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

Al-Ghafri, K. S.; Rezazadeh, Hadi

doi:10.2478/amns.2019.2.00026

Cited by 84 publications

(36 citation statements)

References 27 publications

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“…Thus, we extend and improve the existing theory and previous works on Lotka-Volterra and related models in population biology to the fractional-like case. Indeed, the recent studies and experiments on fractional systems indicated that fractional models are more effective than integer-order models in numerous applications mainly because of their nonlocal properties [1][2][3][4][5][6][7][8][9][10][11][12][13]. In addition, the FLDs have important advantages in computational aspects than classical fractional derivatives, such as Caputo or Riemann-Liouville types [17][18][19][20][21][22][23][24][25][26][27][28][29], which make them more appropriate for applications.…”

Section: There Exists a Function H(t U) Such Thatmentioning

confidence: 99%

“…See, for example, the books [1][2][3] for basic results of systems with fractional derivatives of Riemann-Liouville and Caputo types. Parallel to the development of the theory of fractional systems, numerous definitions of fractional derivatives have been introduced, such as an Atangana-Baleanu fractional derivative, Hadamard-type fractional derivative, Riesz-Miller derivative, and Chen-Machado derivative, just to mention a few [4][5][6][7][8][9][10][11][12][13]. The papers [14][15][16] offered a comprehensive overview and classifications of different types of fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

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Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

2019

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In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka–Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated.

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Section: There Exists a Function H(t U) Such Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

2019

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“…In 2015, Caputo and Fabrizio discovered a new operator of arbitrary order, namely, Caputo‐Fabrizio (CF) operator with arbitrary order and enforced to the several linear and nonlinear physical problems . In 2016, Atangana and Baleanu introduced another nonsingular derivative based on Miitag‐Leffler kernel and applied to the many problems . The parabolic heat equation was first developed and introduced Joseph Fourier in 1822.…”

Section: Introductionmentioning

confidence: 99%

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Kumar

Ghosh

Samet

et al. 2020

Math Methods in App Sciences

174

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The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multi dimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator. KEYWORDS HPTM and RPSM, multidimensional heat equations, new fractional derivative, new YAC operator, Prabhakar or generalized Mittag-Leffler function MSC CLASSIFICATION 26A33; 34A08; 34A34; 60G22 Math Meth Appl Sci. 2020;43:6062-6080. wileyonlinelibrary.com/journal/mma

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“…Thus, a variety of powerful mathematical approaches have been developed to derive soliton solutions for many physical models. For more details, readers are referred to references [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Soliton Behaviours for the Conformable Space–Time Fractional Complex Ginzburg–Landau Equation in Optical Fibers

Al-Ghafri

2020

Symmetry

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In this work, we investigate the conformable space–time fractional complex Ginzburg–Landau (GL) equation dominated by three types of nonlinear effects. These types of nonlinearity include Kerr law, power law, and dual-power law. The symmetry case in the GL equation due to the three types of nonlinearity is presented. The governing model is dealt with by a straightforward mathematical technique, where the fractional differential equation is reduced to a first-order nonlinear ordinary differential equation with solution expressed in the form of the Weierstrass elliptic function. The relation between the Weierstrass elliptic function and hyperbolic functions enables us to derive two types of optical soliton solutions, namely, bright and singular solitons. Restrictions for the validity of the optical soliton solutions are given. To shed light on the behaviour of solitons, the graphical illustrations of obtained solutions are represented for different values of various parameters. The symmetrical structure of some extracted solitons is deduced when the fractional derivative parameters for space and time are symmetric.

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

Cited by 84 publications

References 27 publications

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Soliton Behaviours for the Conformable Space–Time Fractional Complex Ginzburg–Landau Equation in Optical Fibers

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