Fractional two energy groups matrix representation for nuclear reactor dynamics with an external source

Abstract

Nuclear reactor dynamics is the study of the time-dependent behavior of neutron density in an exceeding reactor. Knowledge of however the density in a reactor, which can respond to numerous perturbations, is important to a safe and economical style. Unfortunately, this information is extremely tough to get. To study the behavior of nuclear reactors it is necessary to solve the time-dependent two energy groups neutron diffusion model which is a system of stiff coupled partial differential equations. The two energy groups point kinetics equations with delayed neutrons derived from the neutron diffusion equations in the presence of the time-dependent external neutron source is solved analytically. Since the fractional order differential operator is non-local, then the fractional model of the two energy groups point kinetics equations can be a good representation for the neutron density in the nuclear reactors. In this work, the two energy groups fractional point kinetics equations with multi-group of delayed neutron precursors (2EFPKE) are formulated in the matrix form and two techniques are proposed to solve this system of differential equation analytically, which based on Laplace transformation and eigenvalues and corresponding eigenvectors of the coefficient matrix. The analytical procedure to the proposed fractional model in nuclear reactor dynamics is described and investigated for different types of time-varying reactivity. The performance of developed methods has been tested for both fast and thermal neutron density in the case of the time-dependent external neutron source. In addition, the anomalous diffusion processes, sub-diffusion, and super-diffusion processes are discussed for various fractional order.