The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [31], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This p...

In this paper, a novel wave equation for electromagnetic waves in a medium having anisotropic permittivity has been derived with the help of Maxwell’s curl equations. The x and y components of the Maxwell’s equations are written with the permittivity ( ) being a 3× 3 symmetric matrix. These equations are solved for Ex , Ey, Hx, Hy in terms of Ez , Hz, and the partial derivatives. The Z componen...

In the present paper, a new analytical technique is introduced for obtaining approximate periodic solutions of Helmholtz-Duffing oscillator. Modified Harmonic Balance Method (MHBM) is adopted as the solution method. A classical harmonic balance method does not apply directly for solving Helmholtz-Duffing oscillator. Generally, a set of difficult nonlinear algebraic equations is found when MHBM ...

The exact, well-posed, one-way reformulation of Helmholtz-type wave equations and the generalized Bremmer coupling series are applied to the two-way, scalar, multidimensional Helmholtz equation of mathematical physics. These two constructions provide complementary ways of incorporating one-way wave equation modeling into two-way wave propagation problems. The mathematical framework, for the exp...

The solution of Helmholtz and Maxwell equations by integral formulations (kernel in exp(i kr)/r) leads to large dense linear systems. Using direct solvers requires large computational costs in O(N(3)). Using iterative solvers, the computational cost is reduced to large matrix-vector products. The fast multipole method provides a fast numerical way to compute convolution integrals. Its applicati...

The usual non-linear corrections for a Helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations. We aim to present a systematic derivation of a solution of the non-linear Helmholtz resonator equation, in order to obtain analytically expressions for impedances close to resonance, while including non-linear effects. The amplitude r...

The usual nonlinear corrections for a Helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations. We aim to present a systematic derivation of a solution of the nonlinear Helmholtz resonator equation, in order to obtain analytically expressions for impedances close to resonance, while including nonlinear effects. The amplitude regi...

We present a fast and accurate numerical scheme for the approximation of the primitive equations of the atmosphere. The temporal variable is discretized by using a special semiimplicit scheme which requires only to solve a Helmholtz equation and a nonlocal Stokes problem at each time step; the spatial variables are discretized by a spectral-Galerkin procedure with the horizontal components of v...

Differences are investigated between solutions of the one-dimensional Helmholtz equation for monochromatic waves in a Kerr grating and its approximation, the (generalized) coupled mode (GCM) equations. First it is pointed out that use of the latter can be justified on the basis of averaging theory, and an upper bound is given for the error made this way. Second, the qualitative difference that ...

We develop an efficient algorithm for simulating wave propagation over long distances with both weak and strong scatterers. In domains with weak heterogeneities the wave field is decomposed into forward propagating and back scattered modes using two coupled parabolic equations. In the region near strong scatterers, the Helmholtz equation is used to capture the strong scattering events. The key ...