Inhomogeneous Helmholtz Research Articles

On the Relationship between the Inhomogeneous Wave and Helmholtz Equations in a Fractional Setting

We study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.

Comparative study of a universal theoretical way and 3rd + 2nd-order perturbation density functional theory

A universal theoretical way, proposed recently by Zhou, provides a general scheme which transforms any one of hard sphere density functional approximations (DFAs) into a DFA workable for non-hard sphere fluids. In this article, the universal theoretical way is combined with a hard sphere DFA as a Lagrangian theorem-based DFA to perform calculations for density profiles of a hard core attractive Yukawa model fluid influenced by diverse external fields, and the calculation results are compared with corresponding results due to a 3rd + 2nd-order perturbation density functional theory (DFT) and computer simulation previously reported. Several conclusions are drawn from a detailed analysis on the degree of accuracy of the two approximations relative to the computer simulation and described below. (i) Correspondence between the change of relative performance of the two DFAs and the change of the performance of the bulk 2nd-order direct correlation function (DCF) used as an input as a function of bulk state indicates that the bulk 2nd-order DCF is a key input information determining the accuracy of the approximations using them. (ii) For large inhomogeneity, non-perturbation approximation is evidently more successful than a 3rd-order truncation on the functional perturbation expansion series. (iii) The universal theoretical way is very promising given that a successful hard sphere DFA and accurate bulk 2nd-order DCF are incorporated. Under the present conditions, the universal theoretical way gives an excellent account of the density profiles even if the ‘dangerous’ coexistence bulk states are being under consideration. (iv) The present study helps in formulating new approximations for the inhomogeneous excess Helmholtz free energy functional.

Multidimensional wave field signal theory: Mathematical foundations

Many important physical phenomena are described by wave or diffusion-wave type equations. Since these equations are linear, it would be useful to be able to use tools from the theory of linear signals and systems in solving related forward or inverse problems. In particular, the transform domain signal description from linear system theory has shown concrete promise for the solution of problems that are governed by a multidimensional wave field. The aim is to develop a unified framework for the description of wavefields via multidimensional signals. However, certain preliminary mathematical results are crucial for the development of this framework. This first paper on this topic thus introduces the mathematical foundations and proves some important mathematical results. The foundation of the framework starts with the inhomogeneous Helmholtz or pseudo-Helmholtz equation, which is the mathematical basis of a large class of wavefields. Application of the appropriate multi-dimensional Fourier transform leads to a transfer function description. To return to the physical spatial domain, certain mathematical results are necessary and these are presented and proved here as six fundamental theorems. These theorems are crucial for the evaluation of a certain class of improper integrals which arise in the evaluation of inverse multi-dimensional Fourier and Hankel transforms, upon which the framework is based. Subsequently, applications of these theorems are demonstrated, in particular for the derivation of Green's functions in different coordinate systems.

Inverse Source Problem in an Oblate Spheroidal Geometry

The canonical inverse source problem of reconstructing an unknown source whose region of support is describable as a spheroidal (oblate or prolate) volume from knowledge of the far-field radiation pattern it generates is formulated and solved within the framework of the inhomogeneous scalar Helmholtz equation via a linear inversion framework in Hilbert spaces. Particular attention is paid to the analysis and computer illustration of flat, aperture-like sources whose support is approximated by an oblate spheroidal volume

The role of dyadic Green's functions in the theory of electromagnetic wave propagation

The problem of determining the electromagnetic field of a given monochromatic source current J(r)e−iωt completely surrounded by a limitless, homogeneous, isotropic medium reduces to the mathematical problem of solving the inhomogeneous vector Helmholtz equation ∇×∇×E−k2E=iωμJ (k2=ω2μe)((1)) for the electric field E in terms of the current J. The desired solution of this equation by virtue of its linearity can be written as E(r)=∫Γ(r,r′)·J(r′) dV′((2)) where the integration extends throughout the re,on of finite extent occupied by the current. The quantity Γ (r, r′) is a dyadic function of the observation point r and the integration point r′. If (2) is to be a solufion of (1), then Γ must satisfy ∇×∇×Γ−k2Γ=iωμU(r−r′)((3)) where U is the unit dyadic. Hence, in keeping with the nomenclature of the theory of differential equations, Γ is the Green's function of (1).