scholarly journals An effective technique for the conformable space-time fractional EW and modified EW equations

2019 ◽  
Vol 8 (1) ◽  
pp. 157-163 ◽  
Author(s):  
K. Hosseini ◽  
A. Bekir ◽  
F. Rabiei
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Traveling Wave ◽  
Expansion Method ◽  
Space Time ◽  
Wave Transformation ◽  
Nonlinear Ordinary Differential Equations ◽  
Effective Technique ◽  
Fractional Models

AbstractThe current work deals with the fractional forms of EW and modified EW equations in the conformable sense and their exact solutions. In this respect, by utilizing a traveling wave transformation, the governing space-time fractional models are converted to the nonlinear ordinary differential equations (NLODEs); and then, the resulting NLODEs are solved through an effective method called the exp(−ϕ(ϵ))-expansion method. As a consequence, a number of exact solutions to the fractional forms of EW and modified EW equations are generated.

scholarly journals Regarding on the exact solutions for the nonlinear fractional differential equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 478-482 ◽  
Author(s):  
Melike Kaplan ◽  
Murat Koparan ◽  
Ahmet Bekir
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Simple Equation ◽  
Wave Transformation ◽  
Nonlinear Ordinary Differential Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

AbstractIn this work, we have considered the modified simple equation (MSE) method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW) and the modified equal width (mEW) equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

Application of ODE techniques to spherical gravitational collapse: Methods and results

2016 ◽  
Vol 13 (05) ◽  
pp. 1630005
Author(s):  
Roberto Giambò ◽  
Fabio Giannoni ◽  
Giulio Magli
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Gravitational Collapse ◽  
Equations Of State ◽  
Matter Field ◽  
Nonlinear Ordinary Differential Equations ◽  
Field Equations ◽  
Final State ◽  
Light Rays

The final state of spherical gravitational collapse can be analyzed applying to the geodesic equations governing the behavior of light rays near the singularity relatively simple but powerful techniques of nonlinear ordinary differential equations. In this way, explicit use of exact solutions of Einstein’s field equations is not necessary, and results can be obtained for wide equations of state of the collapsing matter field.

scholarly journals Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Rational Solutions ◽  
Partial Differential ◽  
Wave Solutions

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

scholarly journals Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

2004 ◽  
Vol 59 (7-8) ◽  
pp. 443-454 ◽  
Author(s):  
Nikolai A. Kudryashov
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Integrable Equation ◽  
Main Step ◽  
Nonlinear Ordinary Differential Equations ◽  
Weierstrass Function ◽  
Special Solutions ◽  
Pacs 02.30

A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear differential equations have exact solutions which are general solution of the simplest integrable equation. We use the Weierstrass elliptic equation as building block to find a number of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutionsexpressed via theWeierstrass function are given. - PACS: 02.30.Hq (Ordinary differential equations)

scholarly journals Novel Exact Solutions of the Extended Shallow Water Wave and the Fokas Equations

2018 ◽  
Vol 22 ◽  
pp. 01022
Author(s):  
Serbay DURAN ◽  
Berat KARAAGAC ◽  
Alaattin ESEN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Expansion Method ◽  
Travelling Wave ◽  
Wave Transformation ◽  
Partial Differential ◽  
Shallow Water Wave

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations.

scholarly journals Analytical solitons for the space-time conformable differential equations using two efficient techniques

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmad Neirameh ◽  
Foroud Parvaneh
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Space Time ◽  
Nonlinear Phenomena ◽  
Approximate Analytical Solutions ◽  
Coupled Burgers Equation ◽  
Mathematics And Physics

AbstractExact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

scholarly journals Explicit Exact Solutions of Space-time Fractional Drinfel’d-Sokolov-Wilson Equations

2021 ◽  
Vol 2068 (1) ◽  
pp. 012005
Author(s):  
Hongkua Lin
Keyword(s):  
Partial Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Rational Function ◽  
Expansion Method ◽  
Space Time ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Fractional Partial Differential Equations ◽  
Function Type

Abstract The space-time fractional Drinfel’d-Sokolov-Wilson equations (DSWEs) has attracted many researchers’ attention in recent years. In this study, combining the (G’/G,1/G)-expansion method and a fractional wave transformation, we derive abundant explicit exact solutions of the DSWEs with the conformable fractional derivative. All of the resulting solutions include triangle, hyperbolic and rational function type. It shows this technique is effective and reliable to find exact solutions of other similar nonlinear fractional partial differential equations (NFPDEs).

scholarly journals A Fresh Look To Exact Solutions of Some Coupled Equations

2018 ◽  
Vol 22 ◽  
pp. 01006
Author(s):  
Berat Karaagac ◽  
Nuri Murat Yagmurlu ◽  
Alaattin Esen ◽  
Selcuk Kutluay
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Wave Transformation ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Coupled Equations ◽  
New Exact Solutions

This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine- Gordon expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.

scholarly journals Exact Solutions of the Space-Time Fractional Bidirectional Wave Equations Using the(G′/G)-Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wei Li ◽  
Huizhang Yang ◽  
Bin He
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Wave Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Space Time ◽  
Promising Tool ◽  
Liouville Derivative ◽  
Fractional Differential

Based on Jumarie’s modified Riemann-Liouville derivative, the fractional complex transformation is used to transform fractional differential equations to ordinary differential equations. Exact solutions including the hyperbolic functions, the trigonometric functions, and the rational functions for the space-time fractional bidirectional wave equations are obtained using the(G′/G)-expansion method. The method provides a promising tool for solving nonlinear fractional differential equations.

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