Uncertainty Caused by the Stochastic Process
The interactions and scattering neutrons in reactors follow stochastic processes. In neutron diffusion, the neutron populations show randomness. As such, a birth-death stochastic process is found in these systems. Hence, the governing differential equation for neutron diffusion is converted into stochastic.
The concept of stochastic differential equation (SDE) was initiated by the great philosopher Einstein in 1905 (Sauer 2012). A mathematical connection between the microscopic random motion of particles and the macroscopic diffusion equation was presented. Later, it was seen that the SDE model plays a prominent role in a range of application areas such as mathematics, physics, chemistry, mechanics, biology, microelectronics, economics and finance. Earlier, SDEs were solved using the Ito integral as an exact method, which is discussed in Malinowski and Michta (2011). But using the exact method proved difficult to study nontrivial problems and hence approximation methods came to be used. In 1982, Rumelin (1982) defined general Runge-Kutta approximations for the solution of SDEs and an explicit form of the correction term has been given. Kloeden and Platen (1992) discussed about the numerical solutions of the SDE. Discrete time strong and weak approximation methods were used by Platen (1999) to investigate the solution of SDEs. Next, Higham (2001) gave a major contribution to solve the approximate solutions of SDEs. Furthermore, Higham and Kloeden (2005) investigated nonlinear SDEs numerically. They presented two implicit methods for the Ito SDEs with Poisson-driven jumps. The first method is a split-step extension of the backward Euler method, and the second method arises from the introduction of a compensated, martingale, form of the Poisson process. Hayes and Allen (2005) solved the stochastic point kinetic reactor problem. They modelled the point stochastic reactor problem into the ordinary time-dependent SDE and studied the stochastic behaviour of the neutron flux.