The main goal of the paper is to use a generalized proportional Riemann-Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved such as the fractional derivative of a squared function. Also some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given illustrating the meaning of the equilibrium as well as its stability properties.

This study develops a new definition of fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. Such developed definition encompasses many types of fractional derivatives, such as the Riemann-Liouville and Caputo fractional derivatives for singular kernel type as well as the Caputo-Fabrizio, the Atangana-Baleanu and the generalized Hattaf fractional derivatives for non-singular kernel type. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, newly numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application to computational biology is presented.

In this article, we establish the existence of and then compare smallest eigenvalues for nabla fractional boundary value problems involving a fractional difference boundary condition, using the theory of u0-positive operators with respect to a cone in a Banach space.

The main objective of this present paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. Also, we give some results related to the newly defined fractional operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

Lithium batteries are widely used due to their advantages such as high energy density, stable performance, low pollution, and long recyclable life. Accurate SOC estimation is important for the use of lithium batteries. To address the problem of low accuracy of SOC estimation by traditional methods, this paper proposes a joint method of fractional order robust unscented Kalman filter (FORUKF) and robust unscented Kalman filter (RUKF) to estimate SOC. The method is based on a fractional order model (FOM) that combines the unscented transformation (UT) technique, the H∞ observer and the joint estimator. Specifically, an adaptive genetic algorithm (AGA) was first used to identify the parameters of the FOM for lithium batteries and to verify the accuracy of the model. Estimation and updating of the ohmic resistance R0 and the capacity QN in the model in real-time by RUKF and then estimation of the SOC by FORUKF. Finally, the accuracy of FORUKF-RUKF was verified under the Federal Urban Driving Schedule (FUDS), US06 Highway Driving Schedule and Beijing Dynamic Stress Test (BJDST). According to the results, the FORUKF-RUKF estimated SOC was found to have better accuracy and robustness than the RUKF, FOUKF and EKF.

Exact and soliton type solutions have great importance in propagation of surface waves, fluid dynamics, optics, and many other elds of nonlinear sciences. In this study, the explicit and exact soliton type solutions for two space-time fractional Equal- Width (FEW) equations with conformable derivative are procured via the hyperbolic function approach. The wave type solutions are represented in some hyperbolic and trigonometric functions.

In this paper, we study the fractional pseudo-parabolic equations u_t + (−△)^s u + (−△)^s u_t = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (−△)^s and the fractional Sobolev space H^s and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u₀) < d, the solution is global in time with I(u₀) > 0 or ∥u₀∥X_0(Ω) = 0 and blows up +∞ with I(u₀) < 0; for the critical initial energy J(u₀) = d, the solution is global in time with I(u₀) ≥ 0 and blows up at +∞ with I(u₀) < 0. The decay estimate of the energy functional for the global solution is also given.

In this study, we actually want to explore the time-fractional Phi-four equation via two methods, the exp _a function method and the hyperbolic function method. We transform a fractional order dierential equation into an ordinary differential equation using a wave transformation and the fractional derivative in conformable form. Then, the resulting equation has successfully been explored for new explicit exact solutions. The procured solutions are simply showed the effectiveness and plainness of the projected methods.

We primarily investigate the existence of solutions for fractional neutral integro-differential equations subjected to Neumann-type boundary conditions, which is crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by integrating fixed point theorems, we establish the existence of mild solutions under more relaxed conditions.

In this study, by using Least Square Method, the dataset for the Gross Domestic Product per capita is modeled as a function satisfying the fractional differential equation. The function itself is assumed to be the finite summation of its previous values and the derivatives with unknown coefficients. Then, the prediction for the upcoming years is done by having an approach dividing the dataset into 4 regions corresponding to four different tasks. The mathematical model of the Gross Domestic Product (GDP) per capita of the countries (and union) which are Brazil, China, European Union (EU), India, Italy, Japan, UK, the USA, Spain, and Turkey is constructed with a new methodology called as the deep assessment method which comes from the expressing an arbitrary function modeling the dataset as the finite summation of its previous values and the derivatives with unknown coefficient. The method uses the fractional calculus properties combining with Least Square Method and is compared to Long short-term memory (LSTM) algorithm which is a special type of neural network used for time sequences in general.

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalisedMittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.

We presented a fractional order model to capture the transmission dynamics of Anthrax infection with a nonlinear force of infection and its long-term impacts. Analysis by the standard Fabrizio Caputo approach established the model is well-posed and investigation on the stability of the model, Anthrax-free equilibrium as connected to the basic reproduction number R0

We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semiflexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, as well as that of any other passive particle (or probe) bound to the elastic system. Our study explores the diffusional behavior emerging for both the active particle and a distant probe.

The paper focuses on the admissibility problem of descriptor fractional-order systems (DFOSs). The alternate admissibility criteria are addressed for DFOSs with order in (0,2) which involve a non-strict linear matrix inequality (LMI) method and a strict LMI method, respectively. The forms of non-strict and strict LMIs are brand-new and distinguished with the existing literature, which fill the gaps of studies for admissibility. These approaches are available to the order in (0,2) without separating the order ranges into (0,1) and [1,2). In addition, a method involved least real decision variables in terms of strict LMIs is derived which is more convenient to process the practical solution. Three numerical examples are given to illustrate the validity of proposed results.

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.

New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.

Covid-19 is a lung disease caused by a Coronavirus family virus. Due to its extraordinary prevalence and death rates, it has spread quickly to every country in the world. Thus, achieving peaks and outlines and curing different types of relapses is extremely important. Given the worldwide prevalence of Coronavirus and the participation of physicians in all countries, Information has been gathered regarding the properties of the virus, its diverse types, and the means of analyzing it. Numerous approaches have been used to identify this evolving virus. It is generally considered the most accurate and acceptable method of examining the patient's lungs and chest through a CT scan. As part of the feature extraction process, a method known as fractional Fourier transform (FrFT) has been applied as one of the time-frequency domain transformations. The proposed method was applied to a database consisting of 2481 CT images. Following the transformation of all images into equal sizes and the removal of non-lung areas, multiple combination windows are used to reduce the number of features extracted from the images. In this paper, the results obtained for KNN and SVM classification have been obtained with accuracy values of 99.84% and 99.90%, respectively.

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed and the effects of three typical calculation methods in different test scenarios are compared and studied in this paper. It is proved that Oustaloup’s rational approximation method usually overestimates the peak-to-peak current and brings in pulse-voltage phenomenon in piecewise smooth test scenarios, while the results of two iterative recurrence-form numerical methods are in good agreement with the experimental results. The study results are dedicated to provide a reference for efficiently deploying calculation methods in fractional-order piecewise smooth circuit systems. Some quantitative analysis results are concluded in this paper.

Fractional differential systems have received much attention in recent decades likely due to its powerful ability in modeling memory processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. Obviously, bang-bang property is of a significant importance in optimal control theory. In this paper a Bang-Bang property and time-optimal control problem for time-Fractional Differential System (FDS) is under consideration. First we formulate our problem and we prove the existence theorem. We state and prove the bang-bang theorem. Finally we give the optimality conditions that characterized the optimal control. Some illustrate applications are given to clarify our results.

The paper's main goal is to introduce a novel type of interpolative convex contraction and build up some fresh findings for it using the interpolative convex contractions' progressive approach. We have established certain conclusions using orbitally S-complete and Suzuki type contractions in F-metric spaces. My research aims to examine the fixed point method's efficacy in solving fractional differential equation with boundary conditions.

Fractional differential mathematical model unfolding the dynamics of the COVID-19 pandemic in India is presented and explored in this paper. The purpose of this study is to estimate the future outbreak of disease and potential control strategies using mathematical models in India as a whole country as well as in some of the states of the country. This model is calibrated based on reported cases of infections over the month of April 2020 in India. We have used iterative fractional complex transform method to find approximate solutions of the model having modified Riemann Liouville fractional differential operator. We have also carried out a comparative analysis between actual and estimated cumulative cases graphically, moreover, most sensitive parameters for basic reproduction number$(R_0)$ are computed and their effect on transmission dynamics of COVID-19 pandemic is investigated in detail.

Complex diseases such as cancer are caused by changes in the Gene Regulatory Networks. Systems that model the complex dynamics of these networks along with adapting to real gene expression data are closer to reality and can help understand the creation and treatment of cancer. In this paper, for the first time, modelling of gene regulatory networks is performed using delayed nonlinear variable order fractional systems in the state space by a new tool called GENAVOS. This tool uses gene expression time series data to identify and optimize system parameters. This software has several tools for analyzing system dynamics. The results show that the nonlinear variable order fractional systems have very good flexibility in adapting to real data. We found that regulatory networks in cancer cells actually have a larger delay parameter than in normal cells. It is also possible to create chaos, periodic and quasi-periodic oscillations by changing the delay, degradation and synthesis rates. Our findings indicate a profound effect of time-varying order on these networks, which may be related to a type of cellular memory due to epigenetic and environmental factors. We showed that by changing the delay parameter and the variable order function for a normal cell system, its behavior changes and becomes quite similar to the behavior of a cancer cell. This work also confirms the effective role of the miR-17-92 cluster in the cancer cell cycle. GENAVOS is available at https://github.com/hanif-y/GENAVOS with its user guide and MATLAB codes.

In this paper, we established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained by using the techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we have used the distributed characteristic operators to define the mild solution of the system. Results obtained here improve and extend some known results.

We study the existence of positive solutions for a system of Hadamard fractional differential equations on an infinite interval with nonnegative nonlinearities, subject to coupled nonlocal boundary conditions which contain Hadamard fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Guo-Krasnosel'skii fixed point theorem and the Leggett-Williams fixed point theorem.

Sentinel-2 provides the opportunity to map the snow cover at unprecedented spatial and temporal resolution at global scale. Here we calibrate and evaluate a simple empirical function to estimate the fractional snow cover (FSC) in open terrain using the normalized difference snow index (NDSI) from 20 m resolution Sentinel-2 images. The NDSI is computed from flat surface reflectances after masking cloud and snow-free areas. The NDSI-FSC function is calibrated using Pléiades very high resolution images and evaluated using independent datasets including SPOT 6/7 satellite images, time lapse camera photographs, terrestrial lidar scans and crowd-sourced in situ measurements. The calibration results show that the FSC can be represented with a sigmoid-shaped function 0.5×tanh(a×NDSI+b)+0.5 where a = 2.65 and b = -1.42 yielding a root mean square error of 25%. Similar RMSE are obtained with different evaluation datasets with a high topographic variability. With this function, we estimate that the confidence interval on the FSC retrievals is 38% at the 95% confidence level.

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo type fractional stochastic delay differential systems (FSDDSs) with Poisson jumps by utilizing delayed perturbation of Mittag-Leffler function. Moreover, by using Burkholder-Davis-Gundy's inequality, Doob's martingale inequality and Holder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

In this paper, we investigate the Green measure for a class of non-Gaussian processes in $\mathbb{R}^{d}$. These measures are associated with the family of generalized grey Brownian motions $B_{\beta,\alpha}$, $0

In this research, a computation algorithm is established for a fractional order 2D fuzzy heat equation. In this study, Elzaki transform and HPM fusion is produced. Computing the desired outcome in series yields a fast convergence on an appropriate response. Examples are provided to support the conclusions, which are then compared with a particular approach to show the effectiveness and potential of the suggested approach. Two crisp equations—one for the lower bound solution and one for the upper bound solution are constructed from the input fuzzy fractional heat equation. The contour and surface representations of the approximate and exact results are offered for the lower and upper-bound solutions. The l∞-error norm is used in this study to validate the numerical convergence aspect. Together with the absolute inaccuracy, the approximate and exact solutions are matched. It has been demonstrated that the proposed regime will make it feasible to work with fuzzy fractional partial differential equations in a wide range of dimensions.

The continuing public health issue known as COVID-19 (the 2019 Novel Corona virus infection) has a global emphasis. Despite (or perhaps because of) the fact that there are significant gaps in our understanding of COVID-19 epidemiology, transmission dynamics, research methods, and management breakout poses a new kind of global hazard. The good news is that there is currently enough knowledge about the epidemic process to allow for the creation of mathematical forecasting models. We modify a conventional SEIR epidemic model to the unique dynamic compartments and epidemic features of COVID 19 as it spreads in a population with a diverse age structure. Although many US states and other nations around the world followed lockdown and reopening processes, we perform some analysis on using some techniques of the epidemic course. A new perspective of fractional calculus known as piecewise derivatives of fractional order is used to study the proposed model. Sufficient conditions are established to show the existence theory. In addition, a numerical scheme based on Newton’s polynomials is established to simulate the approximate solutions of the proposed model by using various fractional orders. Some real data results are also shown with comparison of the numerical results.

The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts interests are often in symmetry and in the rates of change of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. Applying these estimates we obtain Chung's laws of the iterated logarithm for spatial surfaces of operator fractional Brownian motion. Our results show that the precise rates of change of spatial surfaces are completely determined by the self-similarity exponent.

This study presents finite-time synchronization of chaotic fractional order systems with disturbance uncertainty and unknown time delay. First, a PID sliding surface is presented. Then, for the finite-time synchronization of the master and slave systems, a robust sliding-adaptive control method is provided. Update rules have been retrieved to estimate the system parameters by using the Lyapunov function while establishing the stability of the suggested mechanism and assuring the convergence of errors to zero. The proposed approach was used to solve the fractional order system of Gensou-Tsei with time-varying parameters, and the simulation results show that the presented approach performs well.

The article is devoted to the research of economic cycles and crises, which are studied within the framework of the theory of long waves by N. D. Kondratiev (K-waves) using the mathematical apparatus of fractional calculus. The object of the study is the mathematical model of S.V. Dubovsky 3 and some of its modifications, which describe the dynamics of the economic system, characterizing the interaction between the efficiency of new technologies and the efficiency of capital productivity. Modifications in the model are determined by the following points: the mathematical model takes into account the dependence of the rate of accumulation on capital productivity, there is an external harmonic influence - the influx of investments and new technological solutions, the economic system has heredity, including time-dependent, which leads to a delayed effect - the system’s response to impact. The property of heredity in the mathematical model is taken into account using fractional derivatives of constant and variable orders. Mathematical model S.V. Dubovsky and its modifications are studied numerically using the Adams-Bashforth-Moulton algorithm. Oscillograms and phase trajectories are constructed, and an interpretation of the simulation results is given.

In this study, we first present a time-fractional L$\hat{e}$vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L$\hat{e}$vy-time fractional diffusion equation of European-style options. Introduce a more general model from the models based on the L$\hat{e}$vy-time fractional diffusion equation and review some recent findings regarding of the Europe option pricing of risk-neutral free.

We here investigate the dynamical behavior of a fractional-order predator-prey system with anti-predator behavior and Holling IV type functional response. First we study the non-negativity, existence, uniqueness, and boundedness of solutions to the system from a mathematical analysis perspective. Then, we analyze the stability of its equilibrium points and the possibility of bifurcations using stability analysis methods and bifurcation theory, and prove the existence of a supercritical Hopf bifurcation in the system. After providing numerical simulations to illustrate the conclusions theoretically derived, by summarizing those various analytical results obtained, we finally present three interesting conclusions that can contribute to better understanding and preservation of ecological systems.

The quality of underground space information has become a major problem that endangers the safety of underground spaces. Currently, the main methods for the high-precision and long-distance transmission of detection information are radar and optical methods. However, in practical applications, we found that the radar method has the shortcomings of large energy loss and poor anti-jamming ability, which limit the accuracy of information data transmission and distance. The optical method has the shortcomings that the weather has a great impact on its accuracy and can only be applied to static objects above ground; therefore, it has the limitation of application objects and use environment. More importantly, the current high-precision information remote detection methods are limited to the detection of overground space objects and are not applicable to the detection of various information data in underground space. In this study, we analyze the spectral properties of the fractional differential operator and find that it is suitable for studying non-linear, non-causal, and non-stationary signals. The theory of fractional calculus is applied to the field of data processing, and a mathematical model of remote transmission and high-precision detection of information based on fractional difference is established, which realizes the functions of high-precision and remote detection of information. By fusing the information data to detect the mathematical model over a long distance and with high accuracy, a mathematical model for stratum data processing used to provide long-distance and high-accuracy data was established. Through application in engineering practice, the effectiveness of this method for underground space information data detection was verified.

Four discrete models using the exact spectral derivative discretization finite difference (ESDDFD) method are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is in part an artifact of the CEFD method as it is absent from the ESDDFD models.

The information data accuracy affects the working reliability of production management system directly. Mobile equipment is an important part of integrated manufacturing system, in the course of its works, due to the complex working environment and the dynamic change of its position, it lead to the important problem of information detection distortion. Because the application of fractional differential operator has the dual functions of enhancing signal strength and reducing data error, we puts forward a data processing method of obtaining production information data of various accuracy by changing the parameters of fractional differential operator. Establish a mobile equipment detection data fusion processing model based on fractional order partial differential equations and apply it in the processing experiment of mobile equipment testing information detection data in an integrated manufacturing system, the feasibility and effectiveness of the method are verified. We conclude that the fractional order partial differential equations used in the processing of mobile equipment monitoring information data of an integrated manufacturing system has the function of obtaining production information data of various accuracy, which are of great significance to improve the working reliability in integrated manufacturing system. which is of great significance to improve the working reliability and scientific decision of integrated manufacturing system.

The Stepped Frequency Waveform (SFW) is commonly employed in radar technology to synthesize wideband signals through the aggregation of narrow-band pulses, thereby achieving high-resolution range profiles without the need to increase the radar's instantaneous bandwidth. However, the inherent large time-bandwidth product of SFW introduces substantial ranging errors and energy dispersion, which significantly hinders its efficacy in detecting high-speed objects. This paper introduces a pioneering velocity estimation technique utilizing the fractional Fourier transform (FrFT) to address these limitations. Leveraging the characteristic of the Doppler signal from a moving target, which manifests as a chirp signal with a rate proportional to the target's velocity, the FrFT is utilized for precise velocity estimation. Subsequent to this, velocity compensation is applied using the deduced metrics, followed by the application of the inverse fast Fourier transform (iFFT) to pinpoint the target's exact location. To optimize the computational efficiency of determining the FrFT's optimal order, we propose an iterative algorithm founded on the golden section search method. The effectiveness of the proposed approach is verified by simulation data, and the results demonstrate that the proposed approach can accurately estimate the velocity and the range of the high-speed targets with a relatively low computational complexity.

In the present work we study the existence of solutions for hybrid nonlinear fractional integrodifferential equations. We developed an algorithm by using the operator theoretical techniques in order to obtain the approximate solutions. The main results depend on the Dhage iteration method that were incorporated with the modern hybrid fixed point theorems. The approximate solutions were obtained by using Lipschitz conditions and weaker form of mixed partial continuity. Further, we provide some examples to explain the hypotheses and the related results.

The fractional diffusion equations with integrals play a significant role in describing anomalous diffusion phenomena. We first construct an appropriate resolvent family, and use the convolution theorem of Laplace transform, the probability density function, Cauchy integral formula and Fubini theorem to study its related equicontinuity, strongly continuity, compactness, etc. Then, a reasonable mild solution is defined. Finally, by combining some fixed point theorems, we obtain some conclusions on the existence and uniqueness of mild solutions.

The major goal of this work is to present a novel fractional temperature-dependent boundary element model (BEM) for solving thermoelastic wave propagation problems in smart nanomaterials. The computing performance of the suggested methodology was demonstrated by using stable communication avoiding S-step – generalized minimal residual method (SCAS-GMRES) to solve discretized linear BEM systems. The benefits of SCAS-GMRES are investigated and compared to those of other iterative techniques. The numerical results are graphed to demonstrate the influence of fractional, piezoelectric, and length scale characteristics on total force-stresses. These findings also demonstrate that the BEM methodology is practical, feasible, effective, and has superiority over domain methods. The results of the present paper help to develop the industrial uses of smart nanomaterials.

L\'evy walks represent important modeling tools for variety of real life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far these scaling limits were derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here we derive the corresponding limiting equations in the case of position depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo-Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalisations are analysed.

This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison principle. This criterion provides a new approach to analyze the finite-time synchronization problem of neural networks with stochasticity. Finally, numerical simulations are provided to demonstrate the effectiveness and superiority of the obtained results.

In this paper, using the two integers that describe the stationary 2-dimensional wave and the charge quantization along with the balance between the Lorentz force and electrical force, we succeed in deriving the fractional quantum Hall effect and the integer quantum Hall effect; we find that the latter exists as a special case of the former. Moreover, using the derived expression describing the fractional quantum Hall effect, a relationship between the plateau in the resistivity of the sample and the applied magnetic field is obtained. The findings of this model agree well with experimental measurements. Because the two integers that describe the stationary 2-dimensional wave and the charge quantization along with the force balance have concrete physical meanings in this work, we could provide a clear picture of the origin of both the integer quantum Hall effect and the fractional quantum Hall effect.

In our increasingly urbanized world, precise and up-to-date maps of human settlements are essential for sustainable urban development policies. The availability of open-access Sentinel-2 data from the Copernicus program presents an opportunity to create a comprehensive global map of human settlements, offering a detailed view of built areas on a large scale. This study estimates large-scale built-up fractions using encoder-decoder deep learning architectures like U-net, Res-U-net, and Attention-U-net in the large and complex urban area of Delhi, India. Openly available datasets like Open Street Map (OSM) and Microsoft building footprint datasets are used to derive built-up fractions at 10×10m resolution cells for over 34,000 km2. Our results show that Attention-U-net with the Huber loss function performs the best in different built-up densities (i.e., urban, semi-urban or rural) with an R2 score of 0.631, while Res-U-net and U-net obtained an R2 score of 0.623 and 0.612, respectively. The investigated networks significantly improve the accuracy over the latest Global Human Settlement Layer product (GHSL-S2), which uses a deep-CNN and reaches an R2 of 0.387 in our case study area. The result of this study yields a valuable spatial layer for examining the spatial distribution of human settlements across the entire spectrum from rural to urban areas.

In this study we used a non-autonomous Chua’s Circuit, and the fractional Lorenz chaos system together with a detection method from Extension theory to analyze the voltage signals. The measured bearing signals by acceleration sensor were introduced into the master and slave systems through a Chua’s Circuit. In a chaotic system minor differences can cause significant changes that generate dynamic errors, and extension matter-element models can be used to judge the bearing conditions. Extension theory can be used to establish classical and sectional domains using the dynamic errors of the fault conditions. The results obtained were compared with those from Discrete Fourier Transform analysis, Wavelet analysis and an integer order chaos system. The diagnostic ratio showed the fractional order master and slave chaos system calculations. The results show that the method presented in this paper is very suitable for monitoring the operational state of ball bearing system to be superior to the other methods. The diagnosis ratio was better and there were other significant advantages such as low cost and few.

The key objective of the present paper is to propose a numerical scheme based on the homotopy analysis transform technique to analyze the time-fractional nonlinear predator-prey population model. The population model is coupled fractional order nonlinear partial differential equations often employed to narrate the dynamics of biological systems in which two species interact, first is a predator and the second is a prey. The proposed scheme provides the series solution with great freedom and flexibility by choosing appropriate parameters. The convergence of results is free from small or large parameters. Three examples are discussed to demonstrate the correctness and efficiency of the used computational approach.

This research paper is about to obtain new soliton solutions to the M-fractional (1+1)-dimensional nonlinear Westervelt model by utilizing $\exp_a$ function and modified simplest equation techniques. The gained solutions involving dark, bright, periodic, dark-bright and other solitons. These results have many applications in wave propagation of sound waves, high amplitude in medical imaging and therapy. Achieved results are verified by Mathematica tool and the effect of M-fractional derivative on the solutions is explained through 2-Dimensional, 3-Dimensional and contour plots. At the end, these techniques are simple, fruitful and effective to deal with nonlinear FPDEs.

Abstract The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the cosine function. Here we generalize this MBF to include a fractional parameter, such that the exponential in the previously mentioned integral is replaced by a Mittag-Leffler function. The necessity for this generalization arises from a problem of communication in networks. We find the power series representation of the fractional MBF of the first kind, as well as some differential properties. We give some examples of its utility in graph/networks analysis and mention some fundamental open problems for further investigation.

The fractional telegraph partial differential equation with fractional Atangana-Baleanu-Caputo (ABC) derivative is studied. Laplace method is used to find the exact solution of this equation. Stability inequalities are proved for the exact solution. Difference schemes for the implicit finite method are constructed. The implicit finite method is used to deal with modelling the fractional telegraph differential equation defined by Caputo fractional of Atangana-Baleanu (AB) derivative for different interval. Stability of difference schemes for this problem is proved by the matrix method. Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the proposed method.

Modern power systems are confronted with a spread concern on the frequency stability issue due to the widespread integration of randomly fluctuating renewable resources. To address the above concern, this work introduces a load frequency control (LFC) scheme based on a hydrogen based energy-storage unit and a parameter tuning strategy for fractional-order proportional-integral-derivative (FOPID) controller. Firstly, a two-area interconnected power system (IPS) model, including thermal, hydro, solar, wind, gas power generator, and hydrogen based energy-storage unit, is established. Then, a FOPID controller is designed for this IPS model, and an improved gradient-based optimizer (IGBO) is developed to adaptively regulate the parameters of the FOPID controllers. Finally, the effectiveness of the offered LFC scheme are tested through load disturbance and renewable energy fluctuations test scenarios, and provides comparison and robustness analysis among different schemes. The test results validated that the offered LFC scheme can effectively suppress the frequency fluctuations of IPS and has excellent robustness.

In engineering practice, various types of information data are affected by many factors during the collection process. For example, information data measurement errors are caused by equipment performance and the working environment. During the transmission of detection information, the signal distortion caused by energy loss and signal interference causes unpredictable detection errors in the collected information data. Through the study of fractional calculus theory, it was found that it is suitable for studying nonlinear, non-causal, and non-stationary signals, and has the dual functions of improving detection information and enhancing signal strength. Therefore, under the influence of many factors, we applied the fractional difference algorithm to the field of information data processing. Multi-sensor detection data fusion algorithm based on the fractional partial differential equation was adopted to establish its online detection data. A multi-sensor detection data fusion algorithm based on a fractional partial differential equation is established, which effectively fuses the information data detection errors caused by various influencing factors and greatly improves the detection accuracy of information data. The effectiveness of this method is proved through its application in an experiment.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

The concepts of fractional calculus and the related techniques have been gaining momentum in circuit system fields, and an increasing body of evidence suggests that fractional-order characteristics are widely distributed in electronic components. In this paper, we consider the fractional characteristics of non-solid aluminum electrolytic capacitors and focus on the nonlinear dynamic characteristics of peak-current-mode controlled (PCMC) dc-dc boost converters with such components. In the work, we establish a piecewise smooth non-commensurate fractional-order model to describe the converter, in which the fractional-order equivalent impedance model of capacitors is adopted and the detailed control loop is considered. In order to analyze the converter, we use the fractional Adams-Bashforth-Moulton typed method (F-ABM) and stroboscopic map technique to obtain the time-domain solutions and the bifurcation diagram. The results of bifurcation diagrams indicate that, with the change of different parameters, the converter enters chaotic state through period doubling bifurcation. In order to provide empirical evidence for the nonlinear dynamic analysis, we consider both circuit-level simulation and experiments in the work, the results of which further confirm the results of theoretical analysis.

This paper describes a proposed fractional filter-based multi-scale underwater and hazy image enhancement algorithm. The proposed system combines a modified global contrast operator with fractional order-based multi-scale filters used to generate several images, which are fused based on entropy and standard deviation. The multi-scale-global enhancement technique enables fully adaptive and controlled colour correction and contrast enhancement without over exposure of highlights when processing hazy and underwater images. This in addition to illumination/reflectance estimation coupled with global and local contrast enhancement. The proposed algorithm is also compared with the most recent available state-of-the-art multi-scale fusion de-hazing algorithm. Experimental comparisons indicate that the proposed approach yields better edge and contrast enhancement results without halo effect, colour degradation and is faster and more adaptive than all other algorithms from the literature.

In this paper, we succeed to discover the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of concerned model. Finally, this technique is simple, fruitful and reliable to handle the nonlinear FPDEs.

Suitable measures of grazing impacts on ground cover, that enable separation of the effects of climatic variations, are needed to inform land managers and policy makers across the arid rangelands of the Northern Territory of Australia. This work developed and tested a time-series, change-point detection method for application to time series of vegetation fractional cover derived from Landsat data to identify irregular and episodic ground-cover growth cycles. These cycles were classified to distinguish grazing impacts from that of climate variability. A measure of grazing impact was developed using a multivariate technique to quantify the rate and degree of ground cover change. The method was successful in detecting both long term (> 3 years) and short term (< 3 years) growth cycles. Growth cycle detection was assessed against rainfall surplus measures indicating a relationship with high rainfall periods. Ground cover change associated with grazing impacts was also assessed against field measurements of ground cover indicating a relationship between both field and remotely sensed ground cover. Cause and effects between grazing practices and ground cover resilience can now be explored in isolation to climatic drivers. This is important to the long term balance between ground cover utilisation and overall landscape function and resilience.

: This paper presents a comparison between direct integer orders or classical model reference adaptive control (IO-DMRAC) versus its fractional orders counterpart (FO-DMRAC) applied to the F-16 aircraft longitudinal model pitch rate control. A substantial improvement is achieved when using fractional order adaptive control, since, in addition to performance indices, such as Integral Square-Error criterion (ISE) and Integral Square-Input criterion (ISU) being lower, the oscillations of the system output (pitch rate) in the transient period, are eliminated. Moreover, it is the first attempt in literature of adaptive control that a FO-DMRAC is applied to the pitch rate control of an aircraft. A F-16 short-period model, whose relative degree is equal to 1 (n^*=1) is analyzed by mean of simulations with good results. A FO-DMRAC is designed, for a specific operating flight conditions and compared with its IO-DMRAC counterpart. Furthermore, the boundedness of all signals, including the internal ones ω(t) are assured.

In this paper, new types of M-fractional wave solutions of mathematical physics model named as truncated M-fractional (1+1)-dimensional non-linear simplified Modified Camassa-Holm model are achieved by applying the modified simplest equation (MSE), Sardar sub-equation and generalized Kudryashov techniques. The gained solutions containing dark, bright, periodic and mixed wave solitons. Effect of fractional order derivative is also discussed. Achieved wave solitons are verified by Mathematica tool. Few of the gained wave solitons are also described through 2-dimensional, 3-dimensional and contour graphs through Mathematica tool. The gained solutions are helpful for the further development of concerned model. Finally, these techniques are simple, fruitful and effective to deal with non-linear FPDEs.

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

This paper deals with the analysis of the temperatures in a group of 36 African countries. By looking at the maximum, minimum and the range (the difference between the maximum and the minimum) and using a long memory model based on fractional integration and cointegration, we first show that all series display a long memory pattern, with a significant positive time trend in 29 countries for the maximum temperatures and in 33 for the minimum ones. Looking at the range, the estimated value for the order of integration is smaller than the one based on maximum or minimum temperatures in 17 countries. Performing fractional cointegration tests between the maximum and minimum temperatures, our results indicate that the two series cointegrate in the classical sense (i.e., with a short memory equilibrium relationship) in a group of 11 countries, and there is another group of eight countries displaying cointegration in a fractional sense. The remaining 17 countries with no evidence of cointegration are therefore at the very high risk of climate change due to the absence of long-term comovement in their maximum and minimum temperatures. Findings in this paper are of tremendous interpretations and relevance for the analysis and climate projections in Africa.

Various factors, such as uncertainty of component parameters and uncertainty of sensor meas-urement values, contribute to the difficulty of fault detection in the landing gear retrac-tion/extension hydraulic system. In this paper, we introduce linear fractional transformation technology and uncertainty analysis theory for the construction of the diagnostic bond graph of the landing gear retraction/extension hydraulic system. In this way, interval analytical redundancy relations and fault signature matrix can be derived. Using the fault signature matrix, existing faults of the system can be preliminarily detected and isolated. Additionally, interval analytical re-dundancy relations can be used to detect system faults in detail, and cases analysis can be carried out to determine if the actuator is externally or internally leaky, and if the landing gear selector valve is reversing stuck. Compared to the traditional analytical redundancy relations, this method takes into account the negative factors of uncertainty; and compared to the traditional absolute diagnostic threshold, the interval diagnostic threshold is more accurate and sensitive.

In this paper, the new optical wave solutions of truncated M-fractional perturbed Kundu-Eckhaus model with full non-linearity are obtained by utilizing the expa function technique and modified extended tanh expansion function technique. The solutions are in the form of dark soliton, bright soliton, singular solitons and other form of solutions. The gained solutions are helpful for the further development of concerned model. The obtained results have also been presented graphically in both two-dimensional and three-dimensional formats to discuss the dynamical features as well as the parametric dependence of the constructed solutions. The study offers a highly spectacular and acceptable techniques to combine various intriguing wave demonstrations for more sophisticated models of the modern day.

TThis paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann-Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts.

In this paper we give approximations to the Mittag-Leffler functions in terms of elementary functions using different methods. This allowed us to establish a practical method we called integerization principle. This principle states that many fractional nonlinear oscillators may be solved by means of the solution to some integer-order Duffing oscillator equation. The accuracy of the obtained results is illsutrated in concrete examples. Formulas for estimating the errors in the approximations are also provided.

The present work is concerned with the study of the generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of Langevin equation depend critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation. By further demanding time reversibility of the drift the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker-Plank equation of the probability density of the differential system, which can be readily estimated from Monte Carlo simulations of the random paths. Based on the Fokker-Plank formalism a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase shift diffusion equation (EPSDE) method and an observing the signal intensity at the origin method. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function type PFG signal attenuation. The methods here provide a clear view of spin evolution under field gradient, and their results will help the analysis of PFG anomalous diffusion.

Natural vegetation restoration can take decades to achieve and requires information on species composition and trend to inform management strategies. Here we demonstrate the potential and portability of drone, dove and Sentinel-2 imagery for species level classification and trajectory monitoring using three study sites with different revegetation histories. Drone imagery (4 cm) was classified using random forests. Dove (3 m) and Sentinel-2 (10 m) images were acquired close to the anniversary of the drone flights and converted into fractional cover maps showing the proportion of exotics, natives and other land covers using ordinary least squares (OLS) and geographically weighted regression (GWR). The time series of fractional cover maps were converted into positive and negative slope and second-order curvature and summarised as one RGB composite to illustrate trend and observe exotic expansion. Drone classification accuracies ranged from 73%-76% when applied to the full plant species list and between 93% and 97% when grouped into natives, exotics, and other land covers. GWR outperformed OLS for all fractional cover mapping and GWR with dove imagery was more accurate than with Sentinel imagery. The combination of very high-resolution drone acquired imagery with the versatility of dove acquired imagery arms land managers with highly accurate species composition and their spatio-temporal dynamics to promote timely intervention.

Complex models have received significant interest in recent years and are being increasingly used to explain the stochastic phenomenon with upward and downward fluctuation such as the stock market. Different from existing semi-variance methods in traditional integer dimension construction for two variables, this paper proposes a simplified multi-factorized fractional dimension derivation with the exact Excel tool algorithm involving the fractional center moment extension to covariance, which is a complex parameter average that is a multi-factorized extension to Pearson covariance. By examining the peaks and troughs of gold price averages, the proposed algorithm provides more insight into revealing underlying stock market trends to see who is the financial market leader during good economic times. The calculation results demonstrate that the complex covariance is able to distinguish subtle differences among stock market performances and gold prices for the same field that the two variable covariance may overlook. We take the London, Tokyo, Shanghai, Toronto and Nasdaq as the representative examples.

Anyon is collective excitation of two dimensional electron gas subjected to strong magnetic field, carrying fractional charges and exotic statistical character beyond fermion and boson. So far, anyons with serial fractional charges only exist in fractional quantum Hall effect. It is still a challenge to find new serial of fractional charges in other physical system and develop an unified mathematical physics theory based on the same root. Here a topological path fusion theory of propagating electrons in magnetic flux lattice is proposed to explore the physical origin of fractional charges based on a generalization of Feynman's path integral theory and Thurston's train track theory. This mathematical physics theory generated the existed serial of fractional charges in fractional quantum Hall effect and predicted new serial of fractional charges. A serial of irrational charges are predicted in one dimensional lattice of magnetic fluxes. Fractionally charged anyons are also generated in two dimensional and three dimensional lattice of train tracks of electric currents, revealing an exact correspondence between knot lattice model and train track model. A new explanation for the modular symmetry of complex Hall conductance and composite fermion in fractional quantum Hall effect is also derived from this topological path fusion theory. Experimental observation of anyon in three dimensions can be realized by constructing three dimensional interlocking magnetic fluxes or mapping magnetic fluxes into forbidden zones in multi-connected space filled by solid state material. A photonic crystal with porous nano-structures is a promising system for detecting fractional charges and paves a new way for topological quantum computation.

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

This paper uses the $\exp(-\Phi(\xi))$-Expansion method to investigate solitons to the M-fractional nonlinear Schrödingers equation with cubic nonlinearity. The results obtained are dark solitons, trigonometric function solutions, hyperbolic solutions and rational solutions. Thus, the constraint relations between the model coefficients and the traveling wave frequency coefficient for the existence of solitons solutions are also derived.

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi univalent function defined in open unit disk connected with a linear q-convolution operator, which are associated with the quasi-subordination. We find coefficients estimate |h_2 |,|h_3 | for functions in these subclasses. Several known and new consequences of these results are also pointed out.

This study investigated whether export performance is correlated with innovation. We explicitly considered five types of innovation: a) process innovation; b) product innovation; c) process and product innovation; d) product innovation new to the firm; and e) product innovation new to the market. We en-larged the findings of R&D-export and R&D-investment relations using a vast sample of cross-European firms. We considered innovation as endogenous and determined by R&D through a process characterized by the role of the regional technological environment. Our analysis enabled us to propose an integrated model incorporating R&D, innovation and export, in a framework of simultaneous equations which consider their mutual correlations. To the best of our knowledge, this work is the first in recent years to empirically compute the relation between R&D and tangible investment at the micro level. Econometric results showed that product and process innovation are positively linked to the export intensity of European manufacturing firms. The results also showed complementarity effects between the two kinds of innovation. The av-erage effect on export intensity from carrying out process innovation is larger than that found for product innovation, except in cases where the product is new to the market. When both types of innovation have been carried out, a larger effect results than that found for product or process innovation alone. Furthermore, the average marginal effect on export intensity from innovations of any kind among those studied here is highly positive and significant.

Brazil, Russia, China, India, and the Republic of South Africa (BRICS) represent developing economies facing different energy and economic development challenges. The current study aims to forecast energy consumption in BRICS at aggregate and disaggregate levels using the annual time series data set from 1992 to 2019 and to compare results obtained from a set of models. The time-series data are from the British Petroleum (BP-2019) Statistical Review of World Energy. The forecasting methodology bases on a novel Fractional-order Grey Model (FGM) with different order parameters. This study contributes to the literature by comparing the forecasting accuracy and the forecasting ability of the FGM(1,1) with traditional ones, like standard GM(1,1) and ARIMA(1,1,1) models. Also, it illustrates the view of BRICS's nexus of energy consumption at aggregate and disaggregates levels using the latest available data set, which will provide a reliable and broader perspective. The Diebold-Mariano test results confirmed the equal predictive ability of FGM(1,1) for a specific range of order parameters and the ARIMA(1,1,1) model and the usefulness of both approaches for energy consumption efficient forecasting.

For solving the fractional differential equations in computational fluid dynamics (CFD), it’s complicated and difficult by the Laplace and Fourier transforms. Based on the Caputo fractional derivative, the analytical solutions for unsteady unidirectional flows of a generalized Oldroyd-B fluid are deduced by the separation of variables method. Results show that the analytical solutions are given easily, and have good university. For some specific parameter values, the well-known analytical solutions for the generalized second grade fluid, the generalized Upper-Convected Maxwell (UCM) fluid as well as the ordinary Oldroyd-B fluid can be obtained.

In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, we succeeded to determine new conditions that sufficiently guarantee the existence of the solutions. The results are obtained by combining techniques of fractional calculus and fixed point theorem for contraction maps. We also characterize the topological structure of the set of solutions. Finally, we provide a demonstration to address the applicability of the theoretical results.

The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as "Kannan-graph-fuzzy contraction." This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled by nonlinear models. It is anticipated that using Kannan-graph-fuzzy contractions will offer a fresh approach to discussing solution existence in situations where these ideas overlap. This strategy might have a significant impact on a lot of real-world applications, proving how practical multidisciplinary techniques are for tackling issues in the real world originating from technology and engineering. The article includes several illustrated examples based on computer simulations, making the results more explicit.

We establish a new Gagliardo-Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behaviour. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, competing with Hs-subcritical nonlinearities, either of local or non-local nature.

In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value i.e.,for some Hölder continuous function onwith smooth function value satisfying and Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in the usual sense

This paper deals with the analysis of trends in road accidents on major highways in Brazil. Using updated time series techniques, our results indicate that a low degree of long memory was detected in the series with shocks having transitory effects over time. We further find that the number of accidents taking place in Brazil has been reducing over time, though in the presence of negative shocks, the recovery is not going to be immediate due to the long memory nature of the data. Despite the absence of relevant investment relating to infrastructure expansion, it is worth mentioning the consolidation of a nationwide tolled road system in Brazil involving concessions to private administrators, alongside more severe traffic laws that can impose limitations on driving licences.

Single-fluid electrospinning creates nanofibers from molten polymer solutions with active ingredients. The study utilized a combination of a fractional factorial design and a Box-Behnken design to examine crucial factors among a multitude of parameters and to optimize the electrospinning conditions that impact the fiber mats' morphology and entrapment efficiency of the Senna alata leaf extract. The findings indicated that the shellac content had the greatest impact on both fiber diameter and bead formation. The optimum electrospinning conditions were identified as a voltage of 24 kV, a solution feed rate of 0.8 mL/h, and a shellac-extract ratio of 38.5:3.8. These conditions produced nanosized fibers with a diameter of 306 nm, a low bead-to-fiber ratio of 0.29, and an extract entrapment efficiency of 96% within the fibers. The biphasic profile of optimized nanofibers was confirmed through an in vitro release study. This profile consisted of an initial burst release of 88% within the first hour, which was succeeded by a sustained release pattern surpassing 90% for the next 12 h, as predicted by zero-order release kinetics. The optimized nanofibers demonstrated antimicrobial efficacy against diverse pathogens, suggesting promising applications in wound dressings and protective textiles.

This article aims to solve the problem of robust supervision of a reverse osmosis desalination system (RO-DS) with two models, external model and bond graph model. The structure of an industrial system from the point of view of the external model operates according to several modes of operation (degraded and normal). For this external model, we cannot locate the faults since we are talking about a global operation of the system. The possible solution for the development of research was to use the multidisciplinary model called bond graph. This model by its graphic nature and using a unified language makes it possible to model the industrial system element by element from where it helps the user not only to detect the faults but also to locate them when they appear in the system. The results suggest that the use of the bond graph model for alarm 02 is a reverse osmosis error (RO1); these phenomena are readable on the bond graph model and can be quantified by equations. The equation of the model of this reverse osmosis (RO1) is found in the residual equations (r5, r6 and r7), so that these residues will be sensitive to this rupture. The possible solution during research development was to use the bond graph model to supplement information with physical knowledge and to locate faults. This article also describes the operating safety (by minimizing false alarms and non-detections as well as delays in fault detection) of the desalination system by using the bond graph model described in Linear Fractional Transformation (LFT) form for enable it to manage the robust supervision of a desalination system. An approach based on (LFT-BG) is developed to monitor tank leakage (Cu) and reverse osmosis (RO1 and RO2) faults or valve level (V1 and V2) closures and membrane clogging reverse osmosis (Rm1 and Rm2) that can occur in the reverse osmosis desalination system (RO-DS).

In clinical diagnostics and longitudinal studies, the reproducibility of MRI assessments is of high importance in order to detect pathological changes, but developments in MRI hard- and software often outrun extended periods of data acquisition and analysis. This could potentially introduce artefactual changes or mask pathological alterations. However, if and how changes of MRI hardware, scanning protocols or preprocessing software affect complex neuroimaging outcomes from e.g. diffusion weighted imaging (DWI) remains largely understudied. We therefore compared DWI outcomes and artefact severity of 121 healthy participants (age range 19-54 years) who underwent two matched DWI protocols (Siemens product and Center for Magnetic Resonance Research sequence) at two sites (Siemens 3T Magnetom Verio and Skyra^fit). After different preprocessing steps, fractional anisotropy (FA) and mean diffusivity (MD) maps,obtained by tensor fitting, were processed with tract-based spatial statistics (TBSS). Inter-scanner and inter-sequence variability of skeletonised FA values reached up to 5% and differed largely in magnitude and direction across the brain. Skeletonised MD values differed up to 14% between scanners. We here demonstrate that DTI outcome measures strongly depend on imaging site and software, and that these biases vary between brain regions. These regionally inhomogeneous biases may exceed and considerably confound physiological effects such as ageing, highlighting the need to harmonise data acquisition and analysis. Future studies thus need to implement novel strategies to augment neuroimaging data reliability and replicability.

Antimicrobial resistance (AMR) has been a serious threat to human health, and combination therapy is proved to be an economic and effective strategy to fight the resistance. However, the abuse of drug combinations would conversely accelerate the spread of AMR. In our previous work, it had been concluded that the mutant selection indexes (SIs) of one agent against a specific bacterial strain are closely related to the proportions of two agents in a drug combination. To discover probable correlations, predictors and laws for further proposing feasible principles and schemes guiding the AMR-preventing practice, here three aspects were further explored. First, the power function (y=ax^b, a > 0) correlation between the SI (y) of one agent and the ratio value (x) of two agents in a drug combination was further established based on the mathematical and statistical analyses for those experimental data, and two rules a₁ × MIC₁ = a₂ × MIC₂ and b₁ + b₂ = -1 were discovered from both equations of y=a₁x^b1 and y=a₂x^b2 respectively for two agents in drug combinations. Simultaneously, it was found that one agent with larger MPC alone for drug combinations show greater potency for narrowing itself MSW and preventing the resistance. Second, a new concept as mutation-preventing selection index (MPSI) was proposed and used for evaluating the mutation-preventing potency difference of two agents in drug combinations, and the positive correlation between the MPSI and the mutant prevention concentration (MPC) or minimal inhibitory concentration (MIC) was subsequently established. Inspired by this, the significantly positive correlation, contrary to previous reports, between the MIC and the corresponding MPC of antimicrobial agents against pathogenic bacteria was established using one hundred and eighty-one of data pairs reported. These results together of above three aspects indicate that the MPCs in alone and combination are very important indexes for drug combinations to predict the mutation-preventing effects and the trajectories of collateral sensitivity, and while the MPC of an agent can be roughly calculated from its corresponding MIC. Subsequently, the former conclusion was further verified and improved by the antibiotic exposure to forty-three groups designed as different drug concentrations and various proportions. The results further proposed that the C/MPC for the agent with larger proportion in drug combinations can be considered as a predictor and is the key to judge whether the resistance and the collateral sensitivity occur to two agents. Based on these above correlations, laws, and their verification experiments, some principles were proposed, and a diagram of the mutation-preventing effects and the resistant trajectories for drug combinations with different concentrations and ratios of two agents was presented. Simultaneously, the reciprocal of MPC alone (1/MPC), proposed as the stress factors of two agents in drug combinations, together with their SI in combination, is the key to predict the mutation-preventing potency and control the trajectories of collateral sensitivity. Finally, a preliminary scheme for antimicrobial combinations preventing the AMR was further proposed for subsequent improvement research and clinic popularization, based on the above analyses and discussion. Moreover, some similar conclusions were speculated for triple or multiple drug combinations.

The evolution of integrable classical systems leads to conserved quantities and vanishing Poisson brackets. In contrast, such invariants do not exist in the dynamics of non-integrable systems, which include (but are not limited to) deterministic models with long-term chaotic behavior. The object of this review is to briefly survey the mathematical background of nonintegrability and its role in the physics unfolding well-above the Standard Model (SM) scale.

COVID-19 outbreak started in the Chinese city of Wuhan and spread around the globe in months due to its high contagious level. Hence, this disease posed a great challenge to researchers and mathematicians. Numerous mathematical models have been suggested to visualize the spreading speed and trends of this pandemic. In this paper, a comparison of a widely established and accepted method, the Susceptible-Exposed-Infected-Quarantined-Recovered-Death-Insusceptible (SEIQRDP) model with a newly-proposed fractional-order SEIQRDP is studied. Densely populated Countries of Asia (Pakistan, India, and Bangladesh) have been chosen as data sets and both algorithms have been applied to their data. The same comparison technique has also been used on the data of two polar countries, New Zealand and Russia, that validated our findings. Error comparison of both algorithms has been recorded in a tabulated form which shows multi-fold erroneous trends over WHO data of selected countries.

A new type of ordinary differential equation is introduced and discussed, namely, the time-dependent order ordinary differential equations. These equations can be solved via fractional calculus and are mapped into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equations smoothly deforms solutions of the classical integer order ordinary differential equations into one-another, and can generate or remove singularities. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers was also proved.

This work is a top-level summary of several contributions published in the last three decades. It makes the case that complex dynamics of nonlinear systems lies at the heart of foundational physics.

In this article we investigate the generalized Kawahara-KdV system. A new topological approach is applied to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results.

The Sanshui Basin (SSB) is located at the northern continental margin of the South China Sea and characterized by a continental rift basin. The bimodal volcanic rocks in SSB record the early Cenozoic magmatic activity in the South China Block, on the magmatic evolution process of bimodal volcanic rocks are poorly understood. Clinopyroxenes in bimodal volcanic rocks in the SSB provide an opportunity to investigate the magma process during magma ascent. We classified nine types of clinopyroxene phenocrysts according to the different compositions and textures types in cogenetic basalt-trachyandesite-comenditic trachyte, the composition of unzoned clinopyroxene have an evolution sequence of diopside- hedenbergite- aegirine with the decrease of Mg#, and the trace element contents of unzoned clinopyroxenes also increase systematically during magma evolution, indicating that the genesis of clinopyroxene dominated by fractional crystallization in a closed magma system; however, the clinopyroxenes with reverse zoning and multiple zoning record the process of magma mixing and recharge indicating an open magma system. Whilst fractional crystallization is the dominated process, magma mixing, recharge, and crystal settling complicate magma evolution. Thermobarometric calculations show that clinopyroxene phenocrysts in bimodal volcanic rocks of SSB are distributed in the whole crust during magma ascent. We have established a magma plumbing system, which provides a new constrain for the complex magmatic evolution history in the SSB by detailed mineral-scale analysis.

The main idea of this paper is to investigate the exact solutions and dynamic properties in optical nanofibers, which is modeled by space-time fractional perturbed nonlinear schr\"odinger equation involving Kerr law nonlinearity with conformal fractional derivative. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely the Hamiltonian is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the prior estimate of the existence of the soliton and periodic solution is established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for polynomial method.

Backgrounds Linear association has widely been assumed for prediction of aging-related fractional anisotropy (FA) decline in white matter of the brain. While useful for testing significance of the aging effect, it fails to identify a threshold age before and after which the age-FA association changes. Identification of such a threshold is often of clinical interest for timely intervention. Methods We employed a sigmoidal growth function to test a threshold effect in age triggering onset of cerebral decline in 21 white matter tracts, and compared its fitting performance to those of linear, and power regression. The study sample was a normal healthy cohort of 106 participants with ages in mid-life ranging from 18 to 60 years. Results Of the 21 white matter tracts analyzed, the posterior thalamic radiation showed better fit with sigmoidal curve model, compared to a linear or power regression. The estimated threshold age in years (95% confidence interval) were 47.2 (44.1-48.4). Conclusion While available evidence regarding the presence of a specific age threshold for cerebral decline in mid-life based on FA was limited, the posterior thalamic radiation exhibited a threshold age of 47.2. Beyond this age point, we observed a significant change in the FA risk pattern.

To obtain drop-in fuel properties from non-feed biomass, we herein report the catalytic hydrotreatment of microalgae biocrude, produced from hydrothermal liquefaction (HTL) of Spirulina. Our contribution focuses on the effect of temperature, initial H2 pressure, and residence time on the removal of heteroatoms (O and N). In contrast to common hydrotreating experimental protocols at batch scale, we devised a set of two-level factorial experiments and studied the most influential parameters affecting the removal of heteroatoms. It was found that up to 350 °C, the degree of deoxygenation (de-O) is mainly driven by temperature, whereas the degree of denitrogenation (de-N) also relies on initial H2 pressure and temperature-pressure interaction.Based on this, complete deoxygenation was obtained at mild operating conditions (350 °C), reaching a concurrent 47 % denitrogenation. Moreover, three optimized experiments are reported with 100 % removal of oxygen. In addition, the analysis by GC-MS and Sim-Dis gives insight to the fuel quality. The distribution of heteroatom N in lower (<340 °C) and higher (>340 °C) fractional cuts is studied by a fractional distillation unit following ASTM D-1160. Final results show that 63-68 % of nitrogen is concentrated in higher fractional cuts.

Complex modeling has received significant attention in recent years and is increasingly used to explain the statistical phenomenon with increasing and decreasing fluctuations such as the similarity or difference of spike protein charge patterns of coronaviruses. Different from the existing covariance or correlation coefficient methods in traditional integer dimension construction, this study proposes a simplified novel fractional dimension derivation with the exact Excel tool algorithm. It involves the fractional center moment extension to covariance, which ends up a complex covariance coefficient that is better than the Pearson correlation coefficient, in the sense that the nonlinearity relationship can be further depicted. The spike protein sequences of coronaviruses were obtained from the GenBank and GISAID database, including the coronaviruses from pangolin, bat, canine, swine (three variants), feline, tiger, SARS-CoV-1, MERS, and SARS-CoV-2 (including the strains of Wuhan, Beijing, New York, German, and UK variant B.1.1.7) were used as the representative examples in this study. By examining the values above and below the average/mean based on the positive and negative charge patterns of the amino acid residues of the spike proteins from coronaviruses, the proposed algorithm provides deep insights into the nonlinear evolving trends of spike proteins for understanding the viral evolution and identifying the protein characteristics associated with viral fatality. The calculation results demonstrate that the complex covariance coefficient analyzed by this algorithm is capable of distinguishing the subtle nonlinear differences in the spike protein charge patterns with reference to Wuhan strain SARS-CoV-2 for which the Pearson correlation coefficient may overlook. Our analysis reveals the unique convergent (positive correlative) to divergent (negative correlative) domain center positions of each virus. The convergent or conserved region may be critical to the viral stability or viability; while the divergent region is highly variable between coronaviruses suggesting high frequency of mutations in this region. The analyses show that the conserved center region of SARS-CoV-1 spike protein is located at amino acid residues 900, but shifted to the amino acid residues 700 in MERS spike protein, and then to amino acid residues 600 in SARS-COV-2 spike protein, indicating the evolvement of the coronaviruses. Interestingly, the conserved center region of the spike protein in SARS-COV-2 variant B.1.1.7 shifted back to amino acid residues 700, suggesting this variant is more virulent than the original SARS-COV-2 strain. Another important characteristic our study reveals is that the distance between the divergent mean and the maximal divergent point in each of the viruses (MERS>SARS-CoV-1>SARS-CoV-2) is proportional to viral fatality rate. This algorithm may help to understand and analyze the evolving trends and critical characteristics of SARS-COV-2 variants, other coronaviral proteins and viruses.

Recently, Shehata et al. [37] introduced the r+1Rs(B;C; z) matrix function and established some properties. The aim of this study established to devote and derive certain basic properties including analytic properties, recurrence matrix relations, dierential properties, new integral representations, k- Beta transform, Laplace transform, fractional k-Fourier transform, fractional integral properties, the k-Riemann{Liouville and k-Weyl fractional integral and derivative operators an extended version of r+1Rs;k matrix function. We establish its relationships with other well known special matrix functions which have some particular cases in the context of three parametric Mittag-Leer matrix function, k- Konhauser and k-Laguerre matrix polynomials. Finally, some special cases of the established formulas are also discussed.

About half of the transcatheter aortic valve replacement (TAVR) recipients exhibit some degree of coronary artery disease (CAD), and controversial results have been reported regarding the impact of the presence and severity of CAD on clinical outcomes post-TAVR. In addition to coronary angiography, promising data has been recently reported on the use of both cardiac computed tomography angiography and the functional invasive assessment of coronary lesions whether by FFR or iFR in the work-up pre-TAVR. Despite mitigated available data, percutaneous revascularization of significant coronary lesions has been the routine strategy in TAVR candidates with CAD. Also, scarce data exists on the incidence, characteristics and management of coronary events post-TAVR, and increasing interest exist on potential coronary access challenges in patients requiring coronary angiography/intervention post-TAVR. This review provides an updated overview of the knowledge of CAD in TAVR recipients, focusing on its prevalence, clinical impact, pre- and post-procedural evaluation and management.

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank-Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver-Wynn-Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained by the two methods.

Permanent magnet synchronous motor (PMSM) AC servo system shows the characteristics of uncertainty, time-varying and non-linearity, which makes it difficult for traditional PID control to achieve ideal control effect. Fuzzy control has strong adaptability to the problems of parameter variation, non-linearity and model inaccuracy of the controlled object. Vector control strategy is used to study its control principle and the realization of SVPWM. Because the motor object has certain fractional order characteristics, the fuzzy parameter self-tuning PIλ control is chosen as the position regulator of servo motor. It combines the accuracy of fractional order PI control and the adaptability of fuzzy control. A simulation model of PMSM three-closed-loop system is built in MATLAB/Simulink environment. The results show that the control method is effective and can satisfy the trajectory tracking of servo control.