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Fractional order neutron point kinetics equations are generalization of the classical neutron point kinetics equations. The system of the fractional neutron point kinetics equations is treated in different context by many authors ignoring an important term related to the derivative of the reactivity. In this paper, we derive a time fractional neutron point kinetics equations (FNPK) model considering a new term for the time derivative of the reactivity. Fractional Leibenz rule is used for such derivation. The effect resulting from the additional term of the reactivity derivative is discussed. It has been found that without this additional term, the modified model will be compliant with the earlier published models for the FNPK. The proposed model of FNPK describes infinite and finite bar reactor cores according to the nonleakage probability and the geometric buckling values. We have applied an effective implicit difference approximation to solve the modified model for finite cylindrical reactor core for both average one group and six groups of delayed neutron precursors. The neutron density results with different values of fractional order for step, ramp and oscillatory reactivities are shown and compared with the classical neutron point kinetics equations.

In this paper, our purpose is to find approximate solutions of fractional neutron point kinetic equations by using non-standard finite difference method. The fractional neutron point kinetic equations are modelled with average one group of delayed neutron precursors and the fractional derivative is given in the form of Grunwald-Letnikov. The efficiency and reliability of the suggested approach are proved by some numerical experiments for critical reactivity, supercritical reactivity and subcritical reactivity for various values of fractional order. It is found that the nonstandard finite difference method (NSFDM) is preferable than the standard finite difference method (SFDM). Also, the stability of the numerical scheme is investigated. The stability range of the step size is introduced for different values of the anomalous diffusion order ( α ) and of the relaxation time ( τ ) . Numerical results and graphs for neutron flux for different values of the anomalous order and of the relaxation time are shown and compared with the classical solutions.

An extensive knowledge of the spatial power distribution is required for the design and analysis of different types of current-generation reactors, and that requires the development of more sophisticated theoretical methods. Therefore, the need to develop new methods for multidimensional transient reactor analysis still exists. The objective of this paper is to develop a computationally efficient numerical method for solving the multigroup, multidimensional, static and transient neutron diffusion kinetics equations. A generalized Runge–Kutta method has been developed for the numerical integration of the stiff space–time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic time step control. In addition, the A ( α )-stability properties of the method are investigated. The analyses of two- and three-dimensional benchmark problems as well as static and transient problems, demonstrate that very accurate solutions can be obtained with assembly-sized spatial meshes. Preliminary numerical evaluations using two- and three-dimensional finite difference codes showed that the presented generalized Runge–Kutta method is highly accurate and efficient when compared with other optimized iterative numerical and conventional finite difference methods.

The point reactor kinetics equations of reactor are solved analytically in the presence of delayed neutron with Newtonian feedback for different types of reactivity input using a straightforward recurrence relation of a power series. Analytical or point-function reactivity variation are introduced together with constant or time-varying reactivity compensation. Numerical evaluation is performed by the developed PWS (Power Series Solution) code, written in Visual FORTRAN for a personal computers. The code solves the general non-linear kinetics problems with six groups of delayed neutron. Practical use of the method is verified through computed reactor response for representative reactivity addition functions of various types. In addition, comparison with the conventional methods confirmed the superiority of the developed power series code with different types of reactivity feedback.

The reactor kinetics equations are reduced to a differential equation in matrix form convenient for explicit power series solution involving no approximations beyond the usual space-independent assumption. The coefficients of the series have been obtained from a straightforward recurrence relation. Numerical evaluation is performed by PWS (power series solution) code, written in Visual FORTRAN for a personal computer. The results are applied to the step reactivity insertion, ramp input, zigzag input, and oscillatory reactivity changes. When the reactivity is given, including the case in which the feedback reactivity is a function of neutron density, the developed method can provide a straightforward procedure for computing reactor dynamics problems. The solution of this method was compared to some other analytical and numerical solutions of the point reactor kinetics equations; the results proved that the approach is both efficient and accurate to several significant figures.