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Initial-Boundary Value Problem for the Non-Stationary Radiative Transfer Equation with Fresnel Reflection and Refraction Conditions

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We consider the initial-boundary value problems for the nonstationary radiative transfer equation in a system of semitransparent bodies with boundary and initial data in the complete scale of Lebesgue spaces. We establish the unique solvability of the first initial-boundary value problem, the problem with “shooting conditions,” and the problem with Fresnel reflection and refraction conditions on the body boundaries.

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References

  1. I. V. Prokhorov, “Boundary value problem of radiation transfer in an inhomogeneous medium with reflection conditions on the boundary,” Differ. Equ. 36, No. 6, 943-948 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  2. I. V. Prokhorov, “On the solvability of a boundary value problem in radiation transfer theory with generalized conjugation conditions at the interface between media,” Izv. Math. 67, No. 6, 1243–1266 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  3. I. V. Prokhorov and I. P. Yarovenko, “Boundary value problem of the for transport theory for multi-layered media with generalized conjugacy conidtions” [in Russian] Sib. Zh. Ind. Mat. 6, No. 1, 93–107 (2003).

  4. I. V. Prokhorov, I. P. Yarovenko, and T. V. Krasnikova, “An extremum problem for the radiation transfer equation”, J. Inverse Ill-Posed Probl. 13, No. 3-6, 365–382 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. E. Kovtanyuk and I. V. Prokhorov, “A boundary-value problem for the polarizedradiation transfer equation with Fresnel interface conditions for a layered medium”, J. Comput. Appl. Math. 235, No. 8, 2006–2014 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. A. Amosov, “Boundary value problem for the radiation transfer equation with reflection and refraction conditions,” J. Math. Sci., New York 191, No. 2, 101–149 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. A. Amosov, “Boundary value problem for the radiation transfer equation with diffuse reflection and refraction conditions,” J. Math. Sci., New York 193, No 2, 151–176 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. A. Amosov, “The radiation transfer equation with reflection and refraction conditions. Continuous dependence of solutions on the data and limit passage to the problem with “shooting conditions”,” J. Math. Sci., New York 195, No. 5, 569–608 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. A. Amosov, “The conjugate boundary value problem for radiation transfer equation with reflection and refraction conditions,” J. Math. Sci., New York 202, No. 2, 113–129 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. A. Amosov, “On some properties of the boundary value problem for the radiation transfer equation with diffuse reflection and refraction conditions,” J. Math. Sci., New York 207, No 2, 118–141 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Amosov, “Boundary value problems for the radiative transfer equation with reflection and refraction conditions,” In: Current Trends in Analysis and its Applications, pp. 625–636, Birkhäuser, Basel (2015).

  12. A. A. Amosov, “Radiative transfer equation with diffuse reflection and refraction conditions in a system of bodies with piecewise smooth boundaries,” J. Math. Sci., New York 216, No 2, 155–181 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. A. Amosov, “Radiative transfer equation with Fresnel reflection and refraction conditions in a system of bodies with piecewise smooth boundaries,” J. Math. Sci., New York 219, No. 6, 821-849 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. A. Amosov and M. G. Shumarov, “Boundary value problem for radiation transfer equation in multilayered medium with reflection and refraction conditions”, Appl. Anal. 95, No. 7, 1581–1597 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. A. Amosov, Boundary Value Problems for the Radiation Transfer Equation with Reflection and Refraction Conditions [in Russian], Tamara Rozhkovskaya Publisher, Novosibirsk (2017).

    Google Scholar 

  16. I. V. Prokhorov, “Solvability of the initial-boundary value problem for an integrodifferential equation,” Sib. Math. J. 53, No. 2, 301–309 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  17. I. V. Prokhorov, “The Cauchy problem for the radiative transfer equation with generalized conjugation conditions,” Comput.Math. Math. Phys. 53, No. 5, 588–600 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  18. I. V. Prokhorov and A. A. Sushchenko, “On the well-posedness of the Cauchy problem for the equation of radiative transfer with Fresnel matching conditions,” Sib. Math. J. 56, No. 4, 736–745 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  19. I. V. Prokhorov, A. A. Sushchenko, and A. Kim, “Initial boundary value problem for the radiative transfer equation with diffusion matching conditions,” J. Appl. Ind. Math. 11, No. 1, 115–124 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. S. Vladimirov, “Mathematical problems in the theory of single-velocity particle transfer” [in Russian], Tr. MIAN SSSR 61, 3–158 (1961).

    Google Scholar 

  21. T. A. Germogenova, Local Properties of Solutions of the Transport Equation [in Russian], Nauka, Moscow (1986).

    MATH  Google Scholar 

  22. T. A. Germogenova, “Generalized solutions of boundary value problems for the transfer equation,” U.S.S.R. Comput. Math. Math. Phys. 9, No. 3, 139–166 (1971).

  23. V. I. Agoshkov, Boundary Value Problems for Transport Equations: Functional Spaces, Variational Statements, Regularity of Solutions, Birkhäuser, Basel etc. (1998).

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Authors and Affiliations

  1. National Research University “Moscow Power Engineering Institute”, 14, Krasnokazarmennaya St, Moscow, 111250, Russia

    A. A. Amosov

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  1. A. A. Amosov
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Correspondence to A. A. Amosov.

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Translated from Problemy Matematicheskogo Analiza 91, 2018, pp. 3-51.

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Amosov, A.A. Initial-Boundary Value Problem for the Non-Stationary Radiative Transfer Equation with Fresnel Reflection and Refraction Conditions. J Math Sci 231, 279–337 (2018). https://doi.org/10.1007/s10958-018-3822-0

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  • DOI: https://doi.org/10.1007/s10958-018-3822-0

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