Geodesic
Initial-boundary value problem for a radiative transfer equation with generalized matching conditions
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1036-1056.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the Cauchy problem for a non-stationary radiative transfer equation in a three-dimensional multicomponent medium with generalized matching conditions. These matching condition describe Fresnel and diffuse reflection and refraction at the interfaces. The existence and uniqueness of a solution of the initial-boundary value problem is proved. We construct a Monte-Carlo numerical method designed to find a solution that accounts for the space-time localization of radiation sources. Computational experiments were carried out and their results presented.
Keywords: radiative transfer equation, a Cauchy problem, Fresnel and diffuse matching conditions, Monte Carlo methods. 
@article{SEMR_2019_16_a94,
     author = {A. Kim and I. V. Prokhorov},
     title = {Initial-boundary value problem for a radiative transfer equation with generalized matching conditions},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1036--1056},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a94/}
}
TY  - JOUR
AU  - A. Kim
AU  - I. V. Prokhorov
TI  - Initial-boundary value problem for a radiative transfer equation with generalized matching conditions
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2019
SP  - 1036
EP  - 1056
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2019_16_a94/
LA  - en
ID  - SEMR_2019_16_a94
ER  - 
%0 Journal Article
%A A. Kim
%A I. V. Prokhorov
%T Initial-boundary value problem for a radiative transfer equation with generalized matching conditions
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2019
%P 1036-1056
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2019_16_a94/
%G en
%F SEMR_2019_16_a94
A. Kim; I. V. Prokhorov. Initial-boundary value problem for a radiative transfer equation with generalized matching conditions. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1036-1056. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a94/

[1] S. Chandrasekhar, Radiative transfer, Oxford University Press, London, 1950 | MR | Zbl

[2] V. S. Vladimirov, “Mathematical problems of the one-velocity theory of particle transport”, Transactions of the V. A. Steklov Mathematical Institute, 61 (1961), 3–158 | MR

[3] K. M. Case, P. F. Zweifel, Linear Transport Theory, Addison-Wesley. Co., Reading, MA, 1967 | MR | Zbl

[4] C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, 1975 | MR | Zbl

[5] A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, New York, 1978 | MR

[6] D. S. Anikonov, A. E. Kovtanyuk, I. V. Prokhorov, Transport Equation and Tomography, VSP, Utrecht–Boston, 2002 | MR | Zbl

[7] I. V. Prokhorov, I. P. Yarovenko, T. V. Krasnikova, “An extremum problem for the radiation transfer equation”, Journal of Inverse and Ill-Posed Problems, 13:4 (2005), 365–382 | DOI | MR | Zbl

[8] A. E. Kovtanyuk, I. V. Prokhorov, “Tomography problem for the polarized-radiation transfer equation”, Journal of Inverse and Ill-Posed Problems, 14:6 (2006), 609–620 | DOI | MR | Zbl

[9] I. V. Prokhorov, I. P. Yarovenko, V. G. Nazarov, “Optical tomography problems at layered media”, Inverse Problems, 24:2 (2008), 025019, 13 pp. | DOI | MR | Zbl

[10] G. Bal, “Inverse transport theory and applications”, Inverse Problems, 25:5 (2009), 025019, 13 pp. | MR | Zbl

[11] I. V. Prokhorov, I. P. Yarovenko, “Analysis of the tomographic contrast during the immersion bleaching of layered biological tissues”, Quantum Electronics, 40:1 (2010), 77–82 | DOI

[12] A. Hussein, M. M. Selim, “Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique”, Applied Mathematics and Computation, 318:13 (2012), 7193–7203 | DOI | MR | Zbl

[13] A. E. Kovtanyuk, A. Yu. Chebotarev, “An iterative method for solving a complex heat transfer problem”, Applied Mathematics and Computation, 219:17 (2013), 9356–9362 | DOI | MR | Zbl

[14] Yong Zhang, Hong-Liang Yi, He-Ping Tan, “Short-pulsed laser propagation in a participating slab with Fresnel surfaces by lattice Boltzmann method”, International Journal of Heat and Mass Transfer, 80 (2015), 717–726 | DOI

[15] Lin-Feng Qian, Guo-Dong Shi, Yong Huang, Yu-Ming Xing, “Backward and forward Monte Carlo method for vector radiative transfer in a two-dimensional graded index medium”, Journal of Quantitative Spectroscopy and Radiative Transfer, 200 (2017), 225–233 | DOI

[16] Zhen W., Shengcheng C., Jun Y., Haiyang G., Chao L., Zhibo Z., “A novel hybrid scattering order-dependent variance reduction method for Monte Carlo simulations of radiative transfer in cloudy atmosphere”, Journal of Quantitative Spectroscopy and Radiative Transfer, 189 (2017), 283–302 | DOI

[17] Cleveland M. A., Wollaber A. B., “Corrected implicit Monte Carlo”, Journal of Computational Physics, 359 (2018), 20–44 | DOI | MR | Zbl

[18] J. Sun, J. A. Eichholz, “Splitting methods for differential approximations of the radiative transfer equation”, Applied Mathematics and Computation, 322 (2018), 140–150 | DOI | MR | Zbl

[19] I. V. Prokhorov, “Boundary value problem of radiation transfer in an inhomogeneous medium with reflection conditions on the boundary”, Differential Equation, 36:6 (2000), 943–948 | DOI | MR | Zbl

[20] I. V. Prokhorov, “On the solubility of the boundary-value problem of radiation transport theory with generalized conjugation conditions on the interfaces”, Izvestiya: Mathematics, 67:6 (2003), 1243–1266 | DOI | MR | Zbl

[21] I. V. Prokhorov, “On the Structure of the Continuity Set of the Solution to a Boundary-Value Problem for the Radiation Transfer Equation”, Mathematical Notes, 86:2 (2009), 234–248 | DOI | MR | Zbl

[22] A. E. Kovtanyuk, I. V. Prokhorov, “A boundary-value problem for the polarized-radiation transfer equation with Fresnel interface conditions for a layered medium”, Journal of Computational and Applied Mathematics, 235:8 (2011), 2006–2014 | DOI | MR | Zbl

[23] A.A. Amosov, “Boundary value problem for the radiation transfer equation with reflection and refraction conditions”, Journal of Mathematical Sciences, 191:2 (2013), 101–149 | DOI | MR | Zbl

[24] A.A. Amosov, “Boundary Value Problem for the Radiation Transfer Equation with Diffuse Reflection and Refraction Conditions”, Journal of Mathematical Sciences, 193:2 (2013), 151–176 | DOI | MR | Zbl

[25] A. Amosov, M. Shumarov, “Boundary value problem for radiation transfer equation in multi-layered medium with reflection and refraction conditions”, Applicable Analysis, 95:7 (2016), 1581–1597 | DOI | MR | Zbl

[26] A.A. Amosov, “Radiative Transfer Equation with Fresnel Reflection and Refraction Conditions in a System of Bodies with Piecewise Smooth Boundaries”, Journal of Mathematical Sciences, 219:6 (2016), 821–849 | DOI | MR | Zbl

[27] I. V. Prokhorov, “Solvability of the initial-boundary value problem for an integro-differential equation”, Siberian Mathematical Journal, 53:2 (2012), 301–309 | DOI | MR | Zbl

[28] “The Cauchy problem for the radiative transfer equation with generalized conjugation conditions”, Computational Mathematics and Mathematical Physics, 53:5 (2013), 588–600 | DOI | MR | Zbl

[29] I. V. Prokhorov, A. A. Sushchenko, “On the well-possessedness of the Cauchy problem for the equation of radiative transfer with Fresnel matching conditions”, Siberian Mathematical Journal, 56:4 (2015), 736–745 | DOI | MR | Zbl

[30] I. V. Prokhorov, A. A. Sushchenko, A. Kim, “Initial boundary value problem for the radiative transfer equation with diffusion matching conditions”, Journal of Applied and Industrial Mathematics, 11:1 (2017), 115–124 | DOI | MR | Zbl

[31] A. A. Amosov, “Initial-Boundary Value Problem for the Non-Stationary Radiative Transfer Equation with Fresnel Reflection and Refraction Conditions”, Journal of Mathematical Sciences, 231:3 (2018), 279–337 | DOI | MR | Zbl

[32] A. A. Amosov, “Initial-Boundary Value Problem for the Non-stationary Radiative Transfer Equation with Diffuse Reflection and Refraction Conditions”, Journal of Mathematical Sciences, 235:2 (2018), 117–137 | DOI | MR | Zbl

[33] A. A. Amosov, “Nonstationary radiation transfer through a multilayered medium with reflection and refraction conditions”, Mathematical Methods in the Applied Sciences, 41:17 (2018), 8115–8135 | DOI | MR | Zbl

[34] A. Kim, I. V. Prokhorov, “Theoretical and Numerical Analysis of an Initial-Boundary Value Problem for the Radiative Transfer Equation with Fresnel Matching Conditions”, Computational Mathematics and Mathematical Physics, 58:5 (2018), 735–749 | DOI | MR | Zbl

[35] I. V. Prokhorov, “The Cauchy Problem for the Radiation Transfer Equation with Fresnel and Lambert Matching Conditions”, Mathematical Notes, 105:1 (2019), 80–90 | DOI | MR | Zbl

[36] G. A. Mikhailov, I. N. Medvedev, Optimization of Weighted Statistical Modeling Algorithms, Omega Print, Novosibirsk, 2011 (in Russian)

[37] S. M. Prigarin, T. V. Aleshina, “Monte Carlo simulation of ring-shaped returns for CCD LIDAR systems”, Russian Journal of Numerical Analysis and Mathematical Modeling, 30:4 (2015), 251–257 | Zbl

[38] The Stanford 3D Scanning Repository, http://graphics.stanford.edu/data/3Dscanrep/