Generalization of the analytical inversion method for the solution of the point kinetics equations

Generalized and Stability Rational Functions for Dynamic Systems of Reactor Kinetics

“…Here, A = [a ] is any point kinetics-like matrix, that is, = 0 except when = 1 or = 1 or = . Consequently, the previous expression is more general compared with that introduced by Aboanber and Nahla [30]. The ℓ in (18) being (generally) real numbers, the quadratic factors will have complex conjugate roots.…”

“…Let us first state in this section the following important theorem before proceeding with the generalized rational function. The stated theorem for the matrices A and A , Abaonber and Nahla [30], provides us with a mathematical formula, which permits to approximate some special functions. Let the matrix A have eigenvalues and eigenvectors ( = 1, 2, .…”

“…Due to the particular structure of , a semianalytical inversion (Analytical Inversion Methods (AIMs) [30]) can be performed for each elementary step resulting eventually in substantial time savings. They developed a general expression for the inverse of [I− A], where for a real and a point kinetic matrix A( ) of (4) with the reactivity ( ) appropriately averaged, the following expression is introduced:…”

“…Several other creative schemes have been implemented as well as da Nóbrega [27], Lawrence and Doring [28], and Aboanber and Nahla [29] use a Padé approximation to the solution of the point kinetics equations. Also, the Padé approximation technique is applied to the nonlinear system including temperature feedback via analytical inversion method [30], CORE [31]. Recently, a very efficient method for obtaining a highly accurate solution of reactor kinetic equation was proposed in [32][33][34].…”

Neutron Density Calculation Using the Generalised Adams-Bashforth-Moulton Method

“…The fast and delayed neutron lifetimes are of different orders of magnitude, converting the equations of point kinetics into a stiff system. Several methods have been proposed to solve the equations of point kinetics: In [1], the confinement method (SCM) is proposed to overcome stiffness, whereas other authors have used the generalised Runge-Kutta (GRK) method [2], the Padé approximations [3], the generalization of the analytical inversion method (AIM) [4], and the analytical method (AEM), which used exponential functions [5]. It has also been demonstrated that the point kinetics equations can be solved numerically using the reactivity Piecewise Constant Approximation method (PCA) [6], using a numerical algorithm called: Constant Reactivity (CORE) to calculate nuclear density [7,8] presented a numerical integral method and investigated the neutron density produced by inserting different forms of reactivity in thermal reactors with multiple groups of the neutrons using the Best Function (BBF).…”