Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas

Dağhan, Durmuş; Dönmez, Orhan

doi:10.1007/s13538-016-0420-9

Cited by 49 publications

(12 citation statements)

References 37 publications

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“…The general form of NLPDEs that contain two or more independent variables which we consider to explore the solutions by using (G ′ /G, 1/G)-expansion method [52,53] can be written as follows:…”

Section: (G ′ /G 1/g)-expansion Methodsmentioning

confidence: 99%

Analytical and numerical approaches to nerve impulse model of fractional‐order

Yavuz

Yokuş

2020

Numerical Methods Partial

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We consider a fractional-order nerve impulse model which is known as FitzHugh-Nagumo (F-N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional-order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave-type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers. KEYWORDS (G ′ /G, 1/G)-expansion method, finite difference scheme, FitzHugh-Nagumo model, Laplace decomposition method, linear stability analysis, numerical analysis 1 INTRODUCTION Nonlinear partial differential equations (NLPDEs) have several applications in engineering and physics. In recent years, the nonlinear fractional-order differential equations have been utilized in

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Section: (G ′ /G 1/g)-expansion Methodsmentioning

confidence: 99%

Analytical and numerical approaches to nerve impulse model of fractional‐order

Yavuz

Yokuş

2020

Numerical Methods Partial

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“…Here we have used the (1⁄G' )-expansion method [12] for calculating the exact solutions. Balancing the terms U' and U 2 in Eq.…”

Section: Exact Solution With (1⁄g' )-Expansion Methodsmentioning

confidence: 99%

Exact solutions of (𝒏+𝟏)-dimensional space-time fractional Zakharov-Kuznetsov equation

Ali

Humanities

Husnine

et al. 2018

Hittite J. Sci. Eng.

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N onlinear partial differential equations (NPDEs) are very useful to model many real world problems in science and engineering. Finding the exact solution of such equations is an important area of research. Fractional differential equations (FDE's) are also getting the attention of the researchers in the recent years. Many real world problems are modelled via FDE's in fluid dynamics. Exact solutions of such models play an important role in the mathematical sciences [1][2][3][4][5][6].Exact solutions for a variety of FDE's are computed by researchers with different techniques including (G'⁄G)-expansion method [7], lie group analysis method [8], new extended trial equation method [9], first integral method [5], exp-function method [10], generalized Kudryashov method [2] and many others are suggested for obtaining the exact solutions of FDE's.In mathematical physics, Zakharov-Kuznetsov (ZK) equation is used to describe the nonlinear development of ion-acoustic waves in magnetized plasma [11]. It is comprised of cold ions and hot isothermal electrons in the presence of a uniform magnetic field. It is also known as the generalizations of the well-known classical KdV equation. In this article, we are interested to

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“…As a pioneer work Li et al [15] has applied the two-variable (G ′ /G, 1/G)-expansion method and found the exact solutions of Zakharov equations. Some applications of the (G ′ /G, 1/G)-expansion method can be seen in [16,17,18,19,20]. (1/G ′ )-expansion method introduced by Yokus [21] firstly.…”

Section: Introductionmentioning

confidence: 99%

“…(1/G ′ )-expansion method introduced by Yokus [21] firstly. Some applications of the (1/G ′ )-expansion method can be seen in [20,22].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for two different non-linear partial differential equations

Dağhan¹,

Esen²

2018

ntmsci

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Travelling wave solutions of the Drinfeld-Sokolov system and Modified Benjamin-Bona-Mahony equations are studied analytically by using two dependent (G ′ /G, 1/G) and (1/G ′)-expansion methods. Solutions are obtained with different forms of functions as hyperbolic, trigonometric and rational functions. These methods are really effective methods, and can be simply applicable to the nonlinear evolution equations encountered in the different physical systems.

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Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas

Abstract: How to citeComplete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative

Cited by 49 publications

References 37 publications

Analytical and numerical approaches to nerve impulse model of fractional‐order

Analytical and numerical approaches to nerve impulse model of fractional‐order

Exact solutions of (𝒏+𝟏)-dimensional space-time fractional Zakharov-Kuznetsov equation

Exact solutions for two different non-linear partial differential equations

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