# Fuzzy Multigroup Neutron Diffusion Equation

Let us consider the 1D reactor core, which is divided into various energy groups and different regions having constant material properties. The discretization of the reactor core into various groups is shown in Figure 7.1.

The general form of the fuzzy neutron diffusion equation may be written as (Glasstone and Sesonke 2004)

where

*g = 1,* 2,. . . *G*

'~' denotes the fuzziness

Using this procedure, the shape functions are multiplied with Equation 8.5 and minimized to get

Integrating Equation 8.6 over the domain, we get

Further simplification of Equation 8.7 gives us a system of algebraic equations. In matrix form, this algebraic equations look like

Here,

[k*g* J is the fuzzy stiffness matrix for the coupled fuzzy neutron diffusion equation is the fuzzy neutron flux vector

In steady case, the [Q J matrix is zero. Equation 8.8 is a fully fuzzy system of equations,

which is tedious to handle. But this difficulty may be overcome by using the method discussed in Nayak and Chakraverty (2013).

Let us consider the two-group fuzzy neutron diffusion equation (Wood and De Oliveira 1984)

where

Using the Galarkin fuzzy finite element formulation for each element having length *l,* we get the following stiffness matrix:

where *mij, i, j =* 1, 2 are fuzzy numbers.