Application of $$\tan (\phi (\xi )/2)$$ tan ( ϕ ( ξ ) / 2 ) -expansion method for the time-fractional Kuramoto–Sivashinsky equation

Manafian, Jalil; Foroutan, Mohammadreza

doi:10.1007/s11082-017-1107-3

Cited by 25 publications

(12 citation statements)

References 52 publications

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“…Seeking different types of solutions to FDEs for such phenomena has become the subject of interest for researchers. Thus, a lot of powerful mathematical methods have been applied to obtain exact analytic solutions of FDEs, namely, the extended tanh-function method [6,7], the exp-function method [8,9], the sub-equation method [10,11], the improved tan(φ /2)-expansion method [12,13], the (G /G)-expansion method [14,15], the modified trial equation method [16,17], the new extended direct algebraic method [18,19], the extended sinh-Gordon equation expansion method [20,21], the unified method [22] and so on.…”

Section: Introductionmentioning

confidence: 99%

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Al-Ghafri

Rezazadeh

2019

Applied Mathematics and Nonlinear Sciences

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In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

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Section: Introductionmentioning

confidence: 99%

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Al-Ghafri

Rezazadeh

2019

Applied Mathematics and Nonlinear Sciences

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“…( ) tan( ) 2   -expansion method has been applied by many authors [19,20,21,22,23]. In [19], ITEM has been simplified and called simplified ITEM (SITEM).…”

Section: Recentlymentioning

confidence: 99%

SITEM for the Conformable Space-Time Fractional Coupled KD Equations

Yaslan¹,

Girgin²

2018

Journal of Engineering Technology and Applied Sciences

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In the present paper, new analytical solutions for the space-time fractional coupled Konopelchenko-Dubrovsky (KD) equations are obtained by using the simplified () tan() 2  -expansion method (SITEM). Here, fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the trigonometric, hyperbolic, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.

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“…Numerous kinds of exact solutions such as solitons, positons, complexitons, dromions, cuspon, rational, kink, periodic, and quasiperiodic solutions have been obtained via solving integrable NLEEs. In the past couple of decades, many robust, efficient, and powerful methods exist that have been developed for finding exact solutions of NLEEs, including the (G /G, 1/G)-expansion method [11], the enhanced (G /G)-expansion method [12], the exp-function method [13], the Jacobi elliptic equation method [14], the generalized Kudryashov's method [15], the sine-Gordon expansion method [16], the sub-equation method [17], the improved tan(φ/2)-expansion method [18], and the extended direct algebraic method [19,20]. More recently, the (G /G 2 )-expansion method [4,[21][22][23][24][25][26][27][28][29] has attracted a remarkable amount of attention of many researchers who employed the method to construct exact solutions of certain NPDEs.…”

Section: Introductionmentioning

confidence: 99%

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

et al. 2021

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The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

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Application of $$\tan (\phi (\xi )/2)$$ tan ( ϕ ( ξ ) / 2 ) -expansion method for the time-fractional Kuramoto–Sivashinsky equation

Cited by 25 publications

References 52 publications

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

SITEM for the Conformable Space-Time Fractional Coupled KD Equations

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

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