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This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power law function that can be approximated in the time domain by a rational function whose pole and zeros are interlaced and linked by geometric laws. The Laplace transform and frequency response of this class of kernel is given and compared with an approximation found in the literature. The comparison reveals less phase oscillation with the solution proposed by the authors. A parameter estimation method is finally proposed to obtain the rational kernel model for general fractional behaviour. An application performed with this estimation method demonstrates the interest in non-singular rational kernels to model fractional behaviours. Another interest is the physical interpretation fractional behaviours that can be implemented with delay distributions.

This paper proposes algorithms to model fractional (dynamical) behaviors using non-singular rational kernels whose interest is first demonstrated on a pure power law function. Two algorithms are then proposed to find a non-singular rational kernel that allows the input-output data to be fitted. The first one derives the impulse response of the modeled system from the data. The second one finds the interlaced poles and zeros of the rational function that fits the impulse response found using the first algorithm. Several applications show the efficiency of the proposed work.

Charge time has become one of the primary issues restricting the development of electric vehicles. To counter this problem, an adapted thermal management system needs to be designed in order to reduce the internal thermal gradient, by predicting the surface and internal temperature responses of the battery. In this work, a pseudo 3D model is developed to simulate battery cell performance and its internal states under various operational scenarios such as temperature and convection conditions as well as the applied current during charge and discharge. An original mesh of the JR is proposed where heat exchanges in the three directions (radial, orthoradial and axial) are considered. The model represents one of the solutions that enable increasing the lifespan of batteries while decreasing charging time. It offers the opportunity to optimize operating parameters to extend battery life. In this paper, attention was paid not only to the core and non-core components, but also to the experiments required to parametrize the thermal and electrochemical models (heat generation). Unlike existing approaches documented in the literature, the model developed in this work achieves an impressive balance between computational efficiency and result accuracy, making it a groundbreaking contribution in the field of electric vehicle technology.

The paper shows that the Caputo definition of fractional differentiation is problematic if it is used in the definition of a time fractional model and if initial conditions are taken into account. The demonstration is done using simple examples (or counterexamples). The analysis is extended to the Riemann-Liouville and Grünwald-Letnikov definitions. These results thus question the validity of results produced in the field of time fractional model analysis in which initial conditions are involved.

This work was greatly influenced by the opinions of one of the authors (JS), who demonstrated in a recent book that it is important to distinguish between “fractal models” and “fractal” (power-law) behaviors. According to the self-similarity principle (SSP), the authors of this study completely distinguish between independent “fractal” (power-law) behavior and the “fractal models”, which result from the solution of equations incorporating non-integer differentiation/integration operators. It is feasible to demonstrate how many random curves resemble one another and how they can be predicted by functions with real and complex-conjugated power-law exponents. Bellman’s inequality can be used to demonstrate that the generalized geometric mean, not the arithmetic mean, which is typically recognized as the fundamental criterion in the signal processing field, corresponds to the global fitting minimum. To highlight the efficiency of the proposed algorithms, they are applied to two sets of data: one without a clearly expressed power-law behavior, the other containing clear power-law dependence.

In the literature, fractional models are commonly approximated by transfer functions with a geometric distribution of poles and zeros, or equivalently, using electrical Foster or Cauer type networks with components whose values also meet geometric distributions. This paper first shows that this geometric distribution is only a particular distribution case and that many other distributions (an infinity) are in fact possible. From the networks obtained, a class of partial differential equations (heat equation with a spatially variable coefficient) is then deduced. This class of equations is thus another tool for power law type long memory behaviour modelling, that solves the drawback inherent in fractional heat equations that was proposed to model anomalous diffusion phenomena.

This paper extends and illustrates a probabilistic interpretation of the fractional derivative operator proposed by Pr. José Tenreiro Machado. While his interpretation concerned the probability of finding samples of the derivate signal in the expression of the fractional derivative, the present paper proposes interpretations for other fractional models and more generally fractional behaviours (without using a model). It also proposes probabilistic interpretations in terms of time constants and time delay distributions. It shows that these probabilistic interpretations in terms of time delay distributions can be connected to the physical behaviour of real systems governed by adsorption or diffusion phenomena.

This paper proposes to model fractional behaviors using Volterra equations. As fractional differentiation-based models that are commonly used to model such behaviors exhibit several drawbacks and are particular cases of Volterra equations (in the kernel definition), it appears legitimate in a modeling approach to work directly with Volterra equations. In this paper, a numerical method is thus developed to identify the kernel associated to a Volterra equation that describes the input–output behavior of a system. This method is used to model a lithium-ion cell using real data. The resulting model is compared to a fractional differentiation-based model with the same number of tunable parameters.

Additive manufacturing of oxide glass enables on-demand, low-cost manufacturing of complex optical components for numerous applications, opening new opportunities to explore functionalities inaccessible otherwise. Here, we report a straightforward extrusion-based 3D-printing approach, deploying the fused deposition modeling (FDM) process, to produce optically transparent phosphate glasses with complex geometries and preserved structural and photoluminescence properties. Using a customized entry-level FDM desktop printer with a layer resolution of 100 μm, highly dense and transparent europium-doped phosphate glass structures can be fabricated from glass filaments pulled using a fiber-drawing tower from the parent glass preform. Combined with the suggested strategies for performance and quality improvement, professional-grade FDM printers can offer better layer resolutions. This direct approach for 3D-printing phosphate glass may open up new horizons not only for developing cutting-edge optical components but also for promoting new biomedical solutions upon making use of alternative biocompatible phosphate compositions.

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1

Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

This paper deals with optimal charging versus aging minimization for lithium-ion batteries. The optimal charging strategy proposed involves charging controllers whose design relies on a battery model. The model, especially designed for automotive battery management systems applications, is recalled in this paper. It provides the voltage response of a cell to an input current. It also models side reactions that produce degradation mechanisms and thus decrease battery performance. Side reaction modelling involves taking into account the temperature cell variations, which are thus also modelled. The association of the three above-mentioned sub-models leads to an electro-thermal battery aging model used to design an optimal charging strategy that simultaneously takes into account the minimization of charging time and maximization of battery lifetime. Thus, to achieve a charging controller that manages battery health, an appropriate charging trajectory was computed by solving an optimization problem minimizing aging. Then, a charge control loop was designed. The nonlinear behavior of the battery was taken into account through the linearization of the electro-thermal aging model in different operating conditions. To take into account the resulting linear model family, the CRONE design methodology was used. The principles of this methodology are recapped and the design of the charging control loop is explained. The efficiency of the resulting charge controller is illustrated by several simulations.

To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. Some limitations still remain, however, with these kernels, whereas solutions prior to the introduction of fractional models exist in the literature. This paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of the Volterra equations, equations introduced nearly a century ago. Volterra equations can thus be viewed as a serious alternative to fractional pseudo state space descriptions for modelling power law type long memory behaviours. This paper thus presents a new class of model involving a Volterra equation and several kernels associated with this equation capable of generating power law behaviours of various kinds. One is particularly interesting as it permits a power law behaviour in a given frequency band and, thus, a limited memory effect on a given time range (as the memory length is finite, the description does not exhibit infinitely slow and infinitely fast time constants as for pseudo state space descriptions).

The paper first describes a process that exhibits a power law-type long memory behaviour: the dynamical behaviour of the heap top of falling granular matter such as sand. Fractional modelling is proposed for this process, and some drawbacks and difficulties associated to fractional models are reviewed and illustrated with the sand pile process. Alternative models that solve the drawbacks and difficulties mentioned while producing power law-type long memory behaviours are presented.

This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell.

This paper analyses algorithms currently found in the literature for the approximation of fractional order models and based on recursive pole and zero distributions. The analysis focuses on the sub-optimality of the approximations obtained and stability issues that may appear after approximation depending on the pole location of the initial fractional order model. Solutions are proposed to reduce this sub-optimality and to avoid stability issues.