Invariant imbedding and Chandrasekhar's planetary problem of radiative transfer
In connection with Chandrasekhar's planetary problem of radiative transfer the total scattering and the diffuse transmission functions have been discussed by several authors (cf. Chandrasekhar, 1950; van de Hulst, 1948; Sobolev, 1948; Bellman,et al., 1967; Kagiwada and Kalaba, 1971). With the aid of the Bellman-Krein formula for the resolvent kernel of the auxiliary equation governing the source function, we show how the invariant imbedding equations governing the diffuse scattering and transmission functions can readily be obtained. So far as we know, the Cauchy system of the functional equations for the scattering and transmission functions is new and is well-suited for the numerical computation.
KeywordsNumerical Computation Functional Equation Radiative Transfer Source Function Transmission Function
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- Bellman, R. E., Kalaba, R. E., and Prestrud, M. C.: 1963,Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness, American Elsevier Publishing Company, Inc., New York.Google Scholar
- Bellman, R., Kagiwade, H., Kalaba, R., and Ueno, S.: 1967,Icarus 7, 365–71.Google Scholar
- Bellman, R., Kagiwada, H., Kalaba, R., and Ueno, S.: 1968,J. Math. Phys. 9, 906–8.Google Scholar
- Busbridge, Ida W.: 1960,The Mathematics of Radiative Transfer, Cambridge University Press, London.Google Scholar
- Casti, J., Kalaba, R., and Ueno, S.: 1969,J. Quant. Spectr. Rad. Transfer 9, 537–52.Google Scholar
- Chandrasekhar, S.: 1950,Radiative Transfer, Oxford University Press, London.Google Scholar
- Hulst, H. C. van de: 1948,Astrophys. J. 107, 220–46.Google Scholar
- Kagiwada, H. and Kalaba, R.: 1971,J. Quant. Spectr. Rad. Transfer. 11, 1101–9.Google Scholar
- Sobolev, V. V.: 1948,Dokl. Akad. Nauk, SSSR 61, 5.Google Scholar
- Sobolev, V. V.: 1963,A Treatise on Radiative Transfer (English Translation), D. Van Nostrand Company, Inc., New York.Google Scholar