Extended simplest equation method for nonlinear differential equations
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.
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