Nonlocal Effects Research Articles

Correlation between handgrip strength and lower extremity strength amongst Malaysian professional football players

ABSTRACT Background: We investigated the pattern of handgrip strength (HGS) and lower extremity strength in professional football players and how they were correlated. Methods: Secondary data were gathered during pre-participation evaluation sessions from a sample of 87 Selangor state-based Malaysian professional football players. The statistical analysis of the pattern of lower limb strength and HGS was based on the player’s position on the field. The correlation between HGS and lower extremity strength was looked at using Pearson’s correlation coefficient. The factors that affect HGS were analysed using a multivariate regression model. Results: In both dominant (P = 0.026) and non-dominant hands (P = 0.016), goalkeepers had significantly higher HGS than strikers. Dominant HGS had a moderate correlation with knee extensor strength (r = 0.38, P < 0.05). The moderate correlation between HGS and knee extensor strength remains consistent despite being controlled for weight (r = 0.33, P <0.05) and skeletal muscle mass (SMM) (r = 0.31, P < 0.05). Significant correlations were observed despite the lower correlation value between dominant HGS and hip flexor strength (r = 0.31, P < 0.05), hip extensor strength (r = 0.30, P < 0.05) and knee flexor strength (r = 0.29, P < 0.05). Dominant knee extensor strength and SMM were identified as statistically significant predictors for dominant HGS (F [2,84] = 11.60, P < 0.01) with R2 = 0.22. Conclusions: The unique anthropometric traits of professional footballers, especially goalkeepers, improve HGS. Handgrip and knee extensor strength were moderately correlated, supporting previous study findings on the existence of a non-local effect between upper limb and lower limb strength.

The random thermo-elastic nonlinear vibration analysis of porous FGM nano-beams using the first order perturbation theory

The objective of present work is to study the nonlocal nonlinear thermo-elastic vibrations of porous nanobeams made up of functionally graded material (FGM) with random material properties. The mechanical properties of the nano-beam are calculated using a porosity based power law model by assuming the cosine type porosity distributions. The randomness present in the material properties cause the mechanical characteristics of the nano-beams to be random, hence first order perturbation theory has been employed to take care of this randomness. The Reddy’s beam theory in association with the Eringen’s non classical elasticity has been used to account for nonlocal effects, and Von-karman nonlinearity has been employed to take care of geometric nonlinearity. The governing equations are derived from the minimum potential energy principle and solved them using the finite element method. To validate the present model, a convergence and comparison study has been performed. Further, the effects of amplitude ratio, aspect ratio, uncertain material properties, size parameter, porosity, thermal environment, and gradation of material on the stochastic behavior of the porous FGM nano-beams are demonstrated. As a result of the present research, the significance of considering thermal environment, porosity, and random material properties has been revealed for the design of nanodevices and systems to be more reliable.

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

Optoelectronic, mechanical and vibrational properties of rhombohedral boron monosulfide (r-BS): A DFT screening exchange approach (sX-LDA) study

The main objective of this study is to predict and understand several crucial physical properties of a little-known material; I cite rhombohedral boron monosulphide (r-BS). In order to improve the accuracy of the calculations, we selected sX-LDA, which is the most appropriate DFT approach according to its performance to reproduce the crystalline and the electronic structure of r-BS. According to our calculations, we observe a net renormalization of the electronic band structure under non-local correlation effects, which led to an indirect energy band gap of 3.246 eV. The r-BS present a higher degree of crystal anisotropy, a high absorption and a high photoconductivity spectrum in the UV energy region, suggesting its potential use as UV absorbers. In addition, at sufficiently low temperatures (5–50 K) the specific heat of r-BS is similar to that of a two-dimensional solids (Cv∝T2). Furthermore, r-BS is one of the innovative materials that have been left aside by researchers for unknown reasons and that it must be dug up and given it a new life and a new breath and re-injected it into the circuit of scientific research for its potential use in new nanotechnologies.

Modeling pull-in instability of CNT nanotweezers under electrostatic and van der Waals attractions based on the nonlocal theory of elasticity

This work investigates the electromechanical response and pull-in instability of an electrostatically-actuated CNT tweezer taking into consideration a TPNL constitutive behavior of the CNTs as well as the intermolecular forces, both of which provide a significant contribution at the nanoscale. The nonlocal response of the material introduces two additional parameters in the formulation, which are effective in capturing the size effects observed at the nanoscale. The problem is governed by a nonlinear integrodifferential equation, which can be reduced to a sixth-order nonlinear ODE with two additional boundary conditions accounting for the nonlocal effects near to the CNT edges. A simplified model of the device is proposed based on the assumption of a linear or parabolic distribution of the loading acting on the CNTs. This assumption allows us to formulate the problem in terms of a linear ODE subject to two-point boundary conditions, which can be solved analytically. The results are interesting for MEMS and NEMS design. They show that strong coupling occurs between the intermolecular forces and the characteristic material lengths as smaller structure sizes are considered. Considering the influence of the nonlocal constitutive behavior and intermolecular forces in CNT tweezers will equip these devices with reliability and functional sensitivity, as required for modern engineering applications.

He abundance in NGC 1850 A and B: Are we observing the early stage of the formation of multiple populations in a stellar cluster?

ABSTRACT We present the results for a sample of B stars in the Large Magellanic Cloud young double stellar cluster NGC 1850 A and NGC 1850 B, as observed with the integral-field spectrograph at the Very Large Telescope, the Multi Unit Spectroscopic Explorer (MUSE). We compare the observed equivalent widths (EWs) of four He lines (4922, 5015, 6678, and 7065 Å) with those determined from synthetic spectra computed with different He mass fractions (Y = 0.25, 0.27, 0.30, and 0.35) with the code synspec, which takes into account the non-local thermodynamic equilibrium effect. From this comparison, we determine the He mass fraction of the B stars, finding a distribution that is not homogeneous. The stars can be divided in three groups: He-weak (Y < 0.24) and He-normal (0.24 ≤ Y ≤ 0.26) stars, belonging to the main sequence of NGC 1850 A, and He-rich stars (0.33 ≤ Y ≤ 0.38), situated in the main sequence associated with NGC 1850 B. We analyse the stellar rotation as possibly being responsible for the anomalous features of the He lines in the He-rich stars. We provide a simple analysis of the differences between the observed EWs and those obtained from theoretical models with different rotation velocities (Vsini = 0 and 250 km s–1). The resolution of the MUSE spectra does not allow us to obtain a conclusive result; however, our analysis support the He-enhanced hypothesis.

Nonlocal phonon thermal transport in graphene in hydrodynamic regime

The hydrodynamic behavior of phonons is of particular interest and importance owing to the strong demand for highly thermal conductive materials. Thermal transport in hydrodynamic regime becomes essentially nonlocal, which can give rise to a number of new and counterintuitive phenomena. In this work, we present a direct numerical study of nonlocal phonon thermal transport in graphene ribbon with vicinity geometry based on the phonon Boltzmann transport equation with first-principles inputs. We demonstrate the viscosity-dominated hydrodynamic transport behaviors with two abnormal thermal transport phenomena: heat current whirlpools and negative nonlocal effect, which originate from phonon viscosity. Phonon viscosity produces the vorticity of shear flows, leading to the backflow of the heat current and the generation of negative nonlocal vicinity response. The system average temperature and the ribbon width as well as the relative positions of the heat sources play a pivotal role in the occurrence of heat current whirlpools and negative nonlocal temperature response. The present work provides solid evidence for phonon hydrodynamic transport in graphene and a potential avenue for experimental detection in the future.

Terahertz large-area unidirectional surface magnetoplasmon and its applications

Unidirectional surface waves in nonreciprocal plasmonic platforms with nonlocal effects have been a topic of significant interest and some controversy. In this study, we present a scheme to achieve unidirectional surface magnetoplasmons (USMPs) with large modal areas at terahertz frequencies. Such large-area USMPs (LUSMPs) exist in a metal-UENZ (uniaxial-ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-near-zero)-Si-InSb structure under external magnetic field, where the effect of nonlocality is included. The field of the LUSMP extends almost uniformly in the UENZ layer with a thickness of wavelength scale, thus its modal size can be represented by the UENZ-layer thickness. Due to the modal energy primarily distributed in the thick UENZ layer, the nonlocality-induced leakage of the LUSMP is significantly reduced by an order of magnitude, compared to previous USMP existing at interface between InSb and opaque isotropic medium. Due to their large modal sizes, such LUSMPs can be efficiently excited by terahertz radiations directly from free space. In addition, LUSMPs offer high degree of freedom for manipulating terahertz waves, such as energy squeezing and trapping. Based on LUSMPs, a terahertz free-space isolator is also developed. Our findings have important implications to the development of innovative plasmonic devices in terahertz regime.

Small scale analysis of porosity-dependent functionally graded triply periodic minimal surface nanoplates using nonlocal strain gradient theory

In recent years, the triply periodic minimal surface (TPMS) has emerged as a remarkable solution for constructing structures, drawing inspiration from natural architectures. TPMS offers several outstanding features, including porous architectures with high interconnectivity, smooth surfaces and the ability to achieve mathematically controllable geometry features. However, it is evident that the current research has not fully harnessed the extensive potential and benefits of TPMS structures. In this paper, a groundbreaking approach for analyzing functionally graded triply periodic minimal surface (FG-TPMS) nanoplate, which is utilized a novel nonlocal strain gradient isogeometric analysis, is provided. Three patterns of the FG-TPMS nanoplate, namely Primitive (P), Gyroid (G) and I-gragh and Wrapped Package-graph (IWP), are utilized in this study. The primary focus is to investigate size dependent problems with two types of density distributions. The proposed model successfully incorporates both nonlocal effects and strain gradient effects into nanoplate structures. The paper demonstrates how the mechanisms responsible for both reducing and enhancing stiffness in the nanoplate can be understood by fine-tuning the nonlocal and strain gradient parameters. The findings of this study offer promising prospects for future design and optimization as they provide a robust approach to address the complex mechanical behavior observed in the FG-TPMS nanoplate. The proposed model not only captures the behavior accurately but also opens up new avenues for exploring the capabilities of FG-TPMS structures.

An alternative method to mimic mode conversion for ion cyclotron resonance heating

Ion cyclotron range of frequency waves in hot plasmas exhibit spatial dispersion effects and the wave equation takes the integro-differential form. Under the local plasma model assumption, the wave equation can be simplified to the differential form and adapts to the numerical scheme of the finite element method (FEM). Even though direct absorption of fast waves by ions and electrons can be described well by the local plasma model, linear mode conversion associated with non-local effects is absent. To deal with this issue, an alternative method is put forward in this paper where quasi-electrostatic fluid waves based on the multi-fluid warm plasma model are employed to take the place of ion Bernstein waves in mode conversion. On this basis, an interative fluid-kinetics (INTFLUK) code based on the FEM is developed for full-wave simulation in hot plasmas. Derivation of the wave equations as well as benchmarking of the INTFLUK code against other wave simulation codes are carried out. In both one- and two-dimensional cases, the validity of the INTFLUK code was verified by comparison of the wave field distributions and power deposition. As a useful illustration of the INTFLUK code including the scrape-off layer and a realistic antenna, the influence of the poloidal antenna phasing difference on ion cyclotron resonance heating is analyzed. Finally, it should be noted that the method in this paper has the potential to be extended to the three-dimensional case, which will be considered in the near future.

Nonlocal effective medium theory for phononic temporal metamaterials

We have developed a nonlocal effective medium theory (EMT) for phononic temporal metamaterials using the multiscale technique. Our EMT yields closed-form expressions for effective constitutive parameters and reveals these materials as reciprocal media with symmetric band dispersion. Even without spatial symmetry breaking, nonzero Willis coupling coefficients emerge with time modulation and broken time-reversal symmetry, when the nonlocal effect is taken into account. Compared to the local EMT, our nonlocal version is more accurate for calculating the bulk band at high wavenumbers and essential for understanding nonlocal effects at temporal boundaries. This nonlocal EMT can be a valuable tool for studying and designing phononic temporal metamaterials beyond the long-wavelength limit.

Hyperelastic constitutive relations for soft elastomers with thermally-induced residual stress

Residual stress widely exists in soft materials. Besides growth, inhomogeneous thermal expansion is also a primary cause of residual stress. However, establishing a proper hyperelastic constitutive relation is a great challenge since the existing theories cannot capture the change of underlying mechanical responses triggered by temperature variations. In this paper, a general hyperelastic constitutive relation for soft elastomers with thermally-induced residual stress is developed. We first reveal the initial temperature dependence of conventional thermoelastic models. This property attributes the alteration of the underlying thermoelastic response to free thermal expansions. Then, a compatibility-broken curvature compensation (CBCC) framework is established based on finite thermoelasticity. It generates a free thermal expansion to eliminate the Riemannian curvatures of the virtual stress-free configuration derived from the isothermal stress release. Such a mechanism indicates the non-local effect of the residual stress, which fundamentally modifies the traditional view that invariant formulations cover all the possible functional dependence of residual stress. Also, the obtained governing equations are similar to Einstein field equations of the general theory of relativity. This similarity may deeply imply a standard mechanism concerning the curvature compensation leading to residual stress genesis. We finally conduct comparative analyses of the spherically symmetric and axisymmetric problems between the current constitutive relation and the existing models. The influences of adopting distinct residual stresses, the performance of the non-local effect, and the availability of the new constitutive relation are investigated in detail. This framework can shed some light on the constitutive modeling of soft materials.

Microscopic origin of granular fluidity: An experimental investigation.

Granular fluidity has been central to the development of nonlocal constitutive equations, which are necessary for characterizing nonlocal effects observed in experimental granular flow data. However, validation of these equationshas been largely computational due to challenges in laboratory experiments. Specifically, the origin of the fluidity on a microscopic, single-particle level is still unproven. In this work, we present an experimental validation of a microscopic definition of granular fluidity, and show the importance of basal boundary conditions to the validity of the theory.

On the physical significance of non-local material parameters in optical metamaterials

When light interacts with a material made from subwavelength periodically arranged constituents, non-local effects can emerge. They occur because of either a complicated response of the constituents or possible lattice interactions. In lowest-order approximations of a general non-local response function, phenomena like an artificial magnetism and a bi-anisotropic response emerge. However, investigations beyond these lowest-order descriptions of non-local effects are needed for optical metamaterials (MMs) where a significant long-range interaction becomes evident. This highlights the need for additional material parameters to account for spatial non-locality in an effective medium description. These material parameters emerge from a Taylor expansion of the general and exact non-local response function. Even though these non-local parameters improve the effective description, their physical significance is yet to be understood. To improve the situation, we consider a conceptional MM consisting of scatterers characterized by a prescribed multipolar response arranged on a square lattice. Lorentzian polarizabilities describe the scatterers in the electric dipolar, electric quadrupolar, and magnetic dipolar terms. A slab of such a MM is homogenized while considering an increasing number of non-local terms in the constitutive relations at the effective level. We show that the effective permittivity and permeability are linked to the electric and magnetic dipole moments of the scatterers. The non-local material parameters are related to the higher-order multipolar moments and their interaction with the dipolar terms. Studying the effective material parameters with the knowledge of the induced multipolar moments in the lattice facilitates our understanding of the significance of each material parameter. Our insights aid in deciding on the order to truncate the Taylor expansion of the considered constitutive relations for a given MM.

Tunable all-optical microwave logic gates based on nonreciprocal topologically protected edge modes

All-optical logic gates have been studied intensively owing to their potential to enable broadband, low-loss and high-speed communications. However, poor tunability has remained a key challenge in this field. In this work, we propose a Y-shaped structure composed of Yttrium Iron Garnet (YIG) layers that can serve as tunable all-optical logic gates, including, but not limited to, OR, AND and NOT gates, by applying external magnetic fields to magnetize the YIG layers. Our findings reveal that these logic gates are founded on protected one-way edge modes, where by tuning the wavenumber k of the operating mode to a sufficiently small (or even zero) value, the gates can become nearly immune to nonlocal effects. This not only enhances their reliability but also allows for maintaining extremely high precision in their operations. Furthermore, the operating band itself of the logic gates is also shown to be tunable. We introduce a straightforward and practical method for controlling and switching these gates between "work", "skip", and "stop" modes. These findings have potentially significant implications for the design of high-performance and robust all-optical microwave communication systems.

Landau-fluid simulations of edge-SOL turbulence with GRILLIX

The Landau-fluid closure for parallel heat fluxes is implemented in the edge turbulence fluid code GRILLIX, replacing the previously used collisional Braginskii closure (with limiters). This extends the validity of the model toward lower collisionality, introduces non-local effects, and leads to a more realistic and self-consistent limiting of heat fluxes. Turbulence simulations comparing the Landau-fluid with the Braginskii closure in realistic divertor geometry are carried out. Clear differences between the simulations are observed, most pronounced a spurious up-down ion temperature asymmetry emerges for a strongly limited Braginskii case. For the Landau-fluid case, we demonstrate the presence and relevance of non-local heat fluxes in full-scale turbulence simulations and show that this behavior could only hardly be reproduced with simple flux-limited models. The implementation of the Landau-fluid closure within the flux-coordinate independent approach employed by GRILLIX results in a set of 3D elliptic problems, where magnetic flutter can be incorporated naturally. On reusing the existing solver in GRILLIX, only a moderate additional computational effort is necessary for the higher fidelity Landau-fluid closure.

Nonlocal strain gradient-based isogeometric analysis of graphene platelets-reinforced functionally graded triply periodic minimal surface nanoplates

Recent developments in additive manufacturing (AM) technologies have empowered the design and fabrication of intricate bioinspired engineering structures at the nano/micro scale. However, mathematical modeling and computation of these structures are still challenging. The main target of this study is to address an efficient computational approach for predicting the mechanical behavior of graphene platelets (GPLs)-reinforced functionally graded triply periodic minimal surface (FG-TPMS) nanoplates. The computational model integrates both nonlocal elasticity and strain gradient effects into the NURBS-based isogeometric analysis of these small-scale structures. We establish advanced nanoplate models by combining three sheet-based TPMS architectures with two new porosity distribution patterns and three distribution patterns of GPLs across the thickness direction. Moreover, the present work makes a pioneering attempt to elucidate how the stiffness-hardening and stiffness-softening mechanisms influence FG-TPMS nanoplates reinforced with GPLs. Compared with two common cellular solids, the superiority of TPMS architectures' mechanical performance is demonstrated. Among all, P and IWP TPMS types along with symmetric porosity and GPLs distributions exhibit outstanding behaviors under static bending, free vibration, and dynamic instability. Furthermore, we conducted a performance analysis for the first time, showcasing the superior capabilities of TPMS architectures under dynamic in-plane compressive loads, especially when compared to isotropic plates of equal weight. The findings of this study greatly enhance our understanding of the intricate mechanical responses of GPLs-reinforced TPMS architectures at the small-scale level, contributing to future interdisciplinary applications.

Topological and high-performance nonreciprocal extraordinary optical transmission from a guided mode to free-space radiation

Extraordinary optical transmission (EOT) is a hallmark of surface plasmons and a precursor to nanoplasmonics and metamaterials. However, to the best of our knowledge, this effect has never been topologically protected in three dimensions, leaving it vulnerable to structural imperfections, nonlocal effects, and backreflections. We report broadband, three-dimensional unidirectional structures that allow for EOT (normalized transmission > 1) through deep-subdiffractional single holes, immune to these deleterious effects. These structures avoid unnecessary propagation losses and achieve maximum transmission through a single hole, limited only by unavoidable dissipative losses. In the limit of vanishing losses, the transmission through a deep-subdiffractional hole can approach unity, significantly surpassing existing devices, and rivaling the performance of negative-index ‘perfect’ lenses. The topological stability of these structures renders them robust against surface roughness, defects, and nonlocality, without the need for elaborate meta-structures or tapering.

Nonlocal effects on a 1D generalized Ohta–Kawasaki model

We propose a generalized Ohta–Kawasaki model to study the nonlocal effect on the pattern formation of some binary systems with general long-range interactions. While in the 1D case, the generalized Ohta–Kawasaki model displays similar bubble patterns as the standard Ohta–Kawasaki model, by performing Fourier analysis, we find that the optimal number of bubbles for the generalized model may have an upper bound no matter how large the repulsive strength is. The existence of such an upper bound is characterized by the eigenvalues of the nonlocal kernels. Additionally, we explore the conditions under which the nonlocal horizon parameter may promote or demote the bubble splitting, and apply the analysis framework to several case studies for various nonlocal operators.

Analytical solution of the fractional neutron point kinetic equations using the Mittag-Leffler function

A new analytical solution of the fractional point kinetic equations is developed in the present work, which considers one group of delayed neutrons as well as a constant reactivity. One of the main novelties and theoretical contributions of the developed solution is that it does not require approximating the Inverse Laplace transform by numerical means, as other solutions do, and instead it is obtained analytically using the Green and the Mittag-Leffler functions. A set of algorithms and MATLAB codes are developed with the purpose to implement and compute the developed solution in an adequate way, which represents the major computational contribution. Numerical experiments show that the developed solution is very accurate solving short-time dynamics phenomena, being able to reproduce, in a precise way, data that is reported in literature. The development of the proposed analytic solution represents an important contribution to validate numerical methods that are currently used to solve the neutron fractional point kinetic equations. Program SummaryProgram Title: Mittag-Leffler FNPKECPC Library link to program files:https://doi.org/10.17632/v4kfhzw7sf.1Developer's repository link:https://github.com/Cruz-Lopez-Carlos-Antonio/Mittag-Leffler-FNPKE/tree/mainLicensing provisions: Creative Commons by 4.0Programming language: MATLAB (2021a) / Python 3Nature of problem:Fractional Neutron Point Kinetic Equations describe the neutron population in a nuclear reactor, considering the changes on transport and thermohydraulic properties of the system. For such task, a mass balance approach based on fractional and integral derivatives is used, which improves and generalizes the integer case that uses the classical definitions of these mathematical operators. From a physical point of view, this new approach takes into account memory and non-local effects that are omitted in the standard neutron diffusion theory and which are relevant in scenarios where the current neutron density undergoes strongly variations on time. From a mathematical point of view, using fractional derivatives leads to a new and more general type of differential equations of arbitrary order that requires different methods of solution and approximation, and which must include the integer approach as particular case. In such context, the development of analytical solutions is a fundamental problem because it provides a framework to carry out verifications of numerical methods, as well as an insight to perform theoretical comparisons with the classical model. The developed Mittag-Leffler FNPK codes analytically solve the Fractional Neutron Point Kinetic Equations defined in terms of the Caputo's derivatives and subject to classical initial conditions.Solution method:The developed codes are written in MATLAB and Python programming languages, and solve the mentioned fractional differential equation system using a power series expansion as well as the Laplace transform. Such solution considers a single group of precursors of delayed neutrons and it is expressed in terms of the 2-parameter Mittag-Leffler function, being accurately computed considering a small number of terms of the infinite sum. Unlike other approaches, in the present case it is not necessary to use a numerical procedure to find the inverse Laplace transform, because it is analytically found using the Green function. Even when the developed solution is found for a constant reactivity case, the Mittag-Leffler FNPKE codes can extended it to more general scenarios where the reactivity is a function of time or where it depends on the neutron density itself. In these cases, it is necessary to apply a subinterval methodology where the time is divided in small steps, assuming a constant reactivity in each of them. The smaller of the intervals, the more accurate the approximation. With the purpose to overcome the numerical issues related to the computation of the 2-parameter Mittag-Leffler function, the developed codes integrate the most advanced algorithm reported in literature, which is useful also to estimate the high-order derivatives of such function.Additional comments including restrictions and unusual features:The developed codes do not face issues related to arithmetical precision digits, as other solutions do, because it is not expressed in terms of high order polynomials and instead it is written as a linear combination of the Green function. Additionally, several steps can be precomputed without regard the neutron parameters and initial conditions, which represents an important advantage in terms of the computational time.