Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions
R. E. Kleinman, Boris Rufimovich Vainberg · Mathematical Methods in the Applied Sciences · 1994
Abstract The present paper contains the low‐frequency expansions of solutions of a large class of exterior boundary value problems involving second‐order elliptic equations in two dimensions. The differential equations must coincide with the Helmholtz equation in a neighbourhood of infinity, however, they may depart radically from the Helmholtz equation in any bounded region provided they retain ellipticity. In some cases the asymptotic expansion has the form of a power series with respect to k2 and k2 (ln k + a)−1, where k is the wave number and a is a constant. In other cases it has the form of a power series with respect to k2, coefficients of which depend polynomially on In k. The procedure for determining the full low‐frequency expansion of solutions of the exterior Dirichlet and Neumann problems for the Helmholtz equation is included as a special case of the results presented here.