On numerical evaluation of the H-functions of transport problems by kernel approximation for the albedo \(0<\omega \leq 1\) (Q1058841)

The second author [ibid. 50, 187-203 (1977; Zbl 0372.33012)] represented the H-functions of transport problems for the albedo \(\omega\in [0,1]\) in the form \(H(z)=R(z)-S(z)\) where R(z) is a rational function of z and S(z) is regular on \([-1,0]^ c\). In this paper we represent S(z) through a Fredholm integral equation of the second kind with a symmetric real kernel L(y,z) as \(S(z)=f(z)-\int^{1}_{0}L(y,z)S(y)dy\). The problem is then solved as an eigenvalue problem. The kernel is converted into a degenerate kernel through finite Taylor's expansion and the integral equation for S(z) takes the form: \(S(z)=f(z)-\sum^{N}_{i=1}\chi_ i\int^{1}_{0}F_ i(z)F_ i(y)S(y)dy\) (which is solved by the usual procedure) where \(\chi_ r's\) are the discrete eigenvalues and \(F_ r's\) the corresponding eigenfunctions of the real symmetric kernel L(y,z).