Legendre polynomials and their application in study of one dimensional transport equation and scattering Kernel

Ordinary differential equation is frequently found in physics and other technical fields. In particular it occurs when solving Laplace's equation (and related partial differential equations) in spherical co-ordinates. The Legendre's equation and Legendre's polynomials are used broadly where the directional dependence of some quantity is treated openly such as particular transport problems. In this paper, we describe a new scattering kernel and general theoretical scheme for the evaluation of the discrete and continuum eigenvalue spectrum in one dimensional slab geometry neutron transport equation. Firstly, some useful properties of Legendre polynomial which revealed during the definition of new scattering kernel are discussed. By using the scattering kernel in one-dimensional neutron transport equation we obtained an integral equation for angular part of the angular flux.