scholarly journals SOLVING SOME IMPORTANT NONLINEAR TIME-FRACTIONAL EVOLUTION EQUATIONS BY USING THE (G’/G)-EXPANSION METHOD

2020 ◽  
Vol 20 (4) ◽  
pp. 815-832
Author(s):  
MEDJAHED DJILALI ◽  
ALI HAKEM
Keyword(s):  
Wave Equation ◽  
Software Package ◽  
Water Wave ◽  
Evolution Equations ◽  
Expansion Method ◽  
Fisher Equation ◽  
Fractional Evolution Equations ◽  
Complex Transformation ◽  
Linear Ode ◽  
Mathematica Software

The aim of the present study is to find the exact solutions for three generalized nonlinear time-fractional evolution equations, Kaup-Kupershmidt equation, Burgers-Fisher equation and, Shallow Water Wave equation. By using the (G’/G)-expansion method and depending on second order linear ODE as well as the complex transformation, three kinds of solutions (hyperbolic, trigonometric, and rational) are obtained. With the help of Mathematica software package, difficult algebraic systems are solved and surfaces of some particular solutions of the equations under study are plotted.

New Exact Solutions of (2+1)-Dimensional Generalization of Shallow Water Wave Equation by (G′/G)-Expansion Method

2010 ◽  
Vol 20-23 ◽  
pp. 1516-1521 ◽  
Author(s):  
Bang Qing Li ◽  
Mei Ping Xu ◽  
Yu Lan Ma
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Function Solution ◽  
Shallow Water Wave

Extending a symbolic computation algorithm, namely, (G′/G)-expansion method, a new series of exact solutions are constructed for (2+1)-dimensional generalization of shallow water wave equation. These solutions included hyperbolic function solution, trigonometric function solution and rational function solution. The procedure can illustrate that the new algorithm is concise, powerful and straightforward, and it can also be applied to find exact solutions for other high dimensional nonlinear evolution equations.

scholarly journals New Complex Hyperbolic Structures to the Lonngren-Wave Equation by Using Sine-Gordon Expansion Method

2019 ◽  
Vol 4 (1) ◽  
pp. 129-138 ◽  
Author(s):  
Haci Mehmet Baskonus ◽  
Hasan Bulut ◽  
Tukur Abdulkadir Sulaiman
Keyword(s):  
Wave Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Structures ◽  
Computational Program ◽  
Hyperbolic Function Solutions

AbstractIn this paper, a powerful sine-Gordon expansion method (SGEM) with aid of a computational program is used in constructing a new hyperbolic function solutions to one of the popular nonlinear evolution equations that arises in the field of mathematical physics, namely; longren-wave equation. We also give the 3D and 2D graphics of all the obtained solutions which are explaining new properties of model considered in this paper. Finally, we submit a comprehensive conclusion at the end of this paper.

Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
H. Jafari ◽  
H. Tajadodi ◽  
D. Baleanu
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Telegraph Equation ◽  
Fisher Equation ◽  
Fractional Evolution Equations ◽  
Liouville Derivative ◽  
Nonlinear Telegraph Equation ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn–Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann–Liouville derivative. Some relevant examples are investigated.

scholarly journals Some New Non-Travelling Wave Solutions of the Fisher Equation with Nonlinear Auxiliary Equation

2018 ◽  
Vol 3 (2) ◽  
pp. 92-101
Author(s):  
Anika Tashin Khan ◽  
Hasibun Naher
Keyword(s):  
Evolution Equations ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Evolution Equations ◽  
Fisher Equation ◽  
Auxiliary Equation ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Functional Forms

We have generated many new non-travelling wave solutions by executing the new extended generalized and improved (G'/G)-Expansion Method. Here the nonlinear ordinary differential equation with many new and real parameters has been used as an auxiliary equation. We have investigated the Fisher equation to show the advantages and effectiveness of this method. The obtained non-travelling solutions are expressed through the hyperbolic functions, trigonometric functions and rational functional forms. Results showing that the method is concise, direct and highly effective to study nonlinear evolution equations those are in mathematical physics and engineering.

Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G′/G$\left({\boldsymbol{G}}^{\prime }/\boldsymbol{G}\right)$-expansion method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Onur Alp Ilhan ◽  
Haci Mehmet Baskonus ◽  
M. Nurul Islam ◽  
M. Ali Akbar ◽  
Danyal Soybaş
Keyword(s):  
Acoustic Waves ◽  
Electromagnetic Waves ◽  
Evolution Equations ◽  
Nonlinear Differential Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Ion Acoustic Waves ◽  
Biological Population ◽  
Complex Transformation ◽  
Internal Structures

Abstract The time-fractional generalized biological population model and the (2, 2, 2) Zakharov–Kuznetsov (ZK) equation are significant modeling equations to analyse biological population, ion-acoustic waves in plasma, electromagnetic waves, viscoelasticity waves, material science, probability and statistics, signal processing, etc. The new generalized G ′ / G $\left({G}^{\prime }/G\right)$ -expansion method is consistent, computer algebra friendly, worthwhile through yielding closed-form general soliton solutions in terms of trigonometric, rational and hyperbolic functions associated to subjective parameters. For the definite values of the parameters, some well-established and advanced solutions are accessible from the general solution. The solutions have been analysed by means of diagrams to understand the intricate internal structures. It can be asserted that the method can be used to compute solitary wave solutions to other fractional nonlinear differential equations by means of fractional complex transformation.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

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