Lie Group Analysis of the Integral Equations Related to Neutron Slowing-Down Theory

This work applies the Lie Group Theory (LGT) to the neutron slowing-down equations for the n’th lethargy interval with the goal of defining the symmetry group associated with Dawn’s analytical solution. We also demonstrate two competing methods of the LGT and how they each result in the same solution and symmetry group. The two methods differ by taking advantage of the definition of a symmetry group from either a geometrical perspective or an algebraic perspective. The methods are the Traditional Lie Algorithm, which we apply to the equivalent system of ordinary differential equations for neutrons slowing down, as well as the Grigoriev-Meleshko Method, which we apply directly to the Volterra integral equation for neutrons slowing down. We also discuss the physical meaning of the symmetry group related to Dawn’s solution.