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Zbl 1168.34003
Kudryashov, Nikolai A.; Loguinova, Nadejda B.
Extended simplest equation method for nonlinear differential equations. (English)
[J] Appl. Math. Comput. 205, No. 1, 396-402 (2008). ISSN 0096-3003

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.
[Vladimir L. Makarov (Ky{\"\i}v)]

MSC 2000:: *34A05 Methods of solution of ODE

Keywords: nonlinear evolution equations; exact solutions; simplest equation method; Sharma-Tasso-Olver equation; Burgers-Huxley equation

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Zentralblatt MATH Berlin [Germany]