Application of the theory of Linear Singular Integral Equations and Contour Integration to Riemann Hilbert Problems for determination of new decoupled expressions of Chandrasekhar's X- and Y- functions for slab geometry in Radiative Transfer

Abstract

In Radiative transfer, the intensities of radiation from the bounding faces of finite slab are obtained in terms of X- and Y- functions of Chandrasekhar . Those are non linear non homogeneous coupled integral equations . Those non linear integral equations are meromorphically extended to the complex plane to get linear non homogeneous coupled integral equations. Those linear integral equations are converted to linear singular integral equations with. linear constraints . Those singular integral equations are then transformed to non homogeneous Riemann Hilbert Problems. Solutions of those Riemann Hilbert Problems are obtained using the theory of linear singular integral equations to decouple those X- and Y- functions. New forms of linear non homogeneous decoupled integral equations are derived for X- and Y- function separately with new linear constraints. Those new decoupled integral equations are transformed into linear singular integral equations to get two new separate non homogeneous Riemann Hilbert problems and to find solutions in terms of one known N- function and five new unknown functions in complex plane . All five functions are represented in terms of N-functions using the theory of contour integration >. Those X- and Y- functions are finally expressed in terms of that N - function and also in terms of H- functions of Chandrasekhar and of integrals in Cauchy principal value sense in the complex plane and real plane. both for conservative and non conservative cases . The H - functions for semi infinite atmosphere are derived as a limiting case from the expression of X- function of finite atmosphere.