The~boundary-value problem for the~equations of radiation transfer of polarized light

A. V. Latyshev ; A. V. Moiseev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 97-115.

Résumé

The theory of the solution of half-space boundary-value problems for Chandrasekhar's equations describing the scattering of polarized light in the case of a combination of Rayleigh and isotropic scattering with arbitrary photon survival probability in an elementary scattering is constructed. A theorem on the expansion of the solution in terms of eigenvectors of discrete and continuous spectra is proved. The proof reduces to solving the Riemann–Hilbert vector boundary-value problem with a matrix coefficient. The matrix that reduces the coefficient to diagonal form has eight branch points in the complex plain. The definition of an analytical branch of a diagonalizing matrix gives us the opportunity to reduce the Riemann–Hilbert vector boundary-value problem to two scalar boundary-value problems on the major cut $[0,1]$ and two vector boundary value problems on the supplementary cut. The solution of the Riemann–Hilbert boundary-value problem is given in the class of meromorphic vectors. The solvability conditions enable unique determination of the unknown coefficients of the expansion and the free parameters of the solution.

Bibliographie

[1] Chandrasekar S., Perenos luchistoi energii, Izd. inostrannoi literatury, M., 1953

[2] Latyshev A. V., “O reshenii granichnykh zadach dlya uravnenii perenosa izlucheniya”, ZhVM i MF, 34:2 (1994), 234–245 | MR | Zbl

[3] Latyshev A. V., “Vektornaya kraevaya zadacha Rimana–Gilberta v granichnykh zadachakh rasseyaniya polyarizovannogo sveta”, ZhVM i MF, 35:7 (1995), 1108–1127 | MR | Zbl

[4] Latyshev A. V., “Analiticheskoe reshenie vektornykh modelnykh kineticheskikh uravnenii s postoyannym yadrom i ikh prilozheniya”, Teor. i matem. fiz., 97:2 (1993), 283–303 | MR | Zbl

[5] Burniston E. E., Siewert C. E., “Half-range expansion theorems in studies of polarized light”, J. Math. Phys., 11:12 (1970), 3416–3420 | DOI | MR

[6] Bond G. R., Siewert C. E., “On the nonconservative eqation of transfer for a combination of Rayleigh and isotropic scattering”, The Astrophysical Journal, 164:1 (1971), 96–110 | MR

[7] Siewert C. E., Burniston E. E., “An explicit closed-form result for the discrete eigenvalue in studies of polarized light”, The Astrophysical Journal, 173:2 (1972), 405–406 | DOI

[8] Siewert C. E., Burniston E. E., “On existence and uniqueness theorems concerning the $H$-matrix of radiative transfer”, The Astrophysical Journal, 174:3 (1972), 629–641 | DOI | MR

[9] Siewert C. E., Maiorino J. R., “The complete solution for the scattering of polarized light in a Rayleigh and isotopically scattering atmosphere”, Astrophysics and Space Science, 72 (1980), 189–201 | DOI | MR

[10] Siewert C. E., “On the half-range orthogonality appropriate to the scattering of polarized light”, J.Q.S.R.T, 12:4 (1972), 683–694 | MR

[11] Germogenova T. A., Konovalov N. V., Kuzmina M. G., “Osnovy matematicheskoi teorii perenosa polyarizovannogo izlucheniya (strogie rezultaty)”, Trudy simpoziuma “Printsip invariantnosti i ego prilozheniya” (Byurokan, 26–30 oktyabrya 1981), Izd-vo AN ArmSSR, Erevan, 1989

[12] Keiz K., Tsvaifel P., Lineinaya teoriya perenosa, Mir, M., 1972

[13] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977 | MR