A Generalization of the Wiener–Hopf Method for an Equation in Two Variables with Three Unknown Functions

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 464-476, April 2024.
Abstract. This manuscript presents an analytic solution to a generalization of the Wiener–Hopf equation in two variables and with three unknown functions. This equation arises in many applications, for example, when solving the discrete Helmholtz equation associated with scattering on a domain with perpendicular boundary. The traditional Wiener–Hopf method is suitable for problems involving boundary data on co-linear semi-infinite intervals, not for boundaries at an angle. This significant extension will enable the analytical solution to a new class of problems with more boundary configurations. Progress is made by defining an underlining manifold that links the two variables. This allows one to meromorphically continue the unknown functions on this manifold and formulate a jump condition. As a result the problem is fully solvable in terms of Cauchy-type integrals, which is surprising since this is not always possible for this type of functional equation.