A flnite element technique for solving an integral-difierential equation

In this paper is solved a neutron transport problem by the flnite element simulations. We present a method that replaces a boundary value problem for one-dimensional transport equation with a boundary value problem for a difiusion equation. To solve this new problem we will construct and then we minimize a functional, using the speciflc techniques of the flnite element methods. Numerical examples allow us a detailed analysis of the errors according to the mesh step. This paper deals with a typical problem of the mathematical-physics: the solving of neutron transport equation. In a reactor, the neutrons are yield at the flssion of the nucleus and they are named the fast neutrons. These neutrons have an average velocity equal to 2 ¢ 10 7 m=s, and are subjected to a slowing process: their energy decreases until these are in an equilibrium state with the other atoms of the environ- ment. When the reactor is in a stationary state, the particles have the tendency to move from a region with a great density to another with a small density and thus we get a uniform density. This process is named the difiusion. The main problem in the nuclear reactor theory is to flnd the neutrons distribution in the reactor, hence its density, which is the solution of an integral-difierential equation named the neutron transport equation. An exact solution of integral-difierential equation was found only in the parti- cular cases. We remind the papers of the authors: Brezis ((5)), Cardona and Vilhena ((6)), Davidson and Sykes ((10)), Kadem ((10)), Siewert ((23)), Wilson and Sen ((29)). Generally, these are obtained with the help of the methods of mathematical analysis, abstract functional analysis and the spectral methods. ⁄