Exact traveling wave solutions for two prolific conformable M-Fractional differential equations via three diverse approaches

Siddique, Imran; Jaradat, Mohammed M. M.; Zafar, Asim; Mehdi, Khush Bukht; Osman, M. S.

doi:10.1016/j.rinp.2021.104557

Cited by 94 publications

(34 citation statements)

References 49 publications

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“…Over the last several decades, one of the most significant challenges has been the development of new methods to construct exact solutions for NLPD equations. In recent years, several new, more powerful, and effective approaches have been established to retrieve exact solutions of NLPD equations, the sine-Gordon expansion method [1][2][3][4], the (𝐺 ′ 𝐺 ⁄ )-expansion method [5][6][7][8], the Sardar sub-equation method [9][10][11][12], the Kudryashov method [13][14][15][16][17][18], and the exponential method [19][20][21][22][23][24][25], are examples to mention.…”

Section: Introductionmentioning

confidence: 99%

Optical solitons to the Ginzburg–Landau equation including the parabolic nonlinearity

Hosseini¹,

Mirzazadeh

Akinyemi

et al. 2022

Opt Quant Electron

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The major goal of the present paper is to construct optical solitons of the Ginzburg-Landau (GL) equation including the parabolic nonlinearity. Such an ultimate goal is formally achieved with the aid of symbolic computation, a complex transformation, and Kudryashov and exponential methods. Several numerical simulations are given to explore the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons. To the best of the authors' knowledge, the results reported in the current study, classified as bright and kink solitons, are new and have been acquired for the first time.

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Section: Introductionmentioning

confidence: 99%

Optical solitons to the Ginzburg–Landau equation including the parabolic nonlinearity

Hosseini¹,

Mirzazadeh

Akinyemi

et al. 2022

Opt Quant Electron

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“…Researchers are therefore motivated to present new methods and refine existing approaches. Various significant and powerful methods have been introduced such as Darboux transformation [14], Weierstrass elliptic functions methods [15,16], Bäcklund transformation [17], Lie group [18][19][20][21], Hirota's method [22,23], bifurcation method [24][25][26][27][28][29][30][31][32][33], and for distinct method, as shown in e.g., [34][35][36][37][38][39][40][41][42]. e analytical and numerical solutions for various types of nonlinear partial differential equations were investigated using traditional Lie symmetry approaches; for instance as shown in [43].…”

Section: Introductionmentioning

confidence: 99%

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

Elmandouh

Elbrolosy

2022

Journal of Mathematics

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The Ivancevic option pricing model comes as an alternative to the Black-Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well-known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright-like soliton, kinky-bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D-graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.

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“…There are several definitions of fractional derivatives such as Riemann Liouville [2], conformable fractional derivative [3], beta derivative [4], and new truncated M-fractional derivative [5] are available in literature. Many powerful methods for obtaining exact solutions of nonlinear fractional PDEs have been presented as Hirota's bilinear method [6], sinecosine method [7], tanh-function method [8], exponential rational function method [9], Kudryashov method [10], sine-Gordon expansion method [11], modified ðG ′ /GÞ -expansion method [12], extended ðG ′ /GÞ-expansion method [13], ðG ′ /GÞ-expansion method [14], tanh-coth expansion method [15], Jacobi elliptic function expansion method [16], first integral method [17], sardar-subequation method [18], new subequation method [19], extended direct algebraic method [20], exp ð−ϕðηÞÞ method [21], Exp a function method [22], ð1/G′Þ, ðG′/G, 1/GÞ, and modified ðG′/ G 2 Þ − expansion methods [23,24], Kudryashov method [25], modified expansion function method [26], new auxiliary equation method [27], extended Jacobi's elliptic expansion function method [28], extended sinh-Gordon equation expansion method [29], modified simplest equation method [30], and many more.…”

Section: Introductionmentioning

confidence: 99%

Diverse Exact Soliton Solutions of the Time Fractional Clannish Random Walker’s Parabolic Equation via Dual Novel Techniques

Siddique

Mehdi

Matkan

et al. 2022

Journal of Function Spaces

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In this article, we acquire a variety of new exact traveling wave solutions in the form of trigonometric, hyperbolic, and rational functions for the nonlinear time-fractional Clannish Random Walker’s Parabolic (CRWP) equation in the sense of beta-derivative by employing the two modified methods, namely, modified G ′ / G 2 − expansion method and modified F − expansion method. The obtained solutions are verified for aforesaid equations through symbolic soft computations. To promote the essential propagated features, some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the precise values to the parameters under the constrain conditions. The obtained solutions show that the presented methods are effective, straight forward, and reliable as compared to other methods. These methods can also be used to extract the novel exact traveling wave solutions for solving any types of integer and fractional differential equations arising in mathematical physics.

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Exact traveling wave solutions for two prolific conformable M-Fractional differential equations via three diverse approaches

Cited by 94 publications

References 49 publications

Optical solitons to the Ginzburg–Landau equation including the parabolic nonlinearity

Optical solitons to the Ginzburg–Landau equation including the parabolic nonlinearity

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

Diverse Exact Soliton Solutions of the Time Fractional Clannish Random Walker’s Parabolic Equation via Dual Novel Techniques

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