Extended tanh-function method and its applications to nonlinear equations
TL;DR: In this article, an extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way, and the key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh function.
About: This article is published in Physics Letters A.The article was published on 2000-12-04. It has received 1830 citations till now. The article focuses on the topics: Partial differential equation & Nonlinear system.
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TL;DR: The (G'/G)-expansion method is firstly proposed in this paper, where G = G(xi) satisfies a second order linear ordinary differential equation (LODE for short), by which the travelling wave solutions involving parameters of the KdV equation, the mKdV equations, the variant Boussinesq equations and the Hirota-Satsuma equations are obtained when the parameters are taken as special values.
1,673 citations
TL;DR: In this article, a Jacobi elliptic function expansion method was proposed to construct the exact periodic solutions of nonlinear wave equations, which includes some shock wave solutions and solitary wave solutions.
1,231 citations
TL;DR: In this paper, a new method is presented to look for exact solutions of nonlinear differential equations by using the general solutions of the simplest nonlinear equations and taking into consideration all possible singularities of equation studied.
Abstract: New method is presented to look for exact solutions of nonlinear differential equations. Two basic ideas are at the heart of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another idea is to take into consideration all possible singularities of equation studied. Application of our approach to search exact solutions of nonlinear differential equations is discussed in detail. The method is used to look for exact solutions of the Kuramoto–Sivashinsky equation and the equation for description of nonlinear waves in a convective fluid. New exact solitary and periodic waves of these equations are given.
568 citations
TL;DR: A comparative study of deep techniques in image denoising by classifying the deep convolutional neural networks for additive white noisy images, the deep CNNs for real noisy images; the deepCNNs for blind Denoising and the deep network for hybrid noisy images.
518 citations
Cites background from "Extended tanh-function method and i..."
...The activation functions such as Sigmoid [139, 93] and Tanh [81, 47] can convert the linear input into non-linearity through W and b as follows....
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...The activation functions such as Sigmoid [139, 93] and Tanh [81, 47] can convert the linear input into non-linearity through W and b as follows. f(X;W ; b) = f(W TX + b)....
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TL;DR: In this article, a direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations, which provides a more systematical and convenient handling of the solution process of non-linear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, and the mapping method.
Abstract: A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.
510 citations
Cites methods from "Extended tanh-function method and i..."
...Later it was broadly adopted in the extended tanh-function method [13,14], the tanh–coth method [16], the F-expansion method [19], the mapping method [20,36], and the extended F-expansion method [21,22]....
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...If v is a polynomial and r 1⁄4 0, then u gives an exact solution in the form of a finite series in tanh n or tan n, which is obtainable by the tanh-function type method [9,12,13]....
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References
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TL;DR: In this article, a method for obtaining traveling-wave solutions of nonlinear wave equations that are essentially of a localized nature is proposed based on the fact that most solutions are functions of a hyperbolic tangent.
Abstract: A method is proposed for obtaining traveling‐wave solutions of nonlinear wave equations that are essentially of a localized nature. It is based on the fact that most solutions are functions of a hyperbolic tangent. This technique is straightforward to use and only minimal algebra is needed to find these solutions. The method is applied to selected cases.
1,394 citations
TL;DR: In this article, the solitary wave solutions of the approximate equations for long water waves, coupled KdV equations and the dispersive long wave equations in 2 + 1 dimensions are constructed by using a homogeneous balance method.
865 citations
TL;DR: In this article, the tanh-function method for finding explicit travelling solitary wave solutions to non-linear evolution equations is described, and a Mathematica package ATFM is presented to deal with the tedious algebra and outputs directly the required solutions.
790 citations
573 citations
TL;DR: Based on the homogeneous balance method, a simple and efficient method for obtaining exact solutions of nonlinear partial differential equations is proposed in this paper, where some equations are investigated by this means and new solitary wave solutions or singular traveling wave solutions are found.
543 citations