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Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions
Author(s) -
Kleinman R.,
Vainberg B.
Publication year - 1994
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670171207
Subject(s) - mathematics , helmholtz equation , mathematical analysis , asymptotic expansion , bounded function , power series , boundary value problem , series (stratigraphy) , infinity , dirichlet distribution , neighbourhood (mathematics) , partial differential equation , paleontology , biology
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 k 2 and k 2 (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 k 2 , 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.