# Helmholtz equation

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## Helmholtz equation

[′helm‚hōlts i‚kwā·zhən]
(mathematics)
A partial differential equation obtained by setting the Laplacian of a function equal to the function multiplied by a negative constant.
(optics)
An equation which relates the linear and angular magnifications of a spherical refracting interface. Also known as Lagrange-Helmholtz equation.
(physical chemistry)
The relationship stating that the emf (electromotive force) of a reversible electrolytic cell equals the work equivalent of the chemical reaction when charge passes through the cell plus the product of the temperature and the derivative of the emf with respect to temperature.
References in periodicals archive ?
The components of the general solutions are particular solutions of the Helmholtz equation in Cartesian, cylindrical, and spherical coordinates.
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.

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