The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative

Eslami, Mostafa; Khodadad, Farid Samsami; Nazari, Fakhroddin; Rezazadeh, Hadi

doi:10.1007/s11082-017-1224-z

Cited by 104 publications

(27 citation statements)

References 29 publications

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“…Converting the fractional nonlinear partial differential equation to the integer order nonlinear partial differential equation using the following conformable fractional derivative [u(x, t) = u(Θ), Θ = x + c t α α ] (for more details about this kind of derivatives see References [25][26][27][28][29]), we get:…”

Section: Applicationmentioning

confidence: 99%

Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation

Attia

Khater

2019

MCA

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This paper studies the nonlinear fractional undamped Duffing equation. The Duffing equation is one of the fundamental equations in engineering. The geographical areas of this model represent chaos, relativistic energy-momentum, electrodynamics, and electromagnetic interactions. These properties have many benefits in different science fields. The equation depicts the energy of a point mass, which is well thought out as a periodically-forced oscillator. We employed twelve different techniques to the nonlinear fractional Duffing equation to find explicit solutions and approximate solutions. The stability of the solutions was also examined to show the ability of our obtained solutions in the application. The main goals here were to apply a novel computational method (modified auxiliary equation method) and compare the novel method with other methods via the solutions that were obtained by each of these methods.

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Section: Applicationmentioning

confidence: 99%

Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation

Attia

Khater

2019

MCA

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“…Optical solitons are restrained electromagnetic waves that stretch in nonlinear dispersive media and allow the intensity to remain unchanged due to the balance between dispersion and nonlinearity effects [4]. Various analytical approaches for securing optical solitons and other solutions to different kind of NLSEs have been reported to the literature such as the the sine-Gordon expansion method [5][6][7], the first integral method [8,9], the improved Bernoulli sub-equation function method [10,11], the trial solution method [12,13], the new auxiliary equation method [14], the extended simple equation method [15], the solitary wave ansatz method [16], the functional variable method [17], the sub-equation method [18][19][20] and several others [21][22][23][24][25][26][27][28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Optical solitons to the fractional Schrödinger-Hirota equation

Sulaıman

Bulut

Ataş

2019

Applied Mathematics and Nonlinear Sciences

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This study reaches the dark, bright, mixed dark-bright, and singular optical solitons to the fractional Schrödinger-Hirota equation with a truncated M-fractional derivative via the extended sinh-Gordon equation expansion method. Dark soliton describes the solitary waves with lower intensity than the background, bright soliton describes the solitary waves whose peak intensity is larger than the background, and the singular soliton solutions is a solitary wave with discontinuous derivatives; examples of such solitary waves include compactions, which have finite (compact) support, and peakons, whose peaks have a discontinuous first derivative. The constraint conditions for the existence of valid solutions are given. We use some suitable values of the parameters in plotting 3-dimensional surfaces to some of the reported solutions.

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“…And so far a lot of effective methods have been developed and used by many authors. For example, these methods include the finite difference method [6], the ( / ) method [7][8][9][10], the Jacobi elliptic function method [11], the projective Riccati equation method [12], the modified Kudryashov method [13][14][15][16][17][18][19], the exp method [20][21][22][23][24][25], the ansatz method [26], the first integral method [27][28][29], and the subequation method [30][31][32][33][34]. Based on these methods, a lot of fractional partial differential equations have been investigated.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

Meng

Feng

2018

Advances in Mathematical Physics

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In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

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The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative

Cited by 104 publications

References 29 publications

Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation

Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation

Optical solitons to the fractional Schrödinger-Hirota equation

Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

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