Existence of a traveling wave solution in a free interface problem with fractional order kinetics

In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0<α<1. We turn the free interface problem into scalar boundary coupled an integral equation. The main intermediary step is to reduce study non-Lipschitz vector field in dimension 2. latter treated by qualitative topological methods based on Poincaré-Bendixson Theorem. phase portrait determined existence stable manifold at origin proved. A significant result settling time reach finite, meaning trailing finite contrast case α=1, but accordance α=0. Finally, integro-differential solved via fixed-point method.