Chapter 8 - Fractional neutron diffusion equation in space and time

Abstract

The fractional neutron diffusion equation establishes nonlocal effects because the approximation considers differential operators that are of fractional order. When the operator differential fractional corresponds to spatial coordinates, in addition to nonlocal effects, it considers memory in space due to fractional-order Laplacian, so these equations are known as spatial memory nonlocal. And when the operator differential fractional-order corresponds to time as the independent variable, the equations are known as time memory nonlocal. The aim of this chapter is nuclear reactor analysis with spatial and time memory with nonlocal differential operators. The neutron diffusion with space memory is analyzed in nuclear reactors in 1D, 2D, and 3D with different neutron energy groups, that is, fractional multigroup neutron diffusion equations to obtain the neutron flux and the effective multiplication factor. The numerical experiments showed results that are comparable to the transport theory with codes based on Monte Carlo methods. The behaviors for a very short time, short time, and a very long time of the neutron flux for a purely absorbing medium, a heterogeneous medium, and a highly scattering medium are presented and the results are compared with the theory of neutron transport.