Exact Integral Equation Research Articles

RADIATION-CONDUCTION INTERACTION IN A PLANAR, ANISOTROPICALLY SCATTERING MEDIUM WITH FLUX BOUNDARY

Steady, combined radiation and conduction heat transfer in an absorbing, emitting, and anisotropically scattering planar medium is investigated theoretically. The problem is considered with a constant net heat flux imposed at one boundary and a constant temperature at the other. Both specular and diffuse reflectivities are included. The influence of radiation heat transfer is obtained by solving the exact integral equations of radiative transfer with the nodal approximation technique. The technique reduces the integral equations into a discrete system of algebraic equations and permits obtaining exact numerical solutions for any scattering-phase function. Temperatures are obtained from the energy equation with an iterative procedure. The effects of scattering anisotropy as well as radiation parameters such as albedo and wall reflectivities on temperatures are analyzed.

Exact relation between relativistic and non-relativistic Green functions for independent particles in a given external potential

Well established integral equations already exist for both relativistic (G(r)) and non-relativistic (G(nr)) Green functions for independent particles moving in a given external potential V. An exact integral equation giving G(r) in terms solely of G(nr) and known operators is derived, by eliminating V. The connection with relativistic density functional theory is emphasised.

Combined radiative and conductive heat transfer in two-dimensional emitting, absorbing and anisotropic scattering square media

Combined thermal radiation and conduction in two-dimensional emitting, absorbing and anisotropic scattering square media is studied by solving numerically the exact integral and differential equations. The product integration method is applied for discretizing the radiative transport equations of integral form. For comparison, the zonal method is also used for radiation in some cases. In solving the resulting nonlinear equations, Chen's method is applied and improved. Comparison shows excellent agreement between the results given by the product integration method and the zonal method. The improved Chen's method is found to be efficient. Influences of various radiation-conduction parameter, optical thickness, scattering parameters and wall emissivity on heat transfer and temperature distribution are discussed.

Rough surface scattering with the parabolic wave equation using a split‐step method

The split‐step algorithm introduced by Tappert and Nghiem‐Phu [J. Acoust. Soc. Am. Suppl. 1 77, S101 (1985)] for solving the parabolic wave equation (PE) with 1‐D rough surfaces subject to the pressure release boundary condition has been implemented. For an incident beam approaching the rough surface from below, the computational region is doubled to include an “image ocean” above the real ocean. A complicated index of refraction is constructed in the image ocean (dependent on the rough surface) so that the image beam and the incident beam combine to give a zero total field on the rough surface. The computed field below the rough surface should then be the solution desired. The details of this method will be outlined. Examples will be presented showing that the split‐step solution is identical to that obtained with a PE integral equation approach. Comparisons will also be made between the split‐step and exact integral equation solutions, showing the effect of the PE approximation in rough surface scattering. [Work supported by ONR.]

Lifshitz tails, instantons and long time decay for diffusion in media with random traps

In media with random traps the long time survival has a stretched exponential decay, related to a Lifshitz tail in the density of states. We first discuss exact results on this behavior in one dimension, where exact integral equations can be analyzed. Then we show that the stretched exponential behavior is modified when each site contains a random trap. Finally we consider large but finite, three dimensional systems with a small concentration of traps. Here the long time decay is exponential, with a rate that varies from sample to sample. The tail of the distribution function of these rates is discussed.

The constraint on the integral kernels of density functional theories which results from insisting that there be a unique solution for the density function

All predictive theories for the spatial variation of the density in an inhomogeneous system can be constructed by approximating exact, nonlinear integral equations which relate the density and pair correlation functions of the system. It is shown that the set of correct kernels in the exact integral equations for the density is on the boundary between the set of kernels for which the integral equations have no solution for the density and the set for which the integral equations have a multiplicity of solutions. Thus arbitrarily small deviations from the correct kernel can make these integral equations insoluble. A heuristic model equation is used to illustrate how the density functional problem can be so sensitive to the approximation made to the correlation function kernel and it is then shown explicitly that this behavior is realized in the relation between the density and the direct correlation function and in the lowest order BGYB equation. Functional equations are identified for the kernels in these equations which are satisified by the correct kernels, which guarantee a unique solution to the integral equations, and which provide a natural constraint on approximations which can be used in density functional theory. It is also shown that this sensitive behavior is a general property of density functional problems and that the methodology for constructing the constraints is equally general. A variety of applications of density functional theory are reviewed to illustrate practical consequences of this sensitivity.

The numerical approach to understanding rough surface scattering

Numerical methods for solving simplified rough surface scattering problems “exactly” provide new insights into the rough surface scattering problem. Exact solutions for 1-D surfaces with the pressure release (Dirichlet) boundary condition are found with integral equation and Monte Carlo techniques. These studies provide a better understanding of the validity of the two basic approximations used in rough surface scattering: the Kirchhoff approximation and first-order perturbation theory. From this work, a resolution can be given for the difference in the Kirchhoff and first-order perturbation theory predictions at small surface heights and at very small surface slopes, where both approximations have generally been considered valid. Studies with the Kirchhoff approximation show that shadowing corrections based on geometric optics lead to inaccurate predictions. This work indicates the need for tractable multiple scattering theory for low grazing angle applications. Comparisons with exact results show that “phase perturbation” theory predictions account for a significant part of the multiple scattering associated with shadowing. In addition to better understanding theory with numerical techniques, the accuracy of approximate numerical methods can also be examined. Attempts are now being made to include rough surface scattering in marching solutions of the parabolic wave equation (PE). Methods of verifying the accuracy of marching solutions using exact and PE integral equations will be briefly discussed.

Bond orientational order in simple liquids

The phase transition of an isotropic liquid into an anisotropic phase is described in terms of a bifurcation of the solution of an exact nonlinear integral equation for the pair distribution function. The behaviour of the correlation function of orientational fluctuations for the pair distribution function is analysed.

Diffusion-controlled reactions in disordered porous media—I. Uniform distribution of reactants

Diffusion-controlled reactions in a porous medium, when the distribution of the reactants in the medium is uniform, are investigated. The medium is represented as a network of pores (bonds) with distributed conductances and several analytical techniques for estimating the effective properties of the system are developed. For two- and three-dimensional networks, an effective-medium approximation is developed to predict the effective reaction rate, the effective diffusivity (under reactive conditions), the distribution of the concentrations of the reactants in the porespace, and the effectiveness factor. For porous media which can be represented by a network of interconnected pores with no closed loops, we develop an exact nonlinear integral equation, the solution of which can be used to calculate all of the quantities of interest. The predictions of these are compared with the results of Monte Carlo simulations; good agreement is found in most cases.

Microscopic description of bond orientational order in simple liquids

A microscopic description of the transition of a simple liquid from an isotropic to an anisotropic phase is proposed. An anisotropic phase is obtained in modern phenomenological theories of two-dimensional melting. The approach is based on the exact nonlinear integral equation for the conditional distribution functions. An exact microscopic criterion of instability of the isotropic phase is obtained. The behavior of the correlation function of the orientational fluctuations of the two-particle distribution function is analyzed in the case of two dimensions. An attempt is made to estimate quantitatively the instability point in a system of hard disks and hard spheres.

Relation between multidimensional radiative transfer in cylindrical and rectangular coordinates with anisotropic scattering

An exact integral equation is derived for the source function in a three-dimensional rectangular medium which scatters anisotropically. The upper boundary of the finite medium is exposed to collimated radiation, while the lower boundary has no radiation incident on it. The problem is multidimensional because the incident radiation varies spatially. The scattering phase function is represented by a series of Legendre polynomials. A double Fourier transform reduces the problem to a one-dimensional integral equation for the source function. The transformed equation is compared with the integral equation for a two-dimensional cylindrical medium which scatters anisotropically and is exposed to Bessel-varying collimated radiation. A simple relation is found between the two source functions which will greatly reduce the number of computations required for the three-dimensional case. The relation also illustrates the wide utility of the generalized one-dimensional source function. Simplification of the two-dimensional rectangular case to the generalized source function is also presented. The results are extended to problems with a strong anisotropic phase function which has a diffraction spike in the forward direction.

Scattering of electromagnetic waves by conducting periodic surfaces: A comparison of exact integral equation methods

Exact, integral equation formulations of electromagnetic scattering by corrugated conducting surfaces, resulting from the extinction theorem, electric field boundary condition, and magnetic field boundary condition, are examined from the viewpoint of practical, numerical calculations.

Discrete dynamics and metastability: Mean first passage times and escape rates

The problem of escape from a domain of attraction is applied to the case of discrete dynamical systems possessing stable and unstable fixed points. In the presence of noise, the otherwise stable fixed point of a nonlinear map becomes metastable, due to noise-induced hopping events, which eventually pass the unstable fixed point. Exact integral equations for the moments of the first passage time variable are derived, as well as an upper bound for the first moment. In the limit of weak noise, the integral equation for the first moment, i.e., the mean first passage time (MFPT), is treated, both numerically and analytically. The exponential leading part of the MFPT is given by the ratio of the noise-induced invariant probability at the stable fixed point and unstable fixed point, respectively. The evaluation of the prefactor is more subtle: It is characterized by a jump at the exit boundaries, which is the result of a discontinuous boundary layer function obeying an inhomogeneous integral equation. The jump at the boundary is shown to be always less than one-half of the maximum value of the MFPT. On the basis of a clear-cut separation of time scales, the MFPT is related to the escape rate to leave the domain of attraction and other transport coefficients, such as the diffusion coefficient. Alternatively, the rate can also be obtained if one evaluates the current-carrying flux that results if particles are continuously injected into the domain of attraction and captured beyond the exit boundaries. The two methods are shown to yield identical results for the escape rate of the weak noise result for the MFPT, respectively. As a byproduct of this study, we obtain general analytic expressions for the invariant probability of noisy maps with a small amount of nonlinearity.

Nonlocal effects on the drift cyclotron loss cone dispersion relation in cylindrical geometry

In carrying out a nonlocal analysis of drift cyclotron loss cone (DCLC) modes in a cylindrically symmetric plasma, a new dispersion relation branch not described by local theories has been found. The model assumes a low-β plasma column with a Gaussian density profile in a uniform magnetic field. The exact k-space integral equation is solved using a Laguerre polynomial decomposition of the eigenfunctions. It is these new modes that couple with the diamagnetic drift wave to form unstable DCLC modes, a phenomenon not predicted by local theory. The existence of these modes requires a proper treatment of the spatial variation of the equilibrium distribution function describing the ions.

An Exact Formula for the Accuracy of a Class of Computer Solutions of Integral Equation Formulations of Electromagnetic Scattering Problems

ABSTRACT A method of improving the accuracy of traditional methods (Guru and Chen [6], Hagmann and Gandhi [7]—[8], Hagmann and Levin [9], and Livesay and Chen [13] of solving integral equations has been obtained. We now have exact formulas for the accuracy of protective approximation methods (Gohberg and Feldman [5], Neittaanmaki and Saranen [20]) for solving integral equations. For a class of approximation schemes for estimating the solution of integral equations of electromagnetic scattering, we can develop a systematic procedure for reducing the original infinite rank integral equation to an exact finite rank integral equation capable of being completely analyzed by digital computer. In this paper the method is illustrated for the popular pulse basis function method(Guru and Chen [6], Hagmann and Gandhi [7]—[8], Hagmann and Levin [9], and Livesay and Chen [13]).

Differential and integral methods for multidimensional inverse scattering problems

A layer stripping procedure for solving three-dimensional Schrödinger equation inverse scattering problems is developed. This procedure operates by recursively reconstructing the Radon transform of the potential from the jump in the Radon transform of the scattered field at the wave front. This reconstructed potential is then used to propagate the wave front and scattered field differentially further into the support of the potential. The connections between this differential procedure and integral equation procedures are then illustrated by the derivations of two well known exact integral equation procedures using the Radon transform and a generalized Radon transform. These procedures, as well as the layer stripping procedure, are then reduced to the familiar Born approximation result for this problem by neglecting multiple scattering events. This illustrates the central role of the Radon transform in both exact and approximate inversion procedures.

Iterative integral equation solution for cylindrical dielectric resonator

An exact integral equation describing the substrate-mounted open cylindrical dielectric resonator is derived, and a first approximate solution for the fundamental resonant mode is calculated. Results are presented and are compared with both experiment and another theoretical approach.

Theoretical basis of the empirical reaction field approximations through continuum model

We derive from exact integral equations of classical electrostatics some approximate expressions of the interaction energy of a point charge distribution with a dielectric medium. We show how they can be used for computing the wave functions of solvated species imbedded in cavities formed by interlocking spheres in a polarizable continuum. We discuss the relation between these formulae and the ones proposed earlier on empirical basis and we especially emphasize the improvement that they bring out.

Three-dimensional nodal diffusion and transport methods for the analysis of fast-reactor critical experiments

An interface-current nodal formulation is developed for the solution of the multigroup neutron diffusion and transport equations in three-dimensional Cartesian geometry. The nodal diffusion method uses polynomial approximations equivalent to those used in the nodal expansion method (NEM), while the nodal transport method is based on the approximation of exact one-dimensional integral equations. The transport scheme employs a double P-1 expansion of the surface angular fluxes in conjunction with either isotropic or angle-dependent approximations to the angular dependence of the transverse-leakage terms. The transport formulation does not involve an iteration on the scattering source, and identical iterative strategies are used to solve the interface-current equations obtained in both methods. Numerical results for a ZPPR fast-reactor critical experiment show that the nodal diffusion method produces spatially-converged solutions with significant reductions in computational cost relative to the finite-difference method. Use of the nodal transport scheme eliminates over 80% of the error due to the use of diffusion theory, with computational costs within a factor of two of nodal diffusion theory.

Clustering and percolation in multicomponent systems of randomly centered and permeable spheres

For a multicomponent system of particles in equilibrium, an exact integral equation is derived for the pair connectedness function (which measures the probability that two particles, with centers a distance r apart, are connected). The pair connectedness function, mean cluster size, and percolation thresholds for mixtures of randomly centered (noninteracting) spheres and permeable spheres are then obtained analytically in the Percus–Yevick (PY) approximation. (The permeable-sphere model provides a one-parameter bridge from randomly centered sphere mixtures to hard-sphere mixtures.) For this family of models, connectedness is defined by particle overlap. It is found that, within the PY approximation, a multicomponent mixture of randomly centered spheres percolates at ξ3=π/6∑iρiR3i =1/2, independent of the concentration and size distributions of the particles. For the permeable-sphere case the percolation threshold depends on the relative densities, size, and interparticle permeability among the species. The loci of the percolation thresholds of a binary mixture of permeable spheres are explicitly determined.