Ambartsumyan’s invariance principle is applied to the nonlinear radiative transfer problem of determining the internal radiation field in a one-dimensional, anisotropic, scattering and absorbing medium when both of its boundaries are illuminated by intense radiative fluxes. Formulas are derived for adding and imbedding layers in media with finite geometrical thicknesses. It is shown that to find the internal radiation field in the nonlinear case, as in the linear case, it is not necessary to solve any new equations: it is sufficient to use only the explicit expressions and quantities found by solving the particular problem of the radiation emerging from the medium, i.e., the diffuse reflection and transmission problem. Then a complete set of differential equations is found for invariant imbedding. The standard two-point nonlinear boundary value problem for radiative transfer reduces to an initial value problem— the Cauchy problem. A new Cauchy problem, in which the spatial variables appear only as fixed parameters, is formulated by eliminating derivatives with respect to the layer thickness. In this way we arrive at a semilinear system of the Ambartsumyan’s complete invariance.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. V. Pikichyan, Astrophysics 53, 251 (2010).
V. A. Ambartsumyan, DAN Arm. SSR 38, 225 (1964).
V. A. Ambartsumyan, in: V. V. Sobolev, et al., ed., Theory of Stellar Spectra [in Russian], p. 91, Nauka, Moscow (1966), 388 pp.
R. Bellman, R. Kalaba, and M. Wing, Proc. Nat. Acad. Sci. USA 46, 1646 (1960).
H. V. Pikichian, in: H. Harutyunian, A. Mickaelian, and Y. Terzian, ed., Proc. of the Conf. “Evolution of Cosmic Objects through their Physical Activity” (dedicated to V. Ambartsumian’s 100-th anniversary, Sept 15–18, 2008, Yerevan-Byurakan, Armenia), p. 302, Publ. House of NAS RA, Yerevan (2010), 356p.
H. Pikichian, in: J. J. Carrol and T. A. Goldman, ed., Proc. International conf. “Lasers’97" (New Orleans, LA, December 15–19, 1997), p. 226, STS Press, McLean, VA (1998), 1011pp.
E. G. Yanovitskii, Astron. zh. 56, 833 (1979).
H. V. Pikichyan, Soobshch. Byur. Obs. 5 (1984).
V. A. Ambartsumyan, Astron. zh. 19, 30 (1942).
V. A. Ambartsumyan, DAN SSSR 38, 257 (1943).
V. A. Ambartsumyan, Zh. Eksp. Teor. Fiz. 13 (9–10), 323 (1943).
V. A. Ambartsumyan, DAN SSSR 43, 106 (1944).
V. A. Ambartsumyan, Izv. AN Arm. SSR, Estestv. nauki, Nos. 1–2, 31 (1944).
V. A. Ambartsumyan, DAN Arm. SSR 7, 199 (1947).
V. V. Sobolev, Radiative Energy Transfer in the Atmospheres of Stars and Planets [in Russian], GITTL, Moscow (1956), 391 pp.
S. Chandrasekhar, Radiative Energy Transfer [Russian translation], IL, Moscow (1953), 432 pp.
M. A. Mnatsakanyan and H. V. Pikichyan, ed., The Invariance Principle and its Applications (Proc. Symp., Byurakan, Oct. 26–30, 1981), p. 88, Izd. AN Arm. SSR, Erevan (1989), 522 pp.
R. Bellman and G. M. Wing, An introduction to Invariant Imbedding, (Classics in Appl. Math. vol. 8), John Wiley & Sons, Inc., New York (1975) (Philadelphia: SIAM (1992)), 248 pp.
V. I. Klyatskin, The Imbedding Method in the Theory of Wave Propagation (series: Modern Physics Problems) [in Russian], Nauka-GRFML, Moscow (1986), 256 pp.
J. Casti and R. Kalaba, Imbedding Methods in Applied Mathematics [Russian translation], Mir, Moscow (1976), 224 pp.
R. Bellman, Dynamic Programming [Russian translation], IL, Moscow (1960), 400 pp.
B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and their Applications in Gas Dynamics [in Russian], Nauka, Moscow (1978), 688 pp.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Astrofizika, Vol. 59, No. 1, pp. 131–143 (February 2016).
Rights and permissions
About this article
Cite this article
Pikichyan, H.V. Internal Radiation Field in the Nonlinear Transfer Problem for a One-Dimensional Anisotropic Medium. I.. Astrophysics 59, 114–125 (2016). https://doi.org/10.1007/s10511-016-9421-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10511-016-9421-1