Variable-order Model Research Articles

Stripe Noise Removal Algorithm for Infrared Remote Sensing Images Based on Adaptive Weighted Variable Order Model

In the infrared remote sensing imaging system, the output of the detector is uneven, resulting in noticeable stripe noise in the image, which significantly reduces the image quality. Therefore, in response to the problem of establishing a variational model for the image globally and fixing the fractional order of the regularization term in the existing stripe noise removal algorithms based on model optimization, a stripe noise removal algorithm for infrared remote sensing images based on an adaptive weighted variable order model is proposed. In this algorithm, the vertical and approximate components containing stripe noise in the image are first separated through multi-level and multi-scale wavelet transform, and only these two components are processed; secondly, the global sparsity and the gradient sparsity of stripe noise and the variable order gradient sparsity of the information component are constrained by L1 norm, so as to establish the stripe noise removal model. The order of the fractional derivative is adaptively assigned to each pixel of the information component through local variance; then, an adaptive weight operator is introduced, which can assign different weights to the pixels in the variable order derivative of the information component on the basis of the image gradient information. Finally, through the ADMM algorithm, we can obtain the optimal solution of this model. The processing results of simulated and actual data indicate that the proposed algorithm performs well in all indexes and has obvious advantages in removing stripe noise and preserving image details.

A Novel Mathematical Investigation of Synthetic Precipitation Employing Diverse Variable -Order Non-linear Fractional Derivatives

An inadequate rainfall pattern results in water scarcity, groundwater depletion, pollution escalation, and ecological disruption. Addressing these challenges, comprehensive water management strategies will be necessary to ensure sustainability. This paper proposes a nonlinear fractional variable-order model of artificial rain making in the sense of Liouville-Caputo, Caputo-Fabrizio and Atangana –Baleanu fractional derivatives .The existence and uniqueness of fractional solutions has been established and numerical simulations have been carried out to study the influence of different parameters on the process of artificial rainmaking leading to rainfall.

Numerical study of variable order model arising in chemical processes using operational matrix and collocation method

This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.

Study of fractional variable-order lymphatic filariasis infection model

Abstract Variable-order derivatives are the natural extension of ordinary as well as of fractional-order differentiations and integration, respectively. Numerous suggestions for fractional variable-order operators have been made in the literature over time. Therefore, this is the moment to shine a light on the variable-order fractional calculus, due to the fact that it accurately describes the mathematical underpinnings and emphasizing the modeling utility via using contemporary numerical techniques. This study focuses on investigating a fractional variable-order model of lymphatic filariasis infection using with Atangana–Beleanue–Caputo derivative. Our investigations have led to the development of newly refined results, focusing on both qualitative and numerical aspects of analysis. To achieve our research objectives, we employ the fixed point theorems of Banach and Krasnoselskii. These theorems serve as powerful tools, allowing us to establish results regarding the existence of solutions to the model. Additionally, for precise numerical simulations, we employ the fractional Euler’s method, a sophisticated computational technique that allows us to effectively simulate and interpret the results both numerically and graphically. These graphs illustrate distinct variable-orders, providing a comprehensive understanding of the model’s behavior under different conditions. Here, it should be kept in mind that we have select various continuous functions for variable to present our graphical illustration.

Online Data-driven Input Mapping Mixed Platoon Control with Variable Order Model

Online Data-driven Input Mapping Mixed Platoon Control with Variable Order Model

Numerical treatment for a novel crossover mathematical model of the COVID-19 epidemic

<abstract><p>This paper extends a novel piecewise mathematical model of the COVID-19 epidemic using fractional and variable-order differential equations and fractional stochastic derivatives in three intervals of time. The deterministic models are augmented with hybrid fractional order and variable order operators, while the stochastic differential equations incorporate fractional Brownian motion. To probe the behavior of the proposed models, we introduce two numerical techniques: the nonstandard modified Euler Maruyama method for the fractional stochastic model, and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method for the fractional and variable-order deterministic models. Several numerical experiments corroborate the theoretical assertions and demonstrate the efficacy of the proposed approaches.</p></abstract>

Variable–Order Model of Cardiac Fibrillation

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

Numerical behavior of the variable-order fractional Van der Pol oscillator

In this article, we investigate the behavior of a Van der Pol oscillator based on the variable-order Caputo fractional derivatives. After variable-order fractional modeling, we discretize the obtained equations using the Legendre–Gauss–Lobatto points and employ Lagrange interpolating functions. An algebraic system is gained that approximates the variables and their fractional derivatives. Also, an approach is suggested to calculate the differentiation matrix related to the variable-order Caputo fractional derivative. Moreover, an algorithm is presented for solving the variable-order Caputo fractional Van der Pol equation on large time-interval. Numerical simulations are provided to represent the applicability of the suggested method and to see the treatment of variable-order Caputo fractional Van der Pol oscillator.

Existence theory and numerical simulations of variable order model of infectious disease

This study uses variable order differentiation and integration to investigate the disease dynamical model of COVID-19. Here, we update the results of the qualitative and quantitative analysis. We obtain necessary conclusions for the existence theory of the solution to the suggested model in order to satisfy the aforementioned criteria using fixed point theories of Banach and Schauder. Additionally, we simulate the outcomes mathematically and graphically using the Euler modified technique for numerical purposes. There are several graphs provided that relate to various variable ordering. In addition, we compare our simulated results with the real data results also in case of infected class.

Stable Computational Techniques for the Advection–Dispersion Variable Order Model

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) [Formula: see text] ([Formula: see text]) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

Application of Variable-Order Fractional Calculus to the Modeling of Calendar Aging in Lithium-Ion Batteries

Battery aging is one of the key challenges that electrochemical energy storage faces. Models for both cycling and calendar aging are valuable for quantitatively assessing their contribution to overall capacity loss. Since batteries are stored and employed under varying conditions of temperature and state of charge in their real-life operation, the availability of a suitable model to anticipate the outcome of calendar aging in lithium-ion batteries under dynamic conditions is of great interest. In this article, we extend a novel model to predict the capacity loss due to calendar aging by using variable-order fractional calculus. For this purpose, some theoretical difficulties posed by variable-order definitions are discussed and compared by applying them to fit experimental results with a multi-parameter optimization procedure. We show that employing a variable-order model allows for a significant improvement in accuracy and predictive ability with respect to its constant-order counterpart. We conclude that variable-order models constitute an interesting alternative for reproducing complex behavior in dynamical systems, such as aging in lithium-ion batteries.

A Pressure‐Sinkage Model for Deep‐Sea Sediments Based on Variable‐Order Fractional Derivatives

In order to better describe the pressure‐sinkage process of deep‐sea surface sediments, this article proposes a new four‐element model. First, the pressure‐sinkage process was divided into four components according to the change in the deep‐sea sediment sinkage rate, and the time‐dependent mechanical property of deep‐sea sediments was described. Then, a new four‐element pressure‐sinkage model was established by introducing variable‐order fractional derivatives into the modelling idea of classic element combination to describe the full pressure‐sinkage regions of deep‐sea sediments. Furthermore, by comparing with the experimental results from other literature, the proposed model was verified to predict the pressure‐sinkage process of deep‐sea sediments under static and dynamic loads. Finally, through a sensitivity analysis of model parameters, the effects of ground pressure and time on the law and mechanism of deep‐sea sediment pressure sinkage were revealed. Results indicate that the deep‐sea sediment pressure‐sinkage process has obvious nonlinear characteristics and that the sinkage depth increases directly with ground‐specific pressure and time. Through comparative analysis of both experimental and predicted results, the proposed pressure‐sinkage constitutive model can describe the stress‐strain of the full stages. Thus, the variable order model represents an effective method to predict the pressure‐sinkage process of deep‐sea sediments.

Toward random walk-based clustering of variable-order networks

AbstractHigher-order networks aim at improving the classical network representation of trajectories data as memory-less order $1$ Markov models. To do so, locations are associated with different representations or “memory nodes” representing indirect dependencies between visited places as direct relations. One promising area of investigation in this context is variable-order network models as it was suggested by Xu et al. that random walk-based mining tools can be directly applied on such networks. In this paper, we focus on clustering algorithms and show that doing so leads to biases due to the number of nodes representing each location. To address them, we introduce a representation aggregation algorithm that produces smaller yet still accurate network models of the input sequences. We empirically compare the clustering found with multiple network representations of real-world mobility datasets. As our model is limited to a maximum order of $2$ , we discuss further generalizations of our method to higher orders.

Nonlinear nabla variable-order fractional discrete systems: Asymptotic stability and application to neural networks

Variable-order fractional discrete models are dynamical systems described by non-integer order difference equations where the fractional order changes over discrete-time. This paper makes a contribution to the topic by presenting two nonlinear nabla variable-order models and by rigorously proving their asymptotic stability. In particular, some novel theorems are illustrated, regarding the asymptotic stability of both nonlinear nabla variable-order systems and nonlinear nabla variable-order neural networks. Finally, numerical simulations of discrete systems where the fractional order varies with nonlinear law are carried out, with the aim to show the effectiveness of the conceived theoretical approach.

A Fractional Variable Order Model of COVID-19 Pandemic

A Fractional Variable Order Model of COVID-19 Pandemic

Numerical treatments for the optimal control of two types variable-order COVID-19 model

In this paper, a novel variable-order COVID-19 model with modified parameters is presented. The variable-order fractional derivatives are defined in the Caputo sense. Two types of variable order Caputo definitions are presented here. The basic reproduction number of the model is derived. Properties of the proposed model are studied analytically and numerically. The suggested optimal control model is studied using two numerical methods. These methods are non-standard generalized fourth-order Runge–Kutta method and the non-standard generalized fifth-order Runge–Kutta technique. Furthermore, the stability of the proposed methods are studied. To demonstrate the methodologies’ simplicity and effectiveness, numerical test examples and comparisons with real data for Egypt and Italy are shown.

Modeling and adaptive identification of arterial behavior; a variable order approach

Cardiovascular diseases are the most globally prevailing death cause. Identifying and modeling the human vascular system can facilitate the improvement of its function and the design of novel treatment techniques. In this paper, a non-integer variable-order Windkessel model is used to simulate the flow and pressure in a blood vessel. To do so, lumped modeling method has been utilized with the approach of the two-element Windkessel model. Innovatively, the capacitor deploys a variable-order model. Next, adaptive identification is proposed to estimate model order and parameters. Then, its convergence is proven. Finally, the model is implemented on a real dataset and the results demonstrate a promising accuracy.

On the Fractional Variable Order Thermostat Model: Existence Theory on Cones via Piece-Wise Constant Functions

In the current manuscript, we intend to investigate the existence, uniqueness, and the stability of positive solution in relation to a fractional version of variable order thermostat model equipped with nonlocal boundary values in the Caputo sense. In fact, we will get help from the constant piece-wise functions for transforming our variable order model into an auxiliary standard model of thermostat. By Guo-Krasnoselskii’s fixed point theorem on cones, we derive the required conditions ensuring the existence property for positive solutions. An example is illustrated to examine the validity of the observed results.

A mathematical model for SARS-CoV-2 in variable-order fractional derivative.

A new coronavirus mathematical with hospitalization is considered with the consideration of the real cases from March 06, 2021 till the end of April 30, 2021. The essential mathematical results for the model are presented. We show the model stability when {mathcal {R}}_0<1 in the absence of infection. We show that the system is stable locally asymptotically when {mathcal {R}}_0<1 at infection free state. We also show that the system is globally asymptotically stable in the disease absence when {mathcal {R}}_0<1. Data have been used to fit accurately to the model and found the estimated basic reproduction number to be {mathcal {R}}_0= 1.2036. Some graphical results for the effective parameters are drawn for the disease elimination. In addition, a variable-order model is introduced, and so as to handle the outbreak effectively and efficiently, a genetic algorithm is used to produce high-quality control. Numerical simulations clearly show that decision-makers may develop helpful and practical strategies to manage future waves by implementing optimum policies.