Grunwald Letnikov Research Articles

A numerical scheme for time-fractional Allen–Cahn equation with application in phase separation

The present study proposes a numerical method to solve the time-fractional Allen–Cahn equation. The Grunwald–Letnikov formula is employed for discretizing the time-fractional derivative, whereas the finite difference scheme is used for spatial approximation. Theoretical analysis indicates that the proposed method is stable in the discrete L 2 -norm and the numerical results satisfy the energy dissipation property. Numerical simulations are conducted for validation of the proposed method. It has been observed that the solution profile approaches equilibrium for various fractional-order values in the range of ( 0 , 1 ) . Moreover, the fractional order values have a significant effect on the solution stabilization rate which is faster for larger values and slower for the smaller values.

A novel methodology for anomaly detection in smart home networks via Fractional Stochastic Gradient Descent

With the rising integration of IoT devices within smart home environments, securing these interconnected systems against unauthorized access and cyber threats has become increasingly critical. This study introduces a novel methodology utilizing an advanced Artificial Neural Network (ANN) frame-work to enhance attack and anomaly detection capabilities within smart home networks. The objective is to develop a robust model that outperforms traditional detection methods and provides high accuracy and low false positive rates in identifying potential security threats. The research employs a pioneering approach to train the ANN by incorporating a Fractional Stochastic Gradient Descent optimizer grounded in Grunwald–Letnikov fractional calculus. This method was chosen for its potential to refine learning processes and improve detection accuracy over standard optimizers. The evaluation of the model was performed using the DS2OS traffic traces dataset, applying precision, sensitivity (recall), and specificity metrics to assess performance. The proposed model demonstrated exceptional performance with an accuracy of 0.9951. It significantly surpassed traditional methods like Logistic Regression and Support Vector Machines. The precision achieved was high, indicating a low rate of false positives, while the sensitivity and specificity values underscore the model’s ability to identify both typical and unconventional behaviours within the network accurately. This study introduces a robust and efficient ANN-based methodology for enhancing security in smart home IoT networks. Using a fractional stochastic gradient descent optimizer has proven effective in improving the model’s accuracy and reliability in detecting anomalies and attacks. The findings suggest significant implications for the future of IoT security, highlighting the potential for broader applications of fractional calculus in machine learning to enhance cybersecurity measures in various domains.

A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes

We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α∈(0,1) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. cDtαu(z,t)−Δu=f(z,t)inz∈Ω,t∈(0,T],where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method’s well-posedness, that is the approximate solution’s existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L2-norm and H1-norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method.

Battery lumped fractional‐order hysteresis thermoelectric coupling model for state of charge estimation adaptive to time‐varying core temperature conditions

AbstractAs electric vehicles become more common, there is increasing concern regarding their battery reliability and safety. The estimation accuracy is strongly correlated with the performance of the battery model. The lumped fractional‐order hysteresis thermoelectric coupling model (LFHTCM), considering temperature and hysteresis effects, is established in this paper. Firstly, the fractional‐order hysteresis sub‐model is established based on the Grunwald‐Letnikov (G‐L) fractional calculus principle and the recursive model of hysteresis voltage. Then, the thermal sub‐model is established by integrating the Bernardi heating mechanism and the heat transfer model. The fractional‐order sub‐model provides voltage to the thermal sub‐model for heat generation calculation. The thermal sub‐model provides temperature to the fractional‐order model for parameter correction. Finally, the fractional‐order unscented Kalman filtering (FOUKF) algorithm was developed to estimate the state of charge (SOC) of the battery. Experimental results confirmed the effectiveness of the model, with its estimation enhancing the utilization, operational optimization, and calibration of batteries in practical applications.

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

Discrete sliding mode control based on a new fractional order power reaching law

This paper presents a robust controller design for a discrete linear system, dealing with internal and external disturbances. The main focus is to reduce chattering, minimize undesirable oscillations, and enhance robustness against disturbances using a new discrete sliding mode control. The novel approach involves replacing the classical Gao’s reaching law with a fractional order reaching law based on the Grunwald–Letnikov definition, leading to advantages like a simplified algorithm and improved performance against disturbances. The performances of the suggested method are evaluated through two simulation examples.

Gronwall type inequality on generalized fractional conformable integral operators

Abstract In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 {0<\alpha\leq{1}} (Riemann approach), yielding Hadamard fractional integrals when α = 0 {\alpha=0} . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for generalized fractional operators are established, as well as solutions to initial value problems in several new weighted spaces of functions.

Improved fractional‐order hysteresis‐equivalent circuit modeling for the online adaptive high‐precision state of charge prediction of urban‐electric‐bus lithium‐ion batteries

SummaryAccurate state of charge (SOC) estimation is based on a precise battery model and is the focus of the battery management system (BMS). First, based on the second‐order RC equivalent circuit model and Grunwald–Letnikov (G‐L) definition, the high‐precision fractional‐order hysteresis‐equivalent circuit model (FH‐ECM) is established considering the open‐circuit voltage hysteresis effect. Then, the global parameters of the battery model are identified using a particle swarm algorithm optimized by the genetic algorithm (GA‐PSO). Third, a fractional‐order adaptive unscented Kalman filter (FOAUKF) algorithm is derived to estimate the SOC of lithium‐ion batteries. Finally, the feasibility of the model and algorithm is verified under complex working conditions. Under the dynamic stress test (DST) condition, the accuracy of model terminal voltage has been improved by 37.83%, and the error of SOC estimation has been reduced by 11.28%. Under Beijing bus dynamic stress test (BBDST) condition, the model terminal voltage accuracy has been improved by 51.44%, and the SOC estimation error has been reduced by 35.71%. The experimental results fully confirm the accuracy of the fractional‐order hysteresis‐equivalent circuit modeling method.

Oxygen excess ratio control of PEM fuel cell: fractional order modeling and fractional filter IMC-PID control

Abstract Concerning global warming, an energy-efficient power source must produce low or no pollutant emissions and provide an unlimited fuel supply. Proton Exchange Membrane fuel cell (PEMFC) is an electrochemical device that transforms chemical energy into electrical energy. The performance and durability of PEM fuel cells are affected by voltage reversals and fuel starvation. Oxygen Excess Ratio (OER) is a crucial factor in controlling the fuel starvation of the PEMFC system. First, this work identified the PEMFC as an integer order and fractional-order first order plus time delay models using the predictor error method and Grunwald–Letnikov simulation method based on a trust-region-reflect algorithm, respectively. Fractional order models more accurately represented the PEM fuel cell system dynamics. Then, robust fractional filters cascaded with PID controllers based on the Internal Model Control scheme (IMC) are designed for identified integer and fractional order models to regulate the OER by compressor voltage manipulation. The genetic Algorithm (GA) optimization technique is used to find the optimal fractional filter tuning parameters. The proposed controller’s performance regarding Integral Absolute Error (IAE) and Total Variance (TV) is analyzed. Furthermore, the robustness of a perturbed plant and fragility with perturbed controllers are elucidated. The results show that a fractional filter cascaded with fractional order PID controller improves the performance compared to a fractional filter cascaded with integer order PID controllers.

SIRS epidemic modeling using fractional-ordered differential equations: Role of fear effect

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases.

Stability and numerical analysis via non-standard finite difference scheme of a nonlinear classical and fractional order model

In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever R0<1 and when R0≤1 at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald–Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population’s risk of infection can be reduced by adhering to the WHO’s suggestions, which include keeping social distances, wearing facemasks, washing one’s hands, avoiding crowds, etc.

Intelligent neuro-computing to analyze the awareness programs of fractional epidemic system outbreaks

A new model of fractional for the influence of consciousness initiatives on epidemic incidences (ICIEI) is described in this research. The fractional Caputo operator (FCO) is used to expand this system to the typical awareness program model (ICIEI-FCO). Analytically and quantitatively, the properties of the new system are investigated. This article presents an artificial neural network-based fractional model (ANN-FM) for estimating the efficacy of epidemic outbreak awareness initiatives. The reliability, resolution, durability, and robustness of the proposed model are examined using the suggested ANN-BLM approach for five distinct situations. The population of the model is created by utilizing the power of the explicit Runge-Kutta numerical approach, and it is represented by a system of nonlinear ordinary differential equations. The Grunwald–Letnikov (GL) technique is used to numerically evaluate the modeled differential system of the physical issue for multiple scenarios to anticipate numerical data, and these results are utilized as a reference dataset of the networks. The data needed to answer the fractional model's questions on how awareness campaigns affect epidemic outbreaks is broken down as follows: training takes up 80% of the time, testing 10%, and authorization 10%. There are two aspects to the strategy: First, the fundamental ANN-BLM operator performances are displayed. In the meantime, the ANN-BLM execution approach is used to address the fractional-order problem. The GL-mathematical system was used to compare the numerical findings. The TSs, regression, correlation, EHs, and MSE are used to demonstrate the dependability and competency of ANN-BLM as well as their numerical performances in the presented numerical findings, which were developed using ANN-BLM to lower the MSE. The Levenberg-Marquart training (LMT) algorithm is used to optimize network results in terms of mean-square errors, training states graphs, errors of the histogram, recession analysis, auto-correlation and time series responses, which demonstrates the system's accurate and proficient trend acknowledgment. The mean-square error fitness analysis in the ranges of 10-6 to 10-11 validates the authenticity and effectiveness of the designed solver. The suggested AI-based study is expected to pave the way for new, creative approaches to fractional order modeling and analysis of naturally unpredictable dynamic systems.

Discrete fracmemristor-based chaotic map by Grunwald–Letnikov difference and its circuit implementation

The topic of discrete memristor has attracted more and more attention in recent years, but there is not enough discussion about discrete memristor under the fractional-order difference definition, especially for the system model defined by Grunwald–Letnikov. In this paper, a discrete fractional-order memristor model based on Grunwald–Letnikov definition is derived, and two new chaotic maps are designed based on this fractional-order model. Dynamic behaviors are investigated by the volt–ampere curve, bifurcation diagram, Lyapunov exponent spectrum, attractor phase diagram and fuzzy entropy complexity algorithm. Simulation results show that the two fractional-order systems can produce rich dynamic behaviors with the change of fractional order value, and both of them can present hyperchaos. Finally, the proposed systems are implemented in FPGA digital circuit and analog circuit respectively, and the phenomena in the numerical simulation are verified.

Fractional derivative based weighted skip connections for satellite image road segmentation

Segmentation of a road portion from a satellite image is challenging due to its complex background, occlusion, shadows, clouds, and other optical artifacts. One must combine both local and global cues for an accurate and continuous/connected road network extraction. This paper proposes a model using fractional derivative-based weighted skip connections on a densely connected convolutional neural network for road segmentation. Weights corresponding to the skip connections are determined using Grunwald–Letnikov fractional derivative. Fractional derivatives being non-local in nature incorporates memory into the system and thereby combine both local and global features. Experiments have been performed on two open source widely used benchmark databases viz. Massachusetts Road database (MRD) and Ottawa Road database (ORD). Both these datasets represent different road topography and network structure including varying road widths and complexities. Result reveals that the proposed system demonstrated better performance than the other state-of-the-art methods by achieving an F1-score of 0.748 and the mIoU of 0.787 at fractional order 0.4 on the MRD and a mIoU of 0.9062 at fractional order 0.5 on the ORD.

On the Validation of a Fractional Order Model for Pharmacokinetics Using Clinical Data

This study aims to validate the hypothesis that the pharmacokinetics of certain drug regimes are better captured using fractional order differential equations rather than ordinary differential equations. To support this research, two numerical methods, the Grunwald–Letnikov and the L1 approximation, were implemented for the two-compartment model with Michaelis–Menten clearance kinetics for oral and intravenous administration of the drug. The efficacy of the numerical methods is verified through the use of the method of manufactured solutions due to the absence of an analytic solution to the proposed model. The model is derived from a phenomenological process leading to a dimensionally consistent and physically meaningful model. Using clinical data, the model is validated, and it is shown that the optimal model parameters select a fractional order for the clearance dynamic for certain drug regimes. These findings support the hypothesis that fractional differential equations better describe some pharmacokinetics.

Image Enhancement Using Adaptive Fractional Order Filter

Image enhancement is an important preprocessing task as the contrast is low in most of the medical images, Therefore, enhancement becomes the mandatory process before actual image processing should start. This research article proposes an enhancement of the model-based differential operator for the images in general and Echocardiographic images, the proposed operators are based on Grunwald-Letnikov (G-L), Riemann-Liouville (R-L) and Caputo (Li & Xie), which are the definitions of fractional order calculus. In this fractional-order, differentiation is well focused on the enhancement of echocardiographic images. This provoked for developing a non-linear filter mask for image enhancement. The designed filter is simple and effective in terms of improving the contrast of the input low contrast images and preserving the textural features, particularly in smooth areas. The novelty of the proposed method involves a procedure of partitioning the image into homogenous regions, details, and edges. Thereafter, a fractional differential mask is appropriately chosen adaptively for enhancing the partitioned pixels present in the image. It is also incorporated into the Hessian matrix with is a second-order derivative for every pixel and the parameters such as average gradient and entropy are used for qualitative analysis. The wide range of existing state-of-the-art techniques such as fixed order fractional differential filter for enhancement, histogram equalization, integer-order differential methods have been used. The proposed algorithm resulted in the enhancement of the input images with an increased value of average gradient as well as entropy in comparison to the previous methods. The values obtained are very close (almost equal to 99.9%) to the original values of the average gradient and entropy of the images. The results of the simulation validate the effectiveness of the proposed algorithm.

Approximating Constant-Q Seismic Wave Propagations in Acoustic and Elastic Media Using a Cole–Cole Model

ABSTRACT Seismic wave propagation in the Earth’s interior inevitably encounters attenuation and dispersion effects, which usually can be represented by a constant-Q model. However, solving the constant-Q wave equations formulated by fractional Laplacians is computationally intensive. Alternatively, the Cole–Cole model provides an optimal description of seismic attenuation. Because of the fractional time derivatives of both stress and strain in the expression, this method exhibits good adaptability and flexibility. In this article, we investigate the performance of the Cole–Cole model to approximate constant-Q behaviors with different fractional orders in acoustic and elastic media. The phase velocity and quality factor are compared to determine an optimal fractional order. After that the Cole–Cole model can be easily represented by the other three parameters (i.e., relaxed modulus, minimum angular frequency, and minimum quality factor), which are similar to the constant-Q theory (reference modulus, reference angular frequency, and frequency-independent quality factor). The first-order viscoacoustic and viscoelastic wave equations are derived to implement seismic wavefield simulations by combining conservation equation and stress–displacement relation. Furthermore, a time-domain algorithm is developed to solve the wave equations based on the Grunwald–Letnikov approximation and finite-difference scheme. Numerical modeling results calculated by our proposed method have good consistencies with the reference solutions from the constant-Q theory, suggesting that a small fractional order can well approximate the constant-Q outputs in a broad frequency band.

Special Fractional-Order Map and Its Realization

Recent works have focused the analysis of chaotic phenomena in fractional discrete memristor. However, most of the papers have been related to simulated results on the system dynamics rather than on their hardware implementations. This work reports the implementation of a new chaotic fractional memristor map with “hidden attractors”. The fractional memristor map is developed based on a memristive map by using the Grunwald–Letnikov difference operator. The fractional memristor map has flexible fixed points depending on a system’s parameters. We study system dynamics for different values of the fractional orders by using bifurcation diagrams, phase portraits, Lyapunov exponents, and the 0–1 test. We see that the fractional map generates rich dynamical behavior, including coexisting hidden dynamics and initial offset boosting.

Numerical simulation for nonlinear space-fractional reaction convection-diffusion equation with its application

In this research, we adopt a fully implicit approach with a weighted shifted Grunwald–Letnikov difference operator to find the numerical solution of the one and two-dimensional nonlinear space fractional convection–diffusion-reaction equation over a bounded domain. We employ the Peacemann-Rachford alternative direction implicit technique for 2D problems, which is computationally efficient and is used to reduce a 2D problem to a 1D problem by alternately applying x and y directions. This type of problem allows us to successfully predict and investigate pollutant distribution in groundwater, which is based on two processes: convection and diffusion. Because computing the numerical solution of the nonlinear system of equations at each time step using the fixed-point iteration technique is computationally expensive, we have linearized our problem. The numerical scheme’s unconditional stability and second-order convergence in both space and time are theoretically investigated. Finally, numerical experiments with several variations of the nonlinear reaction term are performed, confirming the theoretical analysis and demonstrating the effectiveness of the suggested strategy.

Analysis of resonance and bifurcation in a fractional order nonlinear Duffing system

In this paper, resonance and bifurcation of a nonlinear damped fractional‐order Duffing system are studied. The amplitude and phase of the steady‐state response of system are obtained by means of the average method, the stability is analyzed, and then the amplitude‐frequency characteristic curves of the system with different parameters are drawn based on the implicit function equation of amplitude. Grunwald–Letnikov fractional derivative is used to discretize the system numerically, the response curve and phase trajectory of the system under different parameters are obtained; meanwhile, the dynamic behavior is analyzed. The bifurcation and saddle bifurcation behavior of the system is studied through numerical simulation with different parameters.