Variational approach to the Milne problem with a specular and diffuse reflecting boundary

An approximate solution of the Milne integral equation for specular and diffuse reflecting boundary is deduced by using variational calculus. An approximate, (four variable parameters) expression, for the particle density and emergent angular distribution has been proposed. Numerical results for extrapolated endpoint, particle density and angular distribution in a region close to the boundary are calculated. The percentage difference for extrapolated endpoint, taking the exact values as a reference, is approximately 1.4x10-10 for a non-reflecting medium and does not exceed 0.01% for all degrees of blackness in the diffusion reflection. The results for a combination of specular and diffuse reflections are also given