Extended simplest equation method for nonlinear differential equations
type: Item
instance of: scholarly article
title: Extended simplest equation method for nonlinear differential equations
zbMATH Open document ID: 1168.34003
author: Nikolai A. Kudryashov
author: Nadejda B. Loguinova
published in: Applied Mathematics and Computation
publication date: 2009-01-14T00:00:00Z
review text: The authors consider the equation \[ P(y,y',y'',\dots)=0, \tag{1} \] where \(y=y(z)\) is an unknown function, \(P\) is a polynomial in the variable \(y\) and its derivatives and look for exact solutions \(y=y(z)\) of the form \[ y(z)=\sum_{k=0}^NA_k\left( \frac{\psi '}{\psi} \right)^k, \tag{2} \] \(A_k= \text{const}\), \(A_N\neq 0\), where the function \(\psi=\psi(z)\) is the general solution of the linear ordinary differential equation \[ \psi ''' +\alpha\psi '' +\beta \psi ' +\gamma \psi=0, \tag{3} \] \(\alpha, \beta, \gamma =\text{const}\). They propose the algorithm for searching the parameters \(N,A_k,\) \(k=1,\dots,N\), \(\alpha,\beta,\gamma\). This approach for the exact solution of the equation (1) the authors call the extended simplest equation method. They apply this method to the Sharma-Tasso-Olver and the Burgers-Huxley equations. New exact solutions of these equations are obtained.
reviewed by: Vladimir L. Makarov
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 5490894
zbMATH Keywords: nonlinear evolution equations
zbMATH Keywords: exact solutions
zbMATH Keywords: simplest equation method
zbMATH Keywords: Sharma-Tasso-Olver equation
zbMATH Keywords: Burgers-Huxley equation
describes a project that uses: ATFM
MaRDI profile type: Publication
full work available at URL: j.amc.2008.08.019
OpenAlex ID: W2101530839
Wikidata QID: Q115361746
cites work: Method for Solving the Korteweg-deVries Equation
cites work: Q3995225
cites work: Exact Solution of the Korteweg—de Vries Equation for Multiple Collisions of Solitons
cites work: The Painlevé property for partial differential equations
cites work: Special polynomials associated with some hierarchies
cites work: Exp-function method for nonlinear wave equations
cites work: New periodic solutions for nonlinear evolution equations using Exp-function method
cites work: Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method
cites work: Application of Exp-function method for nonlinear evolution equations with variable coefficients
cites work: New application of Exp-function method for improved Boussinesq equation
cites work: New periodic wave solutions via Exp-function method
cites work: Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations
cites work: An automated \(\tanh\)-function method for finding solitary wave solutions to nonlinear evolution equations
cites work: Modified extended tanh-function method and its applications to nonlinear equations
cites work: Solitary solutions of some nonlinear evolution equations
cites work: A travelling wave solution to the Ostrovsky equation
cites work: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations
cites work: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations
cites work: Simplest equation method to look for exact solutions of nonlinear differential equations
cites work: Exact solitary waves of the Fisher equation
cites work: Exact solutions for nonlinear evolution equations using Exp-function method
cites work: Q5843703
cites work: Q5784660
cites work: Polygons of differential equations for finding exact solutions
DOI: 10.1016/J.AMC.2008.08.019
description: scientific article