On a New Integral Equation Arising in the Theory of Radiative Transfer

1971 ◽  
Vol 20 (4) ◽  
pp. 703-713 ◽  
Author(s):  
Benjamin J. Martin
Keyword(s):  
Integral Equation ◽  
Radiative Transfer

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1

2002 ◽  
Vol 124 (4) ◽  
pp. 674-684 ◽  
Author(s):  
Zekeriya Altac¸
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Order Approximation ◽  
High Order ◽  
High Order Approximation ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Plane Parallel ◽  
Transfer Problems

A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

Differential Approximation to Radiative Transfer in Semitransparent Media

1985 ◽  
Vol 107 (2) ◽  
pp. 478-481 ◽  
Author(s):  
F. H. Azad
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Exact Formulation ◽  
Differential Approximation ◽  
Reflection And Refraction ◽  
Integrated Intensity ◽  
Dielectric Interfaces ◽  
Semitransparent Medium

Radiative transfer in a semitransparent medium is treated using the differential approximation. Boundary conditions are formulated to accommodate direction-dependent reflection and refraction at a dielectric interfaces. The approximate results are compared to numerical solution of the exact integral equation. Also, a modification based on the exact formulation of the integrated intensity at the interface is presented that significantly improves the accuracy of the differential approximation in the optically thin regimes.

On a non-linear integral equation occurring in diffraction theory

1966 ◽  
Vol 62 (2) ◽  
pp. 249-261 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Radiative Transfer ◽  
Plane Wave ◽  
Admissible Solution ◽  
Diffraction Theory ◽  
Linear Integral Equation ◽  
Non Linear Equation ◽  
Non Linear ◽  
Linear Integral

AbstractThe problem of diffraction of a plane wave by a semi-infinite grating of iso-tropic scatterers leads to the consideration of a non-linear integral equation. This bears a resemblance to Chandrasekhar's integral equation which arises in the study of radiative transfer through a semi-infinite atmosphere. It is shown that methods which have been used with success to solve Chandrasekhar's equation are equally useful here. The solution to the non-linear equation satisfies a more simple functional equation which may be solved by factoring (in the Wiener-Hopf sense) a given function. Subject to certain additional conditions which are dictated by physical considerations, a solution is obtained which is the unique admissible solution of the non-linear integral equation. The factors and solution are found explicitly for the case which corresponds to closely spaced scatterers.

scholarly journals Radiative transfer in Rayleigh–Taylor unstable ICF pellets

1990 ◽  
Vol 8 (4) ◽  
pp. 741-751 ◽  
Author(s):  
G. C. Pomraning
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Markov Model ◽  
Master Equation ◽  
Energy Flow ◽  
Fluid Mixing ◽  
Renewal Processes ◽  
Integral Equation Formulation ◽  
Statistical Effects

A formulation of radiative transfer is discussed describing energy flow in a turbulent mixture in the vicinity of a Rayleigh–Taylor unstable interface, as might be extant in an ICF pellet. Included in this discussion are (1) the method of smoothing and the Liouville master equation approaches in the case of Markovian statistics as the description of the fluid mixing, (2) the use of asymptotics to derive various limiting descriptions of the Markov model, (3) the use of the theory of alternating renewal processes to obtain an integral equation formulation for non-Markovian statistics, and (4) the reduction of this non-Markovian integral formulation to integro-differential equations of the generic transport form, with statistical effects represented by pseudoscattering terms.

Transient Radiative Transfer of Light Pulse Propagation in Three-Dimensional Scattering Media With Finite Volume Method and Integral Equation Model

2003 ◽  
Author(s):  
Xiaodong Lu ◽  
Pei-Feng Hsu ◽  
John C. Chai
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Light Pulse ◽  
Three Dimensional ◽  
Equation Model ◽  
Isotropic Scattering ◽  
Scattering Media ◽  
Volume Method

The transient radiative transfer process is studied with a finite volume method (FVM) and an integral equation (IE) model. Propagation of a short light pulse in the three-dimensional absorbing and isotropic scattering media is considered. Collimated irradiation enters at one side of the rectangular medium. The other five boundaries are cold and black, nonparticipating surfaces. The spatial and temporal distributions of the integrated intensity and radiative flux are obtained.

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