Applications of Fractals and Fractional Calculus in Nuclear Reactors

Dear Colleagues,

Fractional calculus has been applied in several areas in the last 45 years, including physics, electrical engineering, robotics, signal processing, chemical, bioengineering, and mathematics, but mostly in chaos and control theory. In the field of nuclear science and technology, its history is much shorter; however, there has been a significant rise in its application since 2010. Fractional models of nuclear science and technology have been developed to overcome certain limitations related to the classical approaches, considering more general physical scenarios and non-local and memory effects in the modeling of the neutron population. Due to the advances and results achieved in nuclear science and technology in the last 15 years, many researchers have great interest in this field of research, which contributes to the more realistic description of nuclear power reactors. This Special Issue on "Applications of Fractals and Fractional Calculus in Nuclear Reactors" is dedicated to analyzing nuclear reactor dynamics with fractals and fractional modeling.

Prof. Dr. Gilberto Espinosa Paredes
Dr. Carlos Antonio Cruz-López
Guest Editors

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• Nuclear Reactor analysis
• fractal mathematical modeling
• fractional mathematical modeling
• fractional compartmental models
• Mittag-Leffler kernel
• stability analysis
• semi-analytical method
• analytical and numerical methods
• symmetry analysis and conservation laws
• numerical and computational methods

Potential Title 1: The Fractional Neutron Point Kinetics Equations based on the Fractional Cattaneo-Vernotte’s Equation and Transport principles, a Comparative Study with the Telegraph Equation

Description: The first fractional version of the Neutron Point Kinetics Equations was developed from neutron transport principles and using an analogy with the Cattaneo–Vernotte equation. In this way, memory effects were introduced with the purpose of simulating a subdiffusive behavior and providing a more realistic description of the behavior of the neutron population. This model has been applied to study several problems, including the start-up of a nuclear reactor as well as feedback effects, obtaining a more realistic description of the dynamic of a nuclear reactor. This fractional version can be understood as a direct improvement in the telegraph equation of neutron transport, with whose equation it maintains total symmetry. In other words, both equations are identical, except that the fractional model is set in terms of the fractional derivatives. Nevertheless, until now, no direct comparisons between these two models have been reported in the literature, and instead, there are comparisons only with the classical Neutron Point Kinetics Equations, which are inadequate to understand the effects of the fractional operators of the system. In this planned paper, a comparison between these two models is carried out, considering different scenarios, and focusing on the memory effects of the system. 

Potential Title 2:Generalization of Analytical and Semi-analytical Solutions of the Fractional Neutron Point Kinetics Equations.

Description: The analytical and semi-analytical solutions of some versions of the Fractional Neutron Point Kinetics Equations have been developed by considering a single group of precursors of delayed neutrons. This limitation is explained in terms of the complexity of the associated equations when several groups are considered, particularly with the computation of the inverse of the Laplace transform. Indeed, in the case of analytical solutions, the computation of the inverse transform requires expressing a rational function as a power series expansion and using the Mittag–Leffler functions, which, in turn, involves multinomial expressions when the general case of several groups of precursors is considered. For the semi-analytical solution, the inverse transform is approximated by Talbot’s numerical algorithm, which leads to very complicated polynomial expressions when more than one group is included. Nevertheless, more realistic modeling of the behavior of the neutron population requires considering multigroups of delayed neutrons, and, therefore, it is necessary to develop computational strategies with the purpose of extending the reported solutions to this general case. In this second planned article, different approaches to overcome these issues are proposed, and computational and algorithmical implementations are provided.

Potential Title 3: Analysis of the Fractional Neutron Point Kinetics Equations considering different defintions of fractional operators: analytical, numerical and physical implications.

Description: Caputo’s fractional derivatives are useful to model physical systems because initial conditions can be treated in the same way as the classical differential equations. Nevertheless, this definition has some disadvantages because, except for linearity, it does not fulfill any other important properties of integer derivatives as the product, composition and quotient rules, among others. Therefore, other definitions of fractional derivatives have been proposed in the literature that overcome the limitations of Caputo’s approach. Some of these alternative definitions have been applied to build the Fractional Neutron Point Kinetics Equations and the numerical schemes have been developed to solve them. Nevertheless, there is a lack of theoretical justification for the application of these operators to the particular case of modeling the neutron population, which contrasts with the several discussions and analyses related to memory effects that have been carried out in the case of Caputo’s definition. Similarly, no analytical or semi-analytical solutions have been reported, and there is a lack of theoretical discussion on this topic. In this planned article, an analysis of physical justification and analytical approaches is carried out, considering fractional versions of the Neutron Point Kinetics Equations with other fractional derivative definitions.

Potential Title 4: Mass Balance Violations in Some Versions of Fractional Neutron Point Kinetics Equations. 

Description: It is possible to build fractional differential equations systems that involve several fractional orders, allowing for more realistic modeling of physical processes. In the case of the behavior of the neutron population in nuclear reactors, some versions of the Fractional Neutron Point Kinetics Equations have been proposed, considering a fractional order for the neutron balance equation and another one for the precursors of the delayed neutrons. This distinction is justified because prompt and delayed neutrons exhibit different time natures, and, therefore, they are related to different memory effects. Nevertheless, in order to use different fractional orders, it is necessary to apply a specific methodology that guarantees the mass-balance conservation of the system. This methodology was originally developed in the context of multi-compartmental systems. Unfortunately, some of the fractional models related to the Neutron Point Kinetics Equations were built using no formal methodologies that lead to mass-balance violations, which means that they are models without physical meaning. In this fourth planned article, a detailed analysis of different versions of the Fractional Neutron Point Kinetics Equations is carried out with the purpose of determining which of them preserves the mass-balance conservation. 

Potential Title 5: Mass Balance Violations in Some Versions of Fractional Neutron Point Kinetics Equations.
General Solution of the Bateman Equations Using the Multivariate Mittag-Leffler Function.

Description:The time evolution of nuclides in succesive transformations can be modeled using mass-balance relationships known as Bateman Equations, which are coupled first-order differential equations that involve the concentration of nuclides and radioactive decay constants. In the most elementary case, where it is assumed that all the decay constants are different, Bateman Equations can be straightforwardly solved using the Laplace transform, but they require more advanced methods in the general case where the mentioned assumption is not considered. Fractional calculus has previously been used to obtain the General Solution of the Bateman Equations. For such a task, a fractional mass balance was proposed with multivariate fractional orders, which was solved using Cauchy products. After that, the limit when the fractional orders tended to one was taken, and, therefore, the General Solution of the Bateman Equation was obtained. Unfortunately, this solution is expressed in terms of nested sums, which is disadvantageous in computational terms. In the present planned article, a new approach to finding the General Solution is proposed, using fractional calculus, but this time solving the corresponding system using the Mittag–Leffler function, obtaining a more conveniently written solution.

Potential Title 6:An Improved Numerical Implementation of the Mittag-Leffler Functions in Fractional Neutron Point Kinetics Equations Using the Padé Approximation. 

Description:Analytical solutions of the Fractional Neutron Point Kinetics Equations are commonly expressed in terms of the two-parametric Mittag–Leffler Function, whose computation represents a numerical challenge, particularly for large arguments. This last fact is true because the standard definition of such a function, given in terms of power series, exhibits numerical issues related to its slow convergence, requiring the sue of several terms that involve the computation of the Gamma function for large values, leading to precision digits issues. Other approaches have been proposed in the literature, such as approximating this function using the inverse Laplace theory or using its integral representation. Unfortunately, even when these advanced methodologies are applied, analytical solutions require using substep methodologies to provide precise calculations, implying large computational times. Recently, a Padé Approximation methology has been developed to numerically solve Fractional Neutron Point Kinetics Equations, representing an important alternative to the standard methods mentioned previously. In this new approach, a recursion formula is used to estimate the neutron density by using the Mittag–Leffler function in its matrix form. The Padé Approximation is based on the analytical inversion of the corresponding coefficient matrix. In this planned article, the Padé Approximation is used to approximate the Mittag–Leffler function involved in the analytical solution, improving its efficiency and overcoming the mentiones numerical issues.

Potential Title 7:Analytical Analysis of the Mean-Squared Displacement of Fractional Models.

Description:Mean-squared displacement (MSD) measures the average distance traveled by particles in a system. It can be used to describe and characterize different types of movements and phenomena in neutron transport theory. In the case of the neutron subdiffusive process, this mean-square displacement exhibits a nonlinear power-law dependence on time. Therefore, to show that fractional versions of the Neutron Point Kinetics Equations can model this subdiffusive process, its MSD has been studied and its nonlinearity has been analyzed. Fourier and Laplace transforms are involved in this process, being necessary to find the inverse for this last operator. A numerical scheme was previously used to find this task, approximating the residues of a rational function whose denominator contains pseudo-polynomials and applying a partial fraction decomposition. Nevertheless, from the recent development of analytical solutions, it was found that the inverse Laplace function of these rational functions can be computed analytically, and, therefore, it is possible to find general close-expressions for the fractional models of the Neutron Point Kinetics Equations, something that was not previously considered possible. In the present planned article, an analytical procedure to compute the MSD is developed, using a power series expansion in terms of the Mittag–Leffler functions. Additionally, a more extensive study of the different fractional versions of the Neutron Point Kinetics Equations is carried out.