Resolvent method: exact solutions of half-space transport problems by elementary means

Details Resolvent method: exact solutions of half-space transport problems by elementary means Resolvent method: exact solutions of half-space transport problems by elementary means

Jun 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994a%26a...286..328i&link_type=abstract

Astronomy and Astrophysics 286, 328-337 (1994)

Astronomy and Astrophysics

Astrophysics

10

Radiative Transfer, Methods: Analytical

Scientific paper

A new version of the resolvent method for solving half-space transport problems is presented. We solve the integral transfer equation with the {LAMBDA}-operator. The approach is based on the use of a non-linear equation relating the surface Green function G_0_(τ) for the semi-infinite medium and the Green function Ginfinity_(τ) for the infinite medium. This Ginfinity_(τ) is assumed known. The equation is first used to derive, literally in one line, the famous sqrt(ɛ) law. Then, by a short series of elementary transformations, we obtain the Fock-Chandrasekhar explicit expression for the H-function. Assuming that the kernel function of the {LAMBDA}-operator is a Laplace-type integral, we derive the usual explicit expression for G_0_(τ) in the form of a similar integral. The integrand involves only the functions appearing in the explicit expression for G_0_(τ), and the H-function. And again, the derivation is elementary, i.e., without any use of the complex plane, and surprisingly short; the non-linear H-equation is completely by-passed. Alternative equations satisfied by the H-function are presented, among them a linear integral equation with a non-singular kernel. The standard non-linear H-equation is used to find the asymptotic behavior of the solution of the homogeneous transfer equation in deep layers of the semi-infinite atmosphere. In conclusion, a brief comment on the development of the resolvent method is given.

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