On the evaluation of generalized exponential integral functions

This paper deals with generalized exponential integral (GEI) functions arising in the study of anisotropic scattering in a multidimensional media. These functions are represented as finite linear combinations of basic GEIs introduced in this work. The recurrence relations are derived for the linear combination coefficients and basic GEIs. Numerical results are also given. The extensive test calculations show that the work proposed in this algorithm is the most efficient practical computations of GEI functions.     

Uniform convergence and the properties of the exponential and generalized exponential integral functions

The exponential integral function (EIF) and the generalized exponential integral function (GEIF) are defined as

     

Analytical evaluation of cylindrical exponential integrals arising in radiative transfer

New analytical results are presented performing to cylindrical exponential integral (CEI) functions for integer and noninteger values of parameter n. These integrals are often employed of two-dimensional radiative transfer in an absorbing-emitting medium and determination of the radiative flux in cylindrical media. The simple and efficient algorithm for the calculation of these functions is developed. The series expansion relations established in this work are accurate enough in the whole range of parameters.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

Radiative transfer in absorbing, emitting and linearly anisotropic-scattering inhomogeneous cylindrical medium

Radiative integral transfer equations for one-dimensional solid cylinder with absorbing, emitting and linearly anisotropic-scattering inhomogeneous medium were derived by Abulwafa et al. (JQSRT 62 (1999) 755). The anisotropic terms in the integral equations and their results for anisotropic benchmark problems (JQSRT 66 (2000) 487) are inaccurate. In this study, the integral equations for absorbing, emitting and linearly anisotropic-scattering medium are rederived, and the integral equations for one-dimensional solid cylindrical medium are solved. The results are compared with selected cases using the discrete ordinates S₁₆ and the exact solutions available in the literature.