An iterative method for radiative transfer is a slab with specularly reflecting boundary

The purpose of this paper is to present an iterative scheme for solving the radiative-transfer equation in a scattering, absorbing and emitting slab with specularly reflecting boundary. The medium under consideration is anisotropic, nonhomogeneous and azimuthally unsymmetric, and the boundary surface can be either transparent or reflecting. A novelty of this scheme is that it leads to an explicit recursion formula for the determination of the solution as well as an error estimate for the iterations. The recursion formula involves straightforward integrations of a well-behaved function and can be used to calculate numerical results by using a computer. It is shown that the sequence of iterations from the recursion formula converges uniformly to a unique positive solution of the problem, so that this method also gives an existence and uniqueness theorem.     

The exact solution of the time-dependent equation of radiative transfer in a semi-infinite atmosphere

The exact solution of the time-dependent equation of radiative transfer was obtained for the emergent intensity of radiation from a homogeneous semi-infinite atmosphere whose surface is illuminated by a pulsed beam of radiation, allowing for two parameters having dimension of time: the mean time of temporal capture and the mean free time. The solution is expressed in the form of convergent infinite series. Each term is a product of two known functions whose numerical values are known or can be computed. Using the simple expression of the time-dependent emergent intensity, we can derive the fundamental relations for it.     

The radiative transfer equation with a nonhomogeneous reflectivity boundary condition coupled with the heat conduction equation

In this paper, we make a rigorous mathematical analysis of the radiative heating of a semitransparent body made of a glass by a black radiative source surrounding it. This requires the study of the coupling between quasi‐steady radiative‐transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semitransparent electromagnetic spectrum of the glass) and a nonlinear heat‐conduction evolution equation with a nonlinear Robin boundary condition, which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution and give also uniform bounds on the solution, ie, on the absolute temperature distribution inside the body and on the radiative intensities.     

Asymptotic solutions of the equations of radiative transfer for an optically thick layer with reflecting boundaries

An asymptotic analysis is made of the system of equations obtained in a previous paper. This system is equivalent to the radiative transfer equation and is suitable for effective solution in the case of an optically thick layer with arbitrary linear boundary conditions of reflection and transmission. Asymptotic expressions are obtained that make it possible in each specific case to find the explicit form of the laws of reflection and transmission of diffuse radiation. The spatial distribution of the radiation intensity within and on the boudaries of an optically thick layer is also investigated.State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 424–443, September, 1994.     

Adaptive computation for boundary control of radiative heat transfer in glass

This paper is concerned with an optimal boundary control of the cooling down process of glass, an important step in glass manufacturing. Since the computation of the complete radiative heat transfer equations is too complex for optimization purposes, we use simplified approximations of spherical harmonics including a practically relevant frequency bands model. The optimal control problem is considered as a constrained optimization problem. A first-order optimality system is derived and decoupled with the help of a gradient method based on the solution to the adjoint equations. The arising partial differential–algebraic equations of mixed parabolic–elliptic type are numerically solved by a self-adaptive method of lines approach of Rothe type. Adaptive finite elements in space and one-step methods of Rosenbrock-type with variable step sizes in time are applied. We present numerical results for a two-dimensional glass cooling problem.     

Monte Carlo simulation of radiative transfer in a medium with varying refractive index specified at discrete points

A Monte Carlo method with discrete ray tracing is developed to simulate radiative transfer in a medium with a spatially varying refractive-index distribution available merely at a set of arbitrary discrete points. We solve the ray equation by a Runge−Kutta Dormand−Prince method to carry out the numerical ray tracing. To retrieve the refractive-index values and gradients needed in the discrete ray tracing, we apply cubic spline interpolation for one-dimensional simulation and a moving least square (MLS) method for two-dimensional simulation. The influence of the basis vectors and the numbers of sampled data used by the MLS method on ray tracing based on the retrieved refractive-index values and gradients has been examined. The results of radiative equilibrium in a planar medium and radiative transfer in two-dimensional media with different geometries and conditions obtained by the present methods are compared with those obtained by solving the integral equations of radiative transfer and the discrete ordinates method. The comparisons show that the present methods generate accurate results for radiative transfer with various geometries, parameters and refractive-index distributions specified at discrete points.     

Meshless method for solving multidimensional radiative transfer in graded index medium

A diffuse approximation meshless (DAM) method is presented to solve the radiative transfer equation (RTE) in a graded index medium. The meshless method can solve the equation directly without using an upwind scheme. Absorbing, emitting and scattering media with different kinds of graded indices in 1D and 2D geometries are tested. Prediction results obtained by the proposed method are compared with different references in order to illustrate the performance of this solution method.     

Initial boundary value problem for the radiative transfer equation with diffusion matching conditions

Under study is the Cauchy problemfor the nonstationary radiative transfer equation with generalized matching conditions that describes the diffuse reflection and refraction on the interface. The solvability of the initial-boundary value problem is proved. Some stabilization conditions for the nonstationary solution are obtained.     

Shear waves in a semi-infinite visco-elastic medium due to a sudden twist applied at a point on the plane boundary

Zusammenfassung Es werden die Transversalwellen untersucht, welche in einem viskoelastischen Halbraum unter einem plötzlich an der Grenzebene angebrachten Moment entstehen. Insbesondere werden die Verschiebungen an der Grenzfläche für kleine und grosse Distanzen vom Störungszentrum angegeben. Das Medium wird als isotroper Kelvin-Stoff angenommen.     

Polarized light transfer above a reflecting surface

The existence and uniqueness are established for the solution of the equation of transfer of polarized light in a homogeneous atmosphere of finite optical thickness, assuming reflection by the planetary surface. A general L_p-space formulation is adopted. The boundary value problem is first written as a vector-valued integral equation. Using monotonicity properties of the spectral radii of the integral operators involved as well as recent half-range completeness results for kinetic equations with reflective boundary conditions, the present results follow as a corollary.     

Stochastic differential equations for multi-dimensional domain with reflecting boundary

Summary In this paper we prove that there exists a unique solution of the Skorohod equation for a domain inR ^d with a reflecting boundary condition. We remove the admissibility condition of the domain which is assumed in the work [4] of Lions and Sznitman. We first consider a deterministic case and then discuss a stochastic case.     

System of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary

Summary We study a system of particles and the nonlinear McKean-Vlasov diffusion that is its limit for weak interactions in Statistical Mechanics, reflecting in a domain with sticky boundary. The interaction takes place in particular in the sojourn condition. We show existence and uniqueness for the nonlinear martingale problem, by a contraction argument on time-change. Then we construct the system of particles by a limiting procedure, and show propagation of chaos towards the nonlinear diffusion.

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Résumé Nous étudions un système de particules et la diffusion non-linéaire de type McKean-Vlasov qui en est la limite en Mécanique Statistique pour des interactions faibles, en réflexion dans un domaine à bord collant. L'interaction réside en particulier dans la condition de séjour. Nous montrons l'existence et l'unicité pour le problème de martingales non-linéaire, par une méthode de contraction sur le changement de temps. Nous construisons le système de particules en tant que limite en loi, et démontrons la propagation du chaos vers la diffusion non-linéaire.

     

On reflecting boundary conditions for space-fractional equations on a finite interval: Proof of the matrix transfer technique

Even in the one-dimensional case, dealing with the analysis of space-fractional differential equations on finite domains is a difficult issue. On a finite interval coupled with zero flux boundary conditions, different approaches have been proposed to define a space-fractional differential operator and to compute the solution to the corresponding fractional problem, but to the best of our knowledge, a clear relationship between these strategies is yet to be established. Here, by using the theory of α-stable symmetric Lévy flights and the master equation, we derive a discrete representation of the non-local operator embedding in its definition the concept of reflecting boundary conditions. We refer to this discrete operator as the reflection matrix and provide (and prove) a theorem on the analytic expression of its eigenvalues and eigenvectors. We then use this result to compare the reflection matrix to the discrete operator defined via the matrix transfer technique, and establish the validity of the latter technique in producing the correct solution to a space-fractional differential equation on a finite interval with reflecting boundary conditions. We finally discuss and emphasize the challenges in the generalisation of the proposed result to more than one spatial dimension.     

A note on disturbances in a semi-infinite piezoelectric medium coated with a perfectly conducting thin film

Summary The paper makes use of the constitutive equations of a piezoelectric material and the tool of Laplace transform to investigate the disturbances in a semi-infinite piezoelectric medium coated with a perfectly conducting thin film.     

Time-dependent laser ignition of a combustible stagnant boundary layer

A spatially one-dimensional model for the ignition of a combustiblegas layer adjacent to a plane solid surface is considered. Theeffect of an incident laser beam from the gas side is to raisethe surface temperature; the other boundary is taken to be apoor thermal conductor in the form of an inert gaseous medium.It is assumed that the exothermic chemical reaction within thelayer has a negligible reactant consumption. Three examplesfor the effect of the laser on the solid surface are considered:(a) a large instantaneous temperature rise, (b) a temperaturejump at t=0 which is a linear funtion of time t>0, (c) asurface temperature variation which is a linear function oftime. For (a) and (b) conditions for criticality are obtained.For appropriate states of the system, times to ignition for(a), (b), and (c) have been determined. It is shown that thetheoretical results are in reasonably good agreement with experiment.The kind of physical situation envisaged might occur in a mineshaftwhen a stagnant combustible layer is subject to intense transientlight. Although an idealization, our analysis determines conditionsunder which thermal runaway and subsequent explosion could takeplace.     

On semi-infinite spectral elements for Poisson problems with re-entrant boundary singularities

A spectral element method is described which enables Poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi-infinite rectangular regions. Two choices of trial functions are considered, namely the eigenfunctions of the differential operator and Chebyshev polynomials. The coefficients in the series expansions are obtained by imposing weak C¹ matching conditions across element interfaces. Singularities at re-entrant corners are treated by a post-processing technique which makes use of the known asymptotic behaviour of the solution at the singular point. Accurate approximations are obtained with few degrees of freedom.