New exact solutions of one nonlinear equation in mathematical biology and their properties (Q2784885)

The generalized Fischer equation NEWLINE\[NEWLINEu_t=[(\lambda_0u+\lambda_1)u_x]_x+\lambda_2u-\lambda_3u^2,NEWLINE\]NEWLINE where \( \lambda_0, \lambda_1, \lambda_2, \lambda_3 \in {\mathbb R}\), is considered. The author proposes different methods of finding solutions to the equation. First, the Lie method for finding an exact solution to the problem is used. Moreover, the author proves that it is impossible to find new solutions to the problem by means of the non-Lie \(Q\)-condition symmetries. The author also uses the method of generate conditions for finding the solutions. Exact solutions to the nonlinear Neumann problem under zero boundary conditions are found.