Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity

We search for traveling-wave solutions of the class of equations∑p=1N1αp∂pQ∂tp+∑q=1N2βq∂qQ∂xq+∑m=1MμmQm=0where αp,βqαp,βq and μmμm are parameters. We obtain such solutions by the method of simplest equation for the cases when the simplest equation is the the equation of Bernoulli or the equation of Riccati. We modify the methodology of the simplest equation of Kudryashov as follows. Kudryashov uses the first step of the test for Painleve property in order to determine the size of the solution of the studied PDE. We divide the studied PDEs in two parts: part A which contains the derivatives, and part B which contains the rest of the equation. The application of the ansatz or the extended ansatz of Kudryashov, transforms part A and part B to two polynomials. We balance the highest powers of the polynomials for the parts A and B and, thus obtain a balance equation which depends on the kind of the used simplest equation. The balance equations are connected to nonlinear algebraic systems of relationships among the parameters of the equations and the parameters of the solution. On the basis of these systems, we obtain numerous solutions of the studied class of equations.