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Journal ArticleDOI

On the New Solutions of the Conformable Time Fractional Generalized Hirota-Satsuma Coupled KdV System

TL;DR: In this paper, the generalized Hirota Satsuma coupled KdV system is solved with tanh method and q-Homotopy analysis method. But, the problem of conformable fractional derivative is not addressed.
Abstract: Abstract In this paper, generalized Hirota Satsuma coupled KdV system is solved with tanh method and q-Homotopy analysis method. New fractional derivative definition called “conformable fractional derivative” used in the solution procedure. Tanh method with conformable derivative firstly introduced in the literature. By the graphics of analytical and approximate solutions, it is shown that, both methods provide an effective and powerful mathematical tool for solving nonlinear PDEs containing conformable fractional derivative.

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Citations
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Journal ArticleDOI
TL;DR: In this article, the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity was considered and solved by means of a unified method.
Abstract: Abstract In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.

89 citations

Journal ArticleDOI
TL;DR: In this paper, the authors employed Jacobi elliptic function expansion method to build the new wave solutions of time fractional modified Camassa-Holm equation which is completely integrable dispersive shallow-water equation.

23 citations

01 Jan 2004
TL;DR: In this paper, the authors used the tanh (or hyperbolic tangent) method to look for travelling waves when dealing with one-dimensional non-linear wave and evolution equations.
Abstract: The tanh (or hyperbolic tangent) method is a powerful technique to look for travelling waves when dealing with one-dimensional non-linear wave and evolution equations. In particular, this method is well suited for those problems where dispersion, convection and reaction–diffusion play an important role. To show the strength of this method we study a coupled set (the so-called Boussinesq equations) which arises in the theory of non-linear dispersive water waves. As a result, a solitary wave profile is found which generalizes an earlier result, the famous Korteweg-de Vries solitary wave solution. Copyright © 2005 John Wiley & Sons, Ltd.

22 citations

Journal ArticleDOI
TL;DR: In this article, the potential Kadomtsev-Petviashvili (pKP) equation is solved by the new extended direct algebraic method, and the behavior of the obliquely interacting surface waves in two dimensions can be analyzed.
Abstract: In this study, the potential Kadomtsev-Petviashvili (pKP) equation, which describes the oblique interaction of surface waves in shallow waters, is solved by the new extended direct algebraic method. The results of the study show that by applying the new direct algebraic method to the pKP equation, the behavior of the obliquely interacting surface waves in two dimensions can be analyzed. This article fairly clarifies the behaviors of surface waves in shallow waters. In the literature, several mathematical models have been developed in attempt to study these behaviors, with nonlinear mathematics being one of the most important steps; however, the investigations are still at a level that can be called ‘baby steps’. Therefore, every study to be carried out in this context is of great importance. Thus, this study will serve as a reference to guide scientists working in this field.

21 citations

Journal ArticleDOI
TL;DR: In this paper, the authors introduced the concept of conformable variable order derivative (CVO derivative), which is a local operator based on the conformable derivative and has many properties like usual integer derivative's ones.
Abstract: In this paper, based on the conformable derivative, we introduce the concept of conformable variable order derivative. The conformable derivative is a local operator, it has many properties like usual integer derivative’s ones. Similar to the conformable derivative, we study some properties of the conformable variable order derivative. We investigate the fundamental solutions to initial value problem for linear homogeneous and inhomogeneous diffusion differential equations with the conformable variable order derivative. In addition, using upper and lower solutions and monotone iterative method, we consider the existence and uniqueness of solutions to an initial value problem for nonlinear diffusion differential equations involving with the conformable variable order derivative.

20 citations

References
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Book
01 Jan 1999

15,898 citations

Book
19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

7,643 citations

Book
27 Oct 2003
TL;DR: In this paper, a simple bifurcation of a nonlinear problem multiple solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free oscillations with Quadratic nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous flow Boundary-layer Flow Boundarylayer Flow with Exponential Property Boundary Layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGR
Abstract: PART I BASIC IDEAS Introduction Illustrative Description Systematic Description Relations to Some Previous Analytic Methods Advantages, Limitations, and Open Questions PART II APPLICATIONS Simple Bifurcation of a Nonlinear Problem Multiple Solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free Oscillation Systems with Quadratic Nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous Flow Boundary-layer Flow with Exponential Property Boundary-layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGRAPHY INDEX

2,831 citations


Additional excerpts

  • ...Comp., 154, (2004), 713 [19] W. Malfliet, The tanh method: a tool for solving certain classes of non-linear PDEs, Mathematical methods in the applied sciences, 28.17, (2005), 2031-2035 [20] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, Journal of Computational and Applied Mathematics, 164, (2004), 529-541 [21] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003 Yucel Çenesiz Address of first author Department of Mathematics, Selçuk University, Konya Türkiye....

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  • ...We choose an appropriate value of ~ which guarantee that the series solution is convergent, as pointed by Liao [21], by finding the valid region of ~ which corresponds to the line segments nearly parallel to the horizontal axis....

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Journal ArticleDOI
TL;DR: A new definition of fractional derivative and fractional integral is given and it is shown that it is the most natural definition, and the most fruitful one.

2,068 citations

Journal ArticleDOI
TL;DR: The basic concepts in this new simple interesting fractional calculus called conformable fractional derivative are set and the fractional versions of chain rule, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions, Laplace transforms and linear differential systems are proposed and discussed.

1,331 citations


"On the New Solutions of the Conform..." refers background in this paper

  • ...Abdeljawad [13] has deduced fractional versions of the chain rule, exponential functions, Gronwalls inequality, integration by parts, Taylor power series expansions and Laplace transform....

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