Keywords: Fractional differentiation, oscillating singularity, chirp;

Keywords: Artificial Intelligence, Automatic Control & Stability, Variational Principles
Abstract: In this study, a fractional order equation of motion as a continuous limit model for a family of gradient descent algorithms is discussed based on fractional order Euler-Lagrange equation. The aim of this proposed scheme is to search the ability to go beyond Nesterov scheme by introducing the potential of fractional calculus. The discretized version of the ”designed” fractional order equation of motion (FO- EOM) forms new gradient descent algorithm that has been tested on some optimization benchmark functions to fairly assess the performance in comparison with the standard and accelerated form of gradient descent algorithms. Promising results have been obtained with more rigorous mathematical analysis has to be carried out as our future work.

Keywords: Artificial Intelligence, Epidemics, Mathematical methods
Abstract: The main aim of this work, is to investigate the efficiency by presenting a four-compartmental nonlinear mathematical model including the potential smoker mathrm{P}, tobacco smoker mathrm{S}, electric cigarette smoker mathrm{E} and quite smoker mathrm{Q}. The Caputo fractional operator is applied to the aforementioned model to analyze different solutions. For the approximate solution, the fractional Adams-Bashforth numerical technique is applied. In this work, we also introduce the artificial neural network (ANN) method for the numerical simulation which is a significant contribution to this study. We also compare the Adams-Bashforth method with ANN to explain the dynamics of the considered model in a better way.

Keywords: Mathematical methods, Artificial Intelligence
Abstract: The method of fractional variational physics-informed neural network (fVPINN) combined with the idea of the discontinuous Galerkin (DG) method for time-fractional nonlinear conservation laws is presented. In the spatial direction, the computational domain is divided into several subdomains, where the so-called ``numerical fluxes" in the DG method at the interfaces of adjacent subdomains are used. In the temporal direction, the fractional derivative is approximated by the L1 formula. The method is flexible in the choices of activation function, depth and width of the network, optimization method, etc. And its parallel computing ability can reduce computing consumption. Numerical experiments are given to demonstrate the effectiveness of the proposed fVPINN.

Keywords: System identification & Modeling, Mathematical methods, Automatic Control & Stability
Abstract: For Smart Grid applications, establishing battery models with robustness, accuracy, and low complexity is crucial for the effective utilization and management of lithium-ion batteries, as well as for accurately estimating battery states such as state of charge and state of health. This paper begins by analyzing the electrochemical impedance spectrogram of lithium-ion batteries. Subsequently, it integrates impedance elements with fractional-order characteristics into the basic Thevenin circuit model based on fractional calculus theory. This approach allows the models to be embedded in microprocessors, facilitating the provision of accurate real-time results. Time domain identification procedure is developed based on the minimization of the Integral Absolute Error yielding a simple high acceptable global fractional order modeling of the Energy Storage System of a Real Smart Grid system with a very acceptable average of the Nrmse index of fit.

Keywords: Singularities Analysis and Integral Representations
Abstract: In this paper, an attempt has been made to establish the existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential equation subject to the non-local boundary conditions. We have shown that there exist countably many positive solutions with the help of fixed point index theory in a cone on a Banach space.

Keywords: System identification & Modeling, Automatic Control & Stability, System Analysis & Dynamics
Abstract: In real-world situations, many complex processes exhibit fractional order dynamic characteristics. In this context, the common integer order modeling methods may not appropriately capture the physics of these processes, leading to a less satisfactory controller design. This paper conducts a comparative analysis between the First order plus time delay (FOPTD) model and the Time Delay with Single Fractional Pole (TDWFP) model. These two modeling methods are employed to characterize a one-dimensional heat propagation process and two model predictive controllers are developed based on the models. The results demonstrate a better fitting performance in both time and frequency response with the TDWFP model. Furthermore, the TDWFP model-based Model Predictive Control exhibits reduced sensitivity to the prediction horizon and requires low control input energy. This indicates the potential of using fractional order model in MPC for improved control energy efficiency in practical applications.

Keywords: System Analysis & Dynamics, Mathematical methods, Automatic Control & Stability
Abstract: In this paper, the Infinite State representation is applied to the fractional Lü chaotic system. Thanks to a finite dimension approximation, the original fractional order system is converted into a large dimension set of integer order nonlinear equations whose initial conditions allow to test the butterfly effect of the equivalent chaotic system. This sensitivity to initial conditions is quantified thanks to Lyapunov exponents computed with an experimental technique. Then, the largest Lyapunov exponent is used as fractional index to characterize the Infinite State representation.

Keywords: Mathematical methods, Finance and Economics, Special Functions
Abstract: In this work, we use the invariant subspace method to obtain separable solutions for the well-known time-fractional Black-Scholes equation (tF-BSE) under three different kinds of fractional derivatives that are (i) Caputo derivative, (ii) regularized Prabhakar derivative, and (iii) Hilfer derivative. Also, the comparison of the obtained solutions of tF-BSE with those mentioned above three different fractional-order derivatives has been discussed. In addition, this work shows that generalized separable exact solutions include the two- and three-parameters of the Mittag-Leffler, the Euler-gamma, exponential, and polynomial functions. Additionally, we compare the obtained separable solutions in two-dimensional graphically under considered fractional-order derivatives for various values of fractional-order alpha,alphain(0,1].

Keywords: System identification & Modeling, Electrical Engineering & Electromagnetism
Abstract: A fractor is an electronic component similar to a capacitor, but modelled by a fractional derivative. Consequently, while the discharge of a capacitor corresponds to an exponential evolution of voltage with time, the discharge of a fractor can be modelled with the Mittag-Leffler function. In this paper, experimental data of the discharge of a fractor is given, and used to find a model for its dynamic behaviour.

Keywords: Mathematical methods, Finance and Economics, Signal Processing
Abstract: Fourier-Laplace transform technique allows one to represent several classes of important probability distributions and solutions of basic boundary problems for wide classes of fractional differential equations as integrals of functions enjoying two key properties: analytic continuation to a cone or the union of a cone and tube domain, and regular decay at infinity. Integral representations for Wiener-Hopf factors, fractional moments and special functions enjoy these properties as well. In the paper, we present the general methodology which allows one to evaluate the integrals enjoying these properties very fast and accurately. Among applications, we derive new efficient realizations of the Fourier, Laplace and Z-transforms, representations for probability distributions in Levy models, stable ones including, and algorithms for pricing contingent claims, Monte-Carlo simulations and evaluation of special functions.

Keywords: Signal Processing, Automatic Control & Stability
Abstract: Non-smooth optimization has now played an important role in many fields, such as l_1-regulation problem and sparse recovery problem. In this paper, a novel fractional fixed-time proximal gradient method is proposed for solving the composite optimization problems with non-smooth terms. At first, according to the distribution of zeroes of Mittag-Leffler functions, a novel fixed-time convergence theory with a fractional adaptive gain is proposed. Based on the optimal condition analysis of the transformed problem using the augmented Lagrangian function, a novel fixed-time proximal gradient method is then proposed and the fixed-time convergence property is rigorously proven. Furthermore, the proposed algorithm is applied to solve the sparse recovery problem in compressive sensing, and the restricted isometry condition required for the application of the algorithm is analyzed to achieve fixed-time recovery of the signal. Finally, simulation examples are provided to validate all the results.

Keywords: Image Processing
Abstract: The main objective of this paper is to solve the problem of low resolution of target distance images obtained by GM-APD lidar. Towards overcoming this problem, this paper proposes a fractional-order super-resolution reconstruction algorithm for GM-APD lidar distance images based on convex set projection. Firstly, the first low-resolution image is upsampled three times to obtain a high-resolution reference image. Second, starting from the image degradation model, the transformation model between the low-resolution image and the high-resolution reference image is constructed. Then the high-resolution reference image is convolved using the G-L fractional order differential operator, and the estimated low-resolution image is obtained by combining the point spread function. Finally, the difference between the remaining low-resolution images and the estimated low-resolution image is calculated, and the high-resolution reference image is corrected to obtain the final high-resolution distance image. In addition to solving the edge blurring problem of the original convex set projection algorithm, the edge structure of the image is enhanced and the image quality is improved. Simulation and experimental results show that the proposed algorithm improves the information entropy by at least 24.190% and the average gradient by at least 10.812%, and achieves the super-resolution reconstruction of GM-APD LIDAR target distance images.

Keywords: Image Processing
Abstract: The morphological structure of retinal blood vessels plays an important role in diagnosing ophthalmic diseases such as glaucoma and diabetic retinopathy. Diabetic retinopathy is one of the leading causes of blindness worldwide. Because of the need to examine a large number of individuals on a daily basis, it is indeed challenging for ophthalmologists to analyze the complex retinal vasculature of every outpatient. Automated segmentation of retinal blood vessels can be of significant aid in this labor-intensive process. This work proposes an unsupervised approach for retinal blood vessel segmentation. Here, we enhance the performance of the B-COSFIRE filters for segmentation of the retinal vasculature by denoising a retinal image using a fractional filter derived from a left/forward weighted fractional integral prior to the vascular enhancement by B-COSFIRE filters. Also, we utilize hysteresis thresholding rather than a single global threshold to obtain the final segmented vessels enhanced by the B-COSFIRE filters. This helps to avoid the problem of nonuniform illumination and poor contrast in retinal images.