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

The authors consider the equation

P(y,y ' ,y '' ,)=0,(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)= k=0 N A k ψ ' ψ k ,(2)

A k =const, A N 0, where the function ψ=ψ(z) is the general solution of the linear ordinary differential equation

ψ ''' +αψ '' +βψ ' +γψ=0,(3)

α,β,γ=const. They propose the algorithm for searching the parameters N,A k , k=1,,N, α,β,γ. 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.

34A05Methods of solution of ODE