We consider in this paper approximation properties and applications of Mathieu functions. A first set of optimal error estimates are derived for the approximation of periodic functions by using angular Mathieu functions. These approximation results are applied to study the Mathieu-Legendre approximation to the modified Helmholtz equation and Helmholtz equation. Illustrative numerical results co...

and a0 is a constant, b is a continuous periodic function of variables x and y, a1 and g are continuous periodic functions. This integral equation appears in exterior boundary value problems for the two-dimensional Helmholtz equation (see [11,15– 17,27]). Constructions of conformal mapping also lead to integral equations of this kind (see [22]). In recent years, the Helmholtz equation and its n...

We derive an exact equation for the transverse component of the electric field propagating along a given longitudinal z direction in the presence of an isotropic refractive-index distribution n(x,y).

We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ((+k 2)u = 0 on a Lipschitz domain for all wave number k 2C with Im k 0. Following the approach for the case of smooth boundary 3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothne...

Within the framework of shifted-Laplace preconditioners [Erlangga, Vuik, Oosterlee, Appl. Numer. Math., 50(2004), pp.409–425] for the Helmholtz equation, different methods for the approximation of the inverse of a complex-valued Helmholtz operator are discussed. The performance of the preconditioner for Helmholtz problems at high wavenumbers in heterogeneous media is evaluated. Comparison with ...