# Point Kinetic Neutron Diffusion Equation

In this section, the concept of fuzziness along with the stochastic behaviour of the point kinetic neutron diffusion equation has been modelled. The fuzzy stochastic model is investigated numerically by extending the EMM to fuzzy form. The uncertain neutron density and the delayed neutron population are obtained and compared with the Monte Carlo and stochastic principal component analysis (PCA) solutions. Various combinations of fuzzy parameters are considered, and the uncertain neutron density and delayed neutron population are obtained in different cases. Finally, the sensitivity of these fuzzy parameters with stochastic behaviour has been investigated.

## Stochastic Point Kinetic Model with Fuzzy Parameters

The point kinetic equation has been modelled in terms of stochastic (Hayes and Allen 2005) by considering the birth and death processes of the neutron and the precursor population.

The coupled deterministic time-dependent equations for the neutron density and the delayed neutron precursors may be represented as

for *i* = 1,2 , . . . , m.

Here, we have considered uncertain parameters, viz. delayed neutron fraction, source and initial condition as fuzzy (Equations 11.23 and 11.24). The fuzzy delayed neutron fraction b in а-cut form may be written as b = |^b (a), b (a) J.

These fuzzy parameters are introduced in Equations 11.23 and 11.24, which give

for *i* = 1, 2,. . ., m, where '~' represents fuzzy numbers.

The neutron captures are considered as deaths whereas the fission process is taken as a pure-birth process, where v - 1 neutrons are born in each fission along with a precursor contribution. The number of new neutrons born in each fission is (1 - b )v -1. Let us consider N = / *(r )n (t*) and *C _{i} - g_{i}* (r )c

*(t*), where it is assumed that N and Q are separable in time and space. Proceeding further and using the technique given in Chapter 10, we get the following uncertain point kinetic equation:

where

*n* (*t*) is the total number of uncertain neutrons

*C* (*t*) is the total number of uncertain precursors of the ith type at time *t*

m *Za* (1 -fc») + *DV ^{2}*

the reactivity p =

Za +DV^{2} 2 * _{f}* 1 1

the number of neutrons per fission process is a = »—, *l =* and

So*(r, t*) ^{v}

^{11} -^{q (t} hf)

Finally, we obtain the following Ito SDE for one precursor in terms of fuzzy: where

Here, *W _{1}(t)* and

*W*are Wiener processes.

_{2}(t)Generalizing Equation 11.23, the stochastic point kinetic equation for *m* precursors becomes

where

and

Here,

In compact form, Equation 11.30 may be written as

where *x = \_n C _{i}* c

_{2}•••

*c*]

_{m}^{T}.