The components of the general solutions are particular solutions of the

Helmholtz equation in Cartesian, cylindrical, and spherical coordinates.

Advances in Iterative Methods and Preconditioners for the

Helmholtz Equation. Archives of Computational Methods in Engineering, 15(1), 37-66.

This formulation can be obtained by operating over (1a) to yield the

Helmholtz equation, and after an analogous Hamiltonian formulation of the

Helmholtz equation (HFHE), the resultant Hamiltonian operator is reduced to be the Laplacian, and the rest of the terms can be included in the perturbation operator in a very natural manner.

Notice that (8) is a nonhomogeneous

Helmholtz equation with m as solution; it can be written as

is called generalized axially symmetric

Helmholtz equation (GASHE) and the solutions of (1) are called GAShE functions.

In this paper, we mainly follow the idea in [9,24-26] to study the nonsymmetrical scatterers as a series of one-dimensional problems and consider the analytic continuation theorems of the

Helmholtz equation. This paper aims to examine some spectral invariants associated to (1) and analyze the functional correspondence between the variation of the spectral density function and perturbation of the index of refraction.

Several particular classes of problems have been considered: combinations of point sources--see [4-6]; linear/affine classes as in [7, 8]; classes of characteristic sources (e.g., [9-16]); and in particular for the

Helmholtz equation we refer to the papers [17,18], where a full identification result was established, but using instead an interval of frequencies.

The differential transform method is used in many fields and many mathematical physical problems such as a system of differential equations [18], a class of time dependent partial differential equations (PDEs) [19], wave, Laplace and heat equations [20], the fractional diffusion equations [21], two-dimensional transient heat flow [22], nonlinear partial differential equations [23], diffusion-convection equation [24], convection-dispersion problem [25], linear transport equation [26], two-dimension transient atmospheric pollutant dispersion [27],

Helmholtz equation [28].

From the definitions (3) we readily see that the present choices of the effective permittivity and permeability functions are such that the functions ([epsilon]/r) and ([mu]r) occurring in the

Helmholtz Equation (2) respectively are indeed linear functions.

Abuasad, "Application of He's variational iteration method to

Helmholtz equation," Chaos, Solitons & Fractals, vol.

The approximations shown in the previous sections are of great interest, since they give us the possibility to approximate with a very good level of accuracy the WGM solutions of

Helmholtz equations for an ellipsoid, by means of the WGM solutions of the

Helmholtz equation in a toroidal cavity with circular cross section.