Hazza ◽  
Pelumi E. Oguntunde
Keyword(s):  
Stability Analysis ◽  
Traveling Wave ◽  
Physical Meaning ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Exact Results ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
The Stability

     In this research, the Boiti–Leon–Pempinelli (BLP) system was used to understand the physical meaning of exact and solitary traveling wave solutions. To establish modern exact results, considered. In addition, the results obtained were compared with those obtained by using other existing methods, such as the standard hyperbolic tanh function method, and the stability analysis for the results was discussed.

scholarly journals Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

2021 ◽  
Vol 82 (4) ◽  
pp. 197-203
Author(s):  
G. N. Shaikhova ◽  
◽  
B. K. Rakhimzhanov ◽  
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equations ◽  
Integrable Model ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

scholarly journals Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1003-1010
Author(s):  
Asıf Yokuş ◽  
Hülya Durur ◽  
Taher A. Nofal ◽  
Hanaa Abu-Zinadah ◽  
Münevver Tuz ◽  
...  
Keyword(s):  
Traveling Wave ◽  
Analytical Methods ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Dark Soliton ◽  
Mathematical Tool ◽  
Symbolic Calculation ◽  
Advantages And Disadvantages ◽  
Wave Solutions

Abstract In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations.

scholarly journals Fundamental solutions for the long–short-wave interaction system

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1093-1099
Author(s):  
Mustafa Inc ◽  
Samia Zaki Hassan ◽  
Mahmoud Abdelrahman ◽  
Reem Abdalaziz Alomair ◽  
Yu-Ming Chu
Keyword(s):  
Fluid Mechanics ◽  
Traveling Wave ◽  
Inverse Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Wave Interaction ◽  
Fundamental Solutions ◽  
Short Wave ◽  
Wave Solutions ◽  
Physical Environments

Abstract In this article, the system for the long–short-wave interaction (LS) system is considered. In order to construct some new traveling wave solutions, He’s semi-inverse method is implemented. These solutions may be applicable for some physical environments, such as physics and fluid mechanics. These new solutions show that the proposed method is easy to apply and the proposed technique is a very powerful tool to solve many other nonlinear partial differential equations in applied science.

STABILITY OF STEADY STATE AND TRAVELING WAVES SOLUTIONS IN COUPLED MAP LATTICES

2008 ◽  
Vol 18 (01) ◽  
pp. 219-225 ◽  
Author(s):  
DANIEL TURZÍK ◽  
MIROSLAVA DUBCOVÁ
Keyword(s):  
Steady State ◽  
Essential Spectrum ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Basic Tool ◽  
Wave Solutions ◽  
The Stability

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

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