The factorization method for the inverse obstacle scattering problem with oblique derivative boundary condition

We consider a direct and inverse problem for the scattering of an obstacle with a generalized oblique derivative and impedance boundary condition due to the incident plane wave, which arises in some scattering phenomenons such as the scattering of tidal waves by islands under suitable assumptions. The solvability of the direct scattering problem is proven by using the boundary integral equation method. A special technique is proposed in order to show the formulated boundary integral operator is Fredholm with index 0. Then the factorization method is established to reconstruct the shape of the obstacle. We note that the tangential derivative on the obstacle boundary leads to the difficulties on mathematical analysis. Finally, we present some numerical examples in 2D to show the feasibility and effectiveness of the factorization method.