High-frequency uniform asymptotics for the Helmholtz equation in a quarter-plane

Abstract

We demonstrate an approach to derive uniform and point-wise asymptotic formulas based on Fokas' transform method. To this aim, we study the high-frequency uniform asymptotics for the solution of the Helmholtz equation, in a quarter plane and subject to a specific Neumann condition. The analysis is based on the integral representation of the solution derived via Fokas' transform method. In the case of piecewise constant boundary data, full point-wise asymptotic expansions of the solution are obtained by using the method of steepest descents for definite integrals. A uniform asymptotic expansion, holding in the whole quadrant, is also derived in terms of the complementary error function. The uniform expansion exhibits smooth transitions across certain critical vertical lines, along which the point-wise asymptotic approximations have jumps.