scholarly journals Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity

2019 ◽  
Vol 8 (1) ◽  
pp. 559-567 ◽  
Author(s):  
M.S. Osman ◽  
Hadi Rezazadeh ◽  
Mostafa Eslami
Keyword(s):  
Applied Sciences ◽  
Power Law ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Mathematical Tool ◽  
Rational Solutions ◽  
Wave Solutions ◽  
The Unified Method ◽  
Soliton Wave Solutions ◽  
Unified Method

Abstract In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.

scholarly journals Multi-soliton rational solutions for some nonlinear evolution equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 26-36 ◽  
Author(s):  
Mohamed S. Osman
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Mathematical Tool ◽  
Rational Solutions ◽  
Generalized Unified Method ◽  
The Generalized Unified Method ◽  
Unified Method

AbstractThe Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota’s method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.

scholarly journals The Two-Variable -Expansion Method for Solving the Nonlinear KdV-mKdV Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
E. M. E. Zayed ◽  
M. A. M. Abdelaziz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Powerful Mathematical Tool

We apply the two-variable (, )-expansion method to construct new exact traveling wave solutions with parameters of the nonlinear ()-dimensional KdV-mKdV equation. This method can be thought of as the generalization of the well-known ()-expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

scholarly journals Abundant Explicit Solutions to Fractional Order Nonlinear Evolution Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
M. Ayesha Khatun ◽  
Mohammad Asif Arefin ◽  
M. Hafiz Uddin ◽  
Mustafa Inc
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Solitary Wave Solutions ◽  
Dark Solitons ◽  
Wave Solutions ◽  
Liouville Derivative

We utilize the modified Riemann–Liouville derivative sense to develop careful arrangements of time-fractional simplified modified Camassa–Holm (MCH) equations and generalized (3 + 1)-dimensional time-fractional Camassa–Holm–Kadomtsev–Petviashvili (gCH-KP) through the potential double G ′ / G , 1 / G -expansion method (DEM). The mentioned equations describe the role of dispersion in the formation of patterns in liquid drops ensued in plasma physics, optical fibers, fluid flow, fission and fusion phenomena, acoustics, control theory, viscoelasticity, and so on. A generalized fractional complex transformation is appropriately used to change this equation to an ordinary differential equation; thus, many precise logical arrangements are acquired with all the freer parameters. At the point when these free parameters are taken as specific values, the traveling wave solutions are transformed into solitary wave solutions expressed by the hyperbolic, the trigonometric, and the rational functions. The physical significance of the obtained solutions for the definite values of the associated parameters is analyzed graphically with 2D, 3D, and contour format. Scores of solitary wave solutions are obtained such as kink type, periodic wave, singular kink, dark solitons, bright-dark solitons, and some other solitary wave solutions. It is clear to scrutinize that the suggested scheme is a reliable, competent, and straightforward mathematical tool to discover closed form traveling wave solutions.

scholarly journals Application of the new approach of generalized (G' /G) -expansion method to find exact solutions of nonlinear PDEs in mathematical physics

BIBECHANA ◽  
2013 ◽  
Vol 10 ◽  
pp. 58-70 ◽  
Author(s):  
Md. Nur Alam ◽  
M Ali Akbar
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Pdes ◽  
Mathematical Tool ◽  
New Approach ◽  
Wave Solutions

The exact solutions of nonlinear evolution equations (NLEEs) play a crucial role to make known the internal mechanism of complex physical phenomena. In this article, we construct the traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation by means of the new approach of generalized (G′ /G) -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G′ /G) -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations. BIBECHANA 10 (2014) 58-70 DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9312

EXACT TRAVELING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS WITH NONLINEAR TERMS OF ANY ORDER

2003 ◽  
Vol 14 (01) ◽  
pp. 99-112 ◽  
Author(s):  
YONG CHEN ◽  
BIAO LI ◽  
HONG-QING ZHANG
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Rational Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, we improved the tanh method by means of a proper transformation and general ansätz. Using the improved method, with the aid of Mathematica™, we consider some nonlinear evolution equations with nonlinear terms of any order. As a result, rich explicit exact traveling wave solutions for these equations, which contain kink profile solitary wave solutions, bell profile solitary wave solutions, rational solutions, periodic solutions, and combined formal solutions, are obtained.

scholarly journals The(G'/G,1/G)-Expansion Method and Its Applications for Solving Two Higher Order Nonlinear Evolution Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
E. M. E. Zayed ◽  
K. A. E. Alurrfi
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Pdes ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Klein Gordon

The two variable(G'/G,1/G)-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear evolution equations, namely, the nonlinear Klein-Gordon equations and the nonlinear Pochhammer-Chree equations. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations are rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original(G'/G)-expansion method proposed by Wang et al. It is shown that the two variable(G'/G,1/G)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

scholarly journals Some New Traveling Wave Exact Solutions of the (2+1)-Dimensional Boiti-Leon-Pempinelli Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jian-ming Qi ◽  
Fu Zhang ◽  
Wen-jun Yuan ◽  
Zi-feng Huang
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Rational Solutions ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We employ the complex method to obtain all meromorphic exact solutions of complex (2+1)-dimensional Boiti-Leon-Pempinelli equations (BLP system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutionsur,2(z)and simply periodic solutionsus,2–6(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

scholarly journals The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations

2021 ◽  
Vol 22 ◽  
pp. 103979
Author(s):  
Nauman Raza ◽  
Muhammad Hamza Rafiq ◽  
Melike Kaplan ◽  
Sunil Kumar ◽  
Yu-Ming Chu
Keyword(s):  
Local Time ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
The Unified Method ◽  
Unified Method

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.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

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