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Extended simplest equation method for nonlinear differential equations. (English) Zbl 1168.34003

The authors consider the equation

$P\left(y,{y}^{\text{'}},{y}^{\text{'}\text{'}},\cdots \right)=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $y=y\left(z\right)$ is an unknown function, $P$ is a polynomial in the variable $y$ and its derivatives and look for exact solutions $y=y\left(z\right)$ of the form

$y\left(z\right)=\sum _{k=0}^{N}{A}_{k}{\left(\frac{{\psi }^{\text{'}}}{\psi }\right)}^{k},\phantom{\rule{2.em}{0ex}}\left(2\right)$

${A}_{k}=\text{const}$, ${A}_{N}\ne 0$, where the function $\psi =\psi \left(z\right)$ is the general solution of the linear ordinary differential equation

${\psi }^{\text{'}\text{'}\text{'}}+\alpha {\psi }^{\text{'}\text{'}}+\beta {\psi }^{\text{'}}+\gamma \psi =0,\phantom{\rule{2.em}{0ex}}\left(3\right)$

$\alpha ,\beta ,\gamma =\text{const}$. They propose the algorithm for searching the parameters $N,{A}_{k},$ $k=1,\cdots ,N$, $\alpha ,\beta ,\gamma$. 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.

##### MSC:
 34A05 Methods of solution of ODE