Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates

• We classify the Lie symmetries of a generalized Fisher equation.
• We construct the invariant solutions and reduced ODEs from the optimal systems.
• We find some exact solutions by using Riccati and Bessel equations.
• Some of the solutions we have found exhibit instantaneous sources and sinks.

In this paper, a generalized Fisher equation is studied from the point of view of the theory of symmetry reductions in partial differential equations. A group classification is obtained. All the reductions are derived from the optimal system of subalgebras. Some of the reduced equations admit Lie symmetries which yield to further reductions. By applying direct methods such as the simplest equation method we derive some exact wave solutions.