This chapter deals with the solution of the multigroup neutron diffusion equation under an uncertain environment. Here, the multigroup neutron diffusion equation for the steady-state case is considered, and a two-group neutron diffusion equation for an example problem is investigated by using the fuzzy finite element method. An example benchmark problem is demonstrated with uncertain parameters. Various parameters such as thermal conductivity, diffusion, group fission and neutron interaction constants are taken as fuzzy, and uncertain solutions, viz. thermal- and fast-group fluxes, are discussed.

Fuzzy Finite Element for Coupled Differential Equations

Let us consider the standard fuzzy coupled differential equations in the а-cut form as

Equation 8.1 may be written in compact form as follows:
where

'~' represents the fuzzy numbers

Cij, i ? j = 1,2 and f_{i}, i = 1,2 are the coefficients and flux in the system

To find the approximate uncertain flux (ф), Galarkin's weighted residue method has been used here. Considering the linear element discretization, the shape functions would be Г ~ ~ ~]^{т}

[ni N2 J = 1 - у X , where l is the length of each element of the domain. Now, multiplying the shape functions with the residue of Equation 8.2, we get

Integrating Equation 8.3 over the domain Q, we get

The solution of Equation 8.4 gives the uncertain neutron flux (f). This idea has been used for the formulation of multigroup neutron diffusion equations.