Keywords:Meijer G Form, Fox H Form, fractional integro-differentiation, one-dimensional integral transform Abstract: For many years, Oleg has worked with Meijer G and Fox H functions and has helped to build the largest collection of their particular cases, wherein one can find about 150 named functions and their combinations. Recently, in collaboration with Paco Jain, he has implemented his results in the Wolfram Function Repository with the four functions MeijerGForm, FoxHForm, GenericIntegralTransform, and FractionalOrderD, and made corresponding talks at the Wolfram's annual Technology Conference. In this talk, he will describe the structure of majority of the one-dimensional integral transforms (including Riemann-Liouville fractional integro-differentiation as case) in terms of Mellins-Barnes integrals containing Fox H functions in the kernel.

Keywords:Mathematical methods, System identification & Modeling, System Analysis & Dynamics Abstract: This investigation is devoted to control problems that involve fractional derivatives and nonlinearities. As in any control problem, the goal is to determine control that will govern the system from the given initial state to a desired final state. Various concepts related to control theory will be analyzed in this nonlinear fractional framework, such as controllability, observability, the Gramian matrix, the Kalman rank condition or the adjoint system. In order to study all these notions we propose a novel approach that interwine specialized techniques and tools from fractional calculus and functional analysis, such as linearization, a priori estimates and fixed point.

Keywords:System Analysis & Dynamics, System identification & Modeling, Mathematical methods Abstract: This paper presents a new technique for the analog realizations and the numerical evaluation of the second definition given by Lorenzo and Hartley of the variable fractional order integrator Iα(t){.} using an analog dynamical model. The analog realization of Iα(t){.} is obtained using an analog realization based on Charef's approximation of an adjustable fractional order integrator and the numerical evaluation of Iα(t){.} is obtained as the time response of its proposed analog dynamical model. The efficiency and the accuracy of the proposed Iα(t){.} numerical evaluation results obtained are compared to the ones of Lorenzo and Hartley using illustrative examples.

Université De Bordeaux - Bordeaux INP/ENSEIRB-MATMECA

Keywords:System identification & Modeling Abstract: Initially thought of as a mere mathematical object, fractional calculus has proven over the latest years to be useful to well model complex dynamics in diffusion or propagation. Fractional-order models allow to include a vast coexistence of time-constants without having a huge increase in the number of parameters for the model. Two original functions were once used to analyze dielectric phenomena: Cole-Cole and Davidson-Cole functions. A third one generalizes the two previous ones: the Havriliak-Negami function. By using the concept of recursive poles and zeros (see Oustaloup (1995)) and a decomposition in a Davidson-Cole and a complementary function (Sommacal et al. (2008b)), it is possible to approximate and simulate the response of the Havriliak-Negami function in the time domain. This paper deals with the recursive identification in the continuous-time domain for a system whose model is described as a Havriliak-Negami function. The LMRPEM method is then adapted and tested in simulation.

Université De Bordeaux - Bordeaux INP/ENSEIRB-MATMECA

Keywords:System identification & Modeling, Biology & Biomedicine, Thermal Engineering Abstract: In cardiac surgery where extracorporeal circulation is used, the lungs are temporarily disconnected from the body and are connected to a device that provides air and blood. To minimize the risk of tissue damage, the lungs are subjected to mild hypothermia. Heat transfer modeling offers the potential to enhance temperature regulation through a more advanced approach. A thermal model, based on a thermal quadrupoles, also called two-port network, offers a wide frequency range applicability, making it suitable for modeling the human breathing. This modeling approach can also be adapted to incorporate the influence of blood flow, which also serves as a natural temperature regulator in the human body. This is accomplished by combining the thermal two-port network with the bio-heat equation. The main contributions are in the analytical expressions of the thermal impedances and by proposing a Butterworth approximation model for the equivalent impedances of lung thermal transfers.

Keywords:System identification & Modeling, Electrochemistry Abstract: Electrochemical Impedance Spectroscopy (EIS) is a useful tool for selecting a pertinent Equivalent Circuit Model (ECM) of a Li-ion battery. Impedance model is designed to describe low, middle and high frequency electrochemical processes involved. When considering low frequency restricted in the Warburg zone, diffusion impedance is modeled thanks to a Constant Phase Element (CPE) which behaves as a fractional integrator of order n close to 0.5. Phenomena observed in middle frequency are described using specific circuits called Zarc which consist in connecting a CPE in parallel with a resistor. Therefore, the global impedance model is characterized by non integer order operators and parameters can be estimated by a Complex Non linear Least Squares (CNLS) algorithm which requires a proper initialization in order to guarantee the convergence to a global optimum. The paper presents a method to analyze EIS data measurements in order to select automatically the number of middle frequency Zarc circuits required (one or two) and to initialize properly the CNLS algorithm. The method is validated using experimental open source EIS data.

Keywords:System identification & Modeling, Signal Processing Abstract: This paper concerns with the design of algebraic disturbance estimation method for the fractional order Takagi-Sugeno (T-S) fuzzy systems with noisy output measurements. At first, the overall fractional order fuzzy system model is given for the studied system. By constructing a family of fractional order modulating functions, algebraic integral formulas are derived for the disturbance by using the overall fractional order fuzzy system model. Based on the system’s input and output, these formulas can be algebraically computed in both noise-free and noisy cases. Simulation results are shown to validate the efficiency of the proposed algebraic disturbance estimation method against output measurement noises.

Keywords:Automatic Control & Stability, System identification & Modeling, Biology & Biomedicine Abstract: This paper presents the design and application of a fractional order asymptotic adaptive observer coupled to an adaptive controller for the robust operation of high-cell density cultures in fed-batch mode. The control goal is to maximize biomass productivity by controlling the culture’s estimated specific growth rate. Since the specific growth rate cannot be measured, a fractional order asymptotic adaptive observer is proposed, based on the equivalent integer order asymptotic observer proposed before. Simulations are performed to validate the observer and controller, under the assumption that the system is in the oxidative regime under aerobic conditions. Obtained results show that, in close loop operation, the fractional adaptive observer behaves better than the integer order observer in the presence of measurement noise. For fractional orders of the observer in the range alpha ∈ [0.6,0.8], it was observed a 51.71% increase in biomass concentration, compared to the biomass obtained with the classic integer-order observer. Furthermore, the controlled system reaches very low ethanol concentrations (≤ 1 grams per liter), which is desirable in this process.

Keywords:Biology & Biomedicine, System identification & Modeling, Signal Processing Abstract: Atrial fibrillation is a cardiac disorder marked by rapid and disorganized electrical activity, leading to atrial mechanical dysfunction. The alterations in electrophysiological properties during this arrhythmia are not solely attributed to electrical remodeling, structural changes in atrial tissue are also involved. This work aims to formulate a mathematical model for fibrillatory electrical conduction through the implementation of variable--order fractional derivatives. The adoption of such an operator is intended to represent the process of structural remodeling, which has been related to the course of atrial fibrillation. Simulations are performed using a simplified model of the ionic kinetics of the cardiac cell membrane, which allows for distinct electrophysiological properties through its parameterization. For the variable order, a fluctuating function is adopted that can be interpreted as the progression of structural remodeling when the order decreases, and the reverse process when the order increases. Fibrillatory propagation is initiated by generating reentrant conduction, also known as a rotor. We observed that, on the one hand, electrical conduction becomes chaotic as the fractional order decreases, and persists under such dynamics while the order increases. On the other hand, rotational activity persists during the fluctuation of the fractional order. Such mesoscopic outcomes depend on the sensitivity of the microscopic electrophysiological properties to the variations of the fractional order. Moreover, both propagation patterns can be associated with the known course of clinical AF. These results suggest that the fractional order model of cardiac electrophysiology may provide insight into the AF underlying mechanisms.

Keywords:Epidemics, Mathematical methods, Biology & Biomedicine Abstract: We discuss the main issues related to the adequate and correct derivation of epidemiological compartmental models. Several years ago, initiated a surge in published fractional-order epidemiological compartmental models. Nenetheless, most of them are based in textit{ad hoc} approaches, merely generalizing the integer order to an alpha order derivative. This may pose some issues related to conservation of matter. In this discussion paper, we refer some methods derived from notions of stochastic processes, such as survival functions, ensuring physical validity and parameter interpretability.

University of Larbi Ben M’hidi, Oum El-Bouaghi, Algeria

Keywords:Mathematical methods, Epidemics, Biology & Biomedicine Abstract: Fractional-order (F-O) equations have been shown to accurately describe the spread of coronavirus because of their ability to incorporate the effects of memory into the dynamics. The paper introduces a new and unique way of modeling COVID-19 using fractional-order operator. Unlike traditional models, this innovative approach incorporates the variable of vaccinated individuals, making it more comprehensive and accurate in its predictions. A newly derived theorem is put forth, highlighting the conditions necessary for the pandemic to subside. To illustrate the practical implications of the proposed model, the study includes a series of numerical simulations, demonstrating the efficacy of the obtained findings. These simulations serve to substantiate the effectiveness of the results presented in the paper.

Keywords:System identification & Modeling, Biology & Biomedicine, History of Fractional-Order Calculus Abstract: The research field of clinical practice has witnessed a notable increase in the integration of information technology and control systems engineering tools and the paradigm of drug dosing management for general anesthesia is no exception. Progressing from adequate to optimal drug dosing requires suitable models for closed loop control algorithms. High order, complex parameterized models for hypnosis are available but as one cannot measure real drug concentrations, accurate modelling is not possible. Moreover, ethical limitations impose serious restrictions as to type of excitatory signals acceptable to patient effect response evaluations. This paper proposes an innovative approach to determine fractional order models to compactly represent the dynamics inherent to the hypnosis response. A simplifying assumption is being made: instead of multi-compartmental models, a single transfer model is proposed consisting of a fractional-order dynamic that directly connects propofol, the administered drug, and the Bispectral (BIS) index, the measure of hypnosis. The proposed model is validated against clinical data and compared to integer order models to prove its suitability. Results suggest the model may be well used with control algorithms for computerized drug dosing management.

Keywords:System identification & Modeling, System Analysis & Dynamics Abstract: Personalised pharmacokinetic models imply stepping away from the classical assumption of homogeneous drug mixing in various tissue compartments in the body, with a particular impact on obese patients. In this work, the pharmacokinetic compartmental model structure is revisited to account for non-uniform distribution of uptake/clearance time constants in patients as a nonlinear function of body mass index. Simulations are confirming expected patterns of drug distribution in the body and can account for post-anesthesia side effects up to 72 hours. We apply spectroscopy to extract the complex impedance in fat tissue samples. The data is modelled by a Cole-Cole model, where parameters of the fractional order impedance model are optimized using a genetic algorithm. The findings suggest that fat tissue will exhibit anomalous diffusion when drug uptake and clearance are present.

Keywords:Mathematical methods Abstract: In the last decade, the interest in a special function named after the French mathematician Le Roy has provoked several authors to introduce and study its various extensions. Historically, the Le Roy function appeared almost in same time and in similar way of goals like the Mittag-Leffler function that has important role in Fractional Calculus. The Le Roy type functions are also ``fractional exponentials'' but the fractionalizing parameters appear as power indices of the involved Gamma functions. In our recent works, we have introduced rather general multi-index Le Roy type functions, studied their analytical properties and proposed their association to the class of Special Functions of Fractional Calculus.

A newly developed idea is to show that the Le Roy type functions can be represented in terms of the I-functions of Rathie and bar-H-functions of Inayat-Hussain that are further extensions of the Fox H-functions and Fox-Wright p Psi q-functions. It happens that other important mathematical functions as the polylogarithms, Riemann Zeta function and its extensions, also belong to this more general class of special functions. We have solved, at least in some simpler case, the problem to find integral and differential-like operators for which the Le Roy type functions are eigenfunctions. This lead us to a new class of operators with I-functions kernels that can be considered as further extensions of the operators of generalized fractional calculus.

Keywords:Mathematical methods Abstract: This paper deals with studying of a class of special functions called multi-index Mittag-Leffler-Le Roy functions. More precisely, these are 4m-parametric functions of a complex variable, which are generalizations of the Le Roy function from one side, and of the Mittag-Leffler and Prabhakar functions, from another side. The basic properties of them are discussed, for the case when they are entire functions and also, when they are not. Moreover, in the case of entire functions, their order and type are provided, depending on the parameters. The integral representations of the Mellin-Barns type, as well as the Laplace transform and Erd'elyi-Kober fractional integral of such functions are given.

Keywords:Mathematical methods Abstract: Time-space fractional models are proved to be highly efficient in characterizing anomalous diffusion in many intricate systems. This study introduces a numerical scheme employing a matrix transfer technique to discretize the fractional spectral Laplacian and subsequently applying generalized exponential time differencing for the semi-discrete system. The resulting second-order scheme is implemented efficiently using rational approximations for the two-parameter Mittag-Leffler function and the partial decomposition of these approxima- tions. The advantage of this approach is its applicability to different types of homogeneous boundary conditions including Robin boundary condition. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed numerical scheme.

Keywords:Mathematical methods Abstract: A new criterion for evaluating exact values for the maximum angles of A(alpha)-stability for linear multistep methods is proposed. This criterion gives an equation to be solved for a complex point corresponding to the tangent line from the origin to the root locus curve of the stability boundary. The maximum angle for A(alpha)-stability sector is then obtained from the tangent point. This criterion is applied to find the A(alpha)-stability angles for several fractional BDF-type numerical methods for fractional initial value problems.

Keywords:Mathematical methods Abstract: In this study, an efficient approach for solving the two-dimensional time-fractional convection problem with a non-smooth solution in the temporal direction is proposed. The solution exhibits weakly regular at the initial time, so the L1 formula on the nonuniform meshes is applied to discretizing the Caputo derivative. In the spatial direction, the central discontinuous Galerkin (CDG) method involving two approximate solutions defined on overlapping elements is used. The fully discrete scheme is proven to be numerically stable, and the optimal convergence result is attained. A few numerical experiments are conducted to confirm the theoretical conclusions.

Keywords:Mathematical methods Abstract: This article is dedicated to numerically solving the spatially loaded time-fractional diffusion equation with initial and Dirichlet-type boundary conditions. We apply finite difference approximation to the considered problem and use the well-known L1 method to approximate the Caputo fractional derivative. The numerical approximation of the problem yields a loaded implicit difference scheme. To obtain the solution, we employ a special parametric representation of solutions of auxiliary linear systems using the superposition property. In conclusion, we showcase several numerical tests, validate the outcomes, and observe the convergence of errors.

Keywords:Automatic Control & Stability, Electrical Engineering & Electromagnetism Abstract: This paper presents a new design of an adaptive fractional-order MRAC controller for the class of non-integer second-order systems. The update of the control law gains follows a derivative of fractional order equal to the order of the controlled system model. The stability analysis of the control system has been performed based on Lyapunov theorem. We apply this control law to an experimental multivariable lightning system. The results obtained show the efficiency and performance improvement of the proposed control solution compared to the classical MRAC.

Keywords:Automatic Control & Stability, Mathematical methods Abstract: Type 1 diabetes (TID) is a chronic disease which occurs by the destruction of beta cell, insulin producing hormone, in pancreas. Exogenous infusion of insulin, subcutaneously or via insulin infusion pump, is required for the regulation of blood glucose concentration in diabetic patients. In the paper is presented a comparisons among, proportional integral (PI), fractionalorder proportional integral (FO-PI), sliding mode control (SMC) and fractional-order sliding mode control (FO-SMC) controllers, for the regulation of the blood glucose level utilizing the fractional-order Bergman’s minimal model. Genetic algorithms are applied to optimized the listed controllers. In analyzing the performances of the proposed controllers, meal effect is considered as an external disturbance.

Keywords:Automatic Control & Stability, Mathematical methods, System Analysis & Dynamics Abstract: Many control systems show deteriorated performance because of hard nonlinearities that are common in different applications. Fractional-order control can find new results to compensate nonlinearities, but control design methods are still at infancy. This paper addresses the issue of limit cycles determined by saturation elements in nonlinear control loops including plant servo systems. The fractional-order controller consists of two lead networks in series, one shifted with respect to the other on the frequency axis, and both of non-integer order. They are designed in the frequency domain to provide a high phase lead at the resulting gain crossover frequency and a Bode plot of the loop transfer function with phase varying slowly around that frequency. Robust prevention of the limit cycle is obtained such that the Nyquist plot of the linear elements in the loop does not intersect the negative inverse describing function plot.

Université De Bordeaux - Bordeaux INP/ENSEIRB-MATMECA

Keywords:Automatic Control & Stability, Robotics Abstract: In robotics, obstacle avoidance in real-time is a fundamental aspect for path planning of autonomous mobile robots. A dynamic fractional repulsive potential field was recently proposed: the main advantage lies in the consideration the position, velocity and the nature of danger of each obstacle. One drawback is the lack of continuity of the repulsive force when entering the danger zone around an obstacle. Consequently, acceleration oscillations and energy losses results in a less efficient path planning for obstacle avoidance. Our new proposal of continuous dynamic fractional repulsive potential field aims to correct such drawbacks. Simulations illustrates the performances of our method a and a comparison with classic potential field methods is provided.

Keywords:Automatic Control & Stability, System identification & Modeling, Mathematical methods Abstract: Recent studies have shown that complex-order fractional operators allow for compact modeling with simpler structures and less parameters than models based on integer-order operators and even non-integer real-order operators. This work verifies the usefulness and potential of a complex-order fractional model when used to set parameters of a PID controller regulating the common-rail pressure in the injection system of a compressed natural gas engine. Parameters are derived by a general particle swarm optimization that considers a cost function weighting the performance and robustness measures. Simulation results show the effectiveness of the complex-order fractional model-based PID controller through the response to step references, the rejection of disturbances, the low sensitivity to variations of the injection system parameters.

Keywords:System Analysis & Dynamics, Automatic Control & Stability, Others Abstract: In this paper we evaluate the performance achievable with Proportional-Integral-Derivative fractional Double-Derivative controllers, denoted as PIDD^{2alpha}, by comparing it with that achievable with standard PID controllers and with Proportional-Integral-Double-Derivative (PIDD or PIDD^2) controllers, which are also called Proportional-Integral-Derivative-Acceleration (PIDA) controllers. The cost-effectiveness ratio improvement obtained when augmenting the control law with an element proportional to the fractional second derivative of the control error is assessed by considering different process transfer functions and by tuning the controllers minimizing the integrated absolute error with genetic algorithms. Further, a constraint on the maximum sensitivity is also posed so that the system robustness is taken into account. The set-point following and the load disturbance rejection tasks are evaluated separately.

Keywords:Mathematical methods, Physics, History of Fractional-Order Calculus Abstract: This paper is a review of the recent advances in the investigation of the properties of generalized fractional maps. These maps are Volterra difference equations of the convolution type with power-law-like kernels. They include fractional and fractional difference maps of a constant or variable order. The main objective of this article is to describe the criteria of stability, equations defining periodic points and bifurcations, and transition to chaos that are common for all generalized fractional maps.

Keywords:Mathematical methods, History of Fractional-Order Calculus, Others Abstract: This discussion paper presents some parts of the work in progress. It is shown that G.W. Leibniz was the first who raised the question about geometric interpretation of fractional-order operators. Geometric interpretations of the Riemann--Liouville fractional integral and the Stieltjes integral are explained. Then, for the first time, a geometric interpretation of the Stieltjes derivatives is introduced, which holds also for so-called ``fractal derivatives'', which are a particular case of Stieltjes derivatives.

Keywords:Mathematical methods, History of Fractional-Order Calculus Abstract: Many sources on fractional calculus include the assumption that all fractional orders are real. However, the usual definitions of Riemann--Liouville fractional integrals and derivatives can be used without modification in the case that the fractional orders are complex, and allowing complex orders creates a more rich structure in which the immense power of analytic continuation can be brought to bear to make many results on fractional derivatives trivially provable from the corresponding (easier) results on fractional integrals. This short paper summarises, for the benefit of the fractional community, the facts of using complex orders of differintegration, including the care that must be taken over defining the fractional derivative formulae precisely, and the usefulness of analytic continuation in this context.

National University of Sciences and Technology, Islamabad, Pakis

Keywords:Mathematical methods, Special Functions, History of Fractional-Order Calculus Abstract: We consider a very general class of fractional calculus operators, given by transmuting the classical fractional calculus along an arbitrary invertible linear operator S. Specific cases of S, such as shift, reflection, and composition operators, give rise to well-known settings such as that of fractional calculus with respect to functions, and allow simple connections between left-sided and right-sided fractional calculus with different constants of differintegration. We define, for the first time, general transmuted versions of the Laplace transform and convolution of functions, and discuss how these ideas can be used to solve fractional differential equations in more general settings.

Keywords:System Analysis & Dynamics, Special Functions, History of Fractional-Order Calculus Abstract: We study systems of fractional order differential equations involving the Prabhakar derivative of Caputo type. For commensurate systems we obtain their solutions in closed forms using the eigenvevtors and the eigenvalues of the associated square matrix in the system. We discuss the solutions under the cases where the eigenvalues are distinct, repeated or complex. We present several examples to illustrate the efficiency of the obtained results. For incommensurate systems we apply the Laplace transform to obtain their solutions. As the Prabhakar kernels involve many fractional kernels as particular cases, the obtained results will generalize several existing results in the literature.

Keywords:Mathematical methods, Physics, System Analysis & Dynamics Abstract: Equations involving the local fractional derivative operators allow to incorporate fractals into an equation and obtain corresponding solutions leading to some insight into the phenomena involved. Here we review the recent progress in understanding the interplay between the local fractional differential equations (LFDEs) and fractals. The simplest LFDE is considered and its solution is discussed along with possible applications. The stochastic versions of these equations are also considered and solution of one equation has been discussed here for the first time. Several examples, old and new, have been discussed. New directions towards a general theory have been discussed.

Keywords:Mathematical methods, Physics Abstract: Plasma represents one of the four basic states of matter, along with the solid, liquid, and gas states. Plasma oscillations represent a fundamental concept in plasma physics, and their understanding is crucial in various applications. Fractional order Lorentz models for plasma oscillations offer a more flexible framework for describing the complex dynamics and memory effects that are often observed in plasma systems. In this paper, we consider a fractional-order model for plasma oscillations. The model involves Caputo fractional derivative of order between one and two. An exponential time differencing methods is developed to approximate the motion of electrons driven by zero, static, and oscillating electric fields.

Keywords:Mathematical methods, Physics Abstract: This paper proposes a fast numerical scheme for time-space fractional diffusion equation in two spatial dimensions, where the temporal fractional partial derivative is in the Caputo sense of order alphain(0,1) and the spatial fractional derivative is given by fractional Laplacian left(-Deltaright)^{frac{beta}{2}} in two space dimensions with betain(1,2). The temporal Caputo derivative is discretized by a fast evaluation based on the L2-{1_sigma} formula. The spatial discretization adopts fractional centered difference formula in two dimensions. The proposed fully discrete scheme is proved to be stable and convergent with 2nd order accuracy both in time and space. The feasibility, stability, and convergence of the proposed scheme are demonstrated by the numerical example.