The First Integral Method for the time fractional Kaup-Boussinesq System with time dependent coefficient

Kılıç, Bülent; İnç, Mustafa

doi:10.1016/j.amc.2014.12.094

Cited by 24 publications

(10 citation statements)

References 21 publications

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“…In literature, exact solutions of fractional differential equations have attracted the attention of researcher from different fields. Several research works have proposed techniques for solving fractional differential equations, such as G ′ /G−expansion method [3,28,30] , Exp−function method [12,29], first integral method [1,8,19,22], sub−equation method [13,24,32], Jacobi elliptic funtion method [9,11,26,31] , modified Kudryashov method [4,5,6,17,20], extended tanh method [7],modified simple equation method [18] and others.…”

Section: Introductionmentioning

confidence: 99%

A new method for solving nonlinear fractional differential equations

Mısırlı¹,

Ege²

2017

ntmsci

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Abstract:In this paper, a new extended Kudryashov method for solving fractional nonlinear differential equations is proposed. The fractional derivative in this paper is considered in the sense of modified Riemann-Liouville. We also handle the time-fractional fifth−order Sawada−Kotera equation and the time-fractional generalized Hirota−Satsuma coupled KdV equation to illustrate the the simplicity and the effectiveness of this method. Solutions of these equations are obtained in analytical traveling wave solution form including hyperbolic and trigonometric functions.

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Section: Introductionmentioning

confidence: 99%

A new method for solving nonlinear fractional differential equations

Mısırlı¹,

Ege²

2017

ntmsci

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“…if we prove it for every i, j ≥ 1. Finally, observe that items (a) and (c) of Proposition 27 and a straightforward computation ensures the equivalence of (27) and…”

Section: But Item (D) Allows Us To Concludementioning

confidence: 90%

“…In the framework of fractional calculus we cannot expect analogous formulas to (1) and (2), mainly because fractional derivatives have a non-local behavior, sometimes called "memory property", that is not compatible with these identities (see [38] for more details on the concept that underlies this notion). Despite of the fact that some authors proclaim that their fractional versions of derivative satisfy these equalities (see for instance [8,23,27,56] and several others), Tarasov and Liu have already constructed sufficiently convincing arguments that invalidate such claim, as can be seen in [32,48,49,50].…”

Section: Introductionmentioning

confidence: 99%

On the fractional version of Leibniz rule

Neto

Júnior

2020

Mathematische Nachrichten

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This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann–Liouville and Caputo fractional derivatives of order α∈(0,1). In the context of partial differential equations, the aforesaid inequality allows us to address the Faedo–Galerkin method to study several kinds of partial differential equations with fractional derivative in the time variable; particularly, we apply these ideas to prove the existence and uniqueness of solution to the fractional version of the 2D unsteady Stokes equations in bounded domains.

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“…Similarly, the solutions of fractional differential equations have attracted the attention of researchers in many fields. Thus, many methods have been proposed one after another, such as the first integral method [34,35], (G'/G)expansion method [36,37], the Hirota method [38,39], the trial function method [40], the subequation method [41], and others [42][43][44][45][46]. In this paper, using extended Kudryashov method [47], the exact solution of (3+1)-dimensional TSF-ZK equation is obtained.…”

Section: Introductionmentioning

confidence: 99%

An Application of (3+1)‐Dimensional Time‐Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

Yang

2019

Complexity

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Dust plasma is a new field of physics which has developed rapidly in recent decades. The study of dust plasma has received much attention due to its importance in the environment of space and the Earth. Dust acoustic waves are generated because of the inertia of dust mass while the restoring force is provided by the thermal pressure of electrons and ions. Since dust acoustic waves were first reported theoretically in unmagnetized dust plasma by Rao et al., they have become a research hot spot. In this paper, the excitation of dust acoustic waves by a gravity field in a dust plasma is analyzed. According to the control equations of dust plasma motion and employing multiscale analysis and perturbation method, we have obtained a (3+1)-dimensional ZK model. Because of the space property of dust plasma, (3+1)-dimensional ZK equation is more suitable than KdV equation and (2+1)-dimensional ZK equation to describe the real dust acoustic waves. Then, the (3+1)-dimensional time-space fractional ZK (TSF-ZK) equation describing the fractal process of nonlinear dust acoustic waves is given for the first time. To further explore how dust acoustic waves change energy as they travel, we discuss the conservation laws of the new model. Moreover, we study the exact solution of (3+1)-dimensional TSF-ZK equation by using extended Kudryashov method. Finally, based on the exact solution, we further investigate the effect of the parameter k, the charge properties of dust particle Zd0, the fractional order values α, β, γ, and θ, the temperature Td, the gravity g, and the collision frequency β0 and β1 on the properties of dust acoustic waves by a gravity field in dust plasma.

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The First Integral Method for the time fractional Kaup-Boussinesq System with time dependent coefficient

Cited by 24 publications

References 21 publications

A new method for solving nonlinear fractional differential equations

A new method for solving nonlinear fractional differential equations

On the fractional version of Leibniz rule

An Application of (3+1)‐Dimensional Time‐Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves

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