General invariance relations reduction method and its applications to solutions of radiative transfer problems for turbid media of various configurations

Rogovtsov, N. N.

doi:10.1007/978-3-642-10336-0_7

Cited by 6 publications

(4 citation statements)

References 58 publications

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“…This function is a solution of the following canonical BVP for the theory of the radiation transport [10] :…”

Section: Analytic Solutions Of Inhomogeneous Scalar Characteristic Eqmentioning

confidence: 46%

“…Note that, for applications of the TRT, it is especially interesting to develop analytic methods for solving infinite SLAE of the type (4), which would permit one to construct convenient numerical real-time algorithms for "practically arbitrary" phase functions. The use of general ideas and constructions of the GIRRM [9,10] and the method suggested in [2,3] permits one to develop a new well-posed effective analytic method for infinite SLAE of the type (4) and (14). …”

Section: Analytic Representations and Recursion Formulas For Solutionmentioning

confidence: 49%

“…In the derivation of the Green function for μ 1 = ±0 on the basis of relations (31) and (32), one should assume that μ = 0 and treat g ∨ |m| as the sequence {f s+|m| P s+|m|,|m| (μ)} s∈N0 . In conclusion, note that, by using the analytic representation (31) and the GIRRM [9,10], one can efficiently solve BVP for a scalar RTE in the case of practically arbitrary phase functions and disperse media of various configurations.…”

Section: Analytic Solutions Of Inhomogeneous Scalar Characteristic Eqmentioning

confidence: 51%

“…For the derivation of asymptotic solutions of BVP for a scalar RTE with the use of the Case method [8] or the GIRRM [9,10], one needs the normalization constants…”

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confidence: 46%

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Constructive theory of scalar characteristic equations of the theory of radiation transport: II. algorithms for finding solutions and their analytic representations

Rogovtsov

2015

Diff Equat

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Abstract-We present methods for finding discrete spectra and derive analytic expressions for the eigenfunctions of scalar characteristic equations of the theory of radiation transport. We obtain new two-term recursion formulas and analytic representations for solutions of infinite tridiagonal systems of linear algebraic equations. We obtain analytic forms of the resolvents of scalar characteristic equations for phase functions square integrable on the closed interval [−1, 1]. In addition, we derive a general analytic expression for the Green function of a two-dimensional (with respect to the angular variables) integro-differential equation of the radiation transport for the case in which the phase functions satisfy the Hölder condition on the closed interval [−1, 1]. DOI: 10.1134/S0012266115050092The present paper continues the paper [1] and the corresponding numbering of sections, formulas, theorems, corollaries, and remarks. ALGORITHMS FOR FINDING DISCRETE SPECTRA, EIGENFUNCTIONS, AND NORMALIZING CONSTANTS FOR REDUCED SCALAR CHARACTERISTIC EQUATIONS OF THE TRTThe following assertion can be proved with the use of systems (4), (8), and (9), Theorems 3-6, Corollary 1, Remark 3, and constructions used in [2,3] in the analysis and statement of properties of infinite continued fractions.

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“…This function is a solution of the following canonical BVP for the theory of the radiation transport [10] :…”

Section: Analytic Solutions Of Inhomogeneous Scalar Characteristic Eqmentioning

confidence: 46%

Section: Analytic Representations and Recursion Formulas For Solutionmentioning

confidence: 49%

Section: Analytic Solutions Of Inhomogeneous Scalar Characteristic Eqmentioning

confidence: 51%

“…For the derivation of asymptotic solutions of BVP for a scalar RTE with the use of the Case method [8] or the GIRRM [9,10], one needs the normalization constants…”

mentioning

confidence: 46%

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Constructive theory of scalar characteristic equations of the theory of radiation transport: II. algorithms for finding solutions and their analytic representations

Rogovtsov

2015

Diff Equat

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Radiative Transfer in Spherically and Cylindrically Symmetric Media

Kolesov

Kropacheva

2016

Light Scattering Reviews, Volume 11

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15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s H $H$ -functions for four-term phase functions with the double-exponential formula

Kawabata

2017

Astrophys Space Sci

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We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar's H-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (2015, Astrophys. Space Sci. 358:32) and the doubleexponential integration formula (DE-formula) of Takahashi and Mori (1974, Publ. RIMS, Kyoto Univ. 9, 721), which proved highly effective in Kawabata (2016, Astrophys. Space Sci. 361:373). Actual calculations of the H-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo ̟ 0 , and 36 values of an angular variable µ(= cos θ), with θ being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. They indicate that it is important to simultaneously verify accuracy of the numerical values of the H-functions for µ < 0.05, the domain often neglected in tabulation. As a sample application of the isotropic scattering H-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (2016, J. Quant. Spectr. Radiat. Transf. 183, 128), who employed their analytical representations for these quantities and the singleterm and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy. While our results are in satisfactory agreement with theirs, our Kiyoshi Kawabata Department of Physics, Tokyo University of Science, Shinjukuku, Tokyo 162-8601, Japan E-mail: kawabata@rs.kagu.tus.ac.jp procedure is in need of a faster algorithm to routinely deal with problems involving highly anisotropic phase functions giving rise to near-conservative scattering.

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General invariance relations reduction method and its applications to solutions of radiative transfer problems for turbid media of various configurations

Cited by 6 publications

References 58 publications

Constructive theory of scalar characteristic equations of the theory of radiation transport: II. algorithms for finding solutions and their analytic representations

Constructive theory of scalar characteristic equations of the theory of radiation transport: II. algorithms for finding solutions and their analytic representations

Radiative Transfer in Spherically and Cylindrically Symmetric Media

15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s H $H$ -functions for four-term phase functions with the double-exponential formula

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