Fractional-order Operators Research Articles

Long time behavior for the two-dimensional magnetohydrodynamic flows in a general domain

By utilizing spectral analysis methods and properties of fractional order operators, some new estimates have been established for the nonlinear terms appearing from the two-dimensional non-stationary magnetohydrodynamic flows in a general domain. Based on these, a series of energy decay properties are given. Furthermore, the large time decay behavior of the solution and that of its gradient have also been addressed in the L∞ and L2 spaces, respectively.

Bilinear optimal control for a fractional diffusive equation

We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order 0<s<1. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first- and second-order optimality conditions.

Hover Flight Improvement of a Quadrotor Unmanned Aerial Vehicle Using PID Controllers with an Integral Effect Based on the Riemann–Liouville Fractional-Order Operator: A Deterministic Approach

The hovering flight of a quadrotor Unmanned Aerial Vehicle (UAV) refers to maintaining the aircraft in a fixed position in the air, without lateral, vertical, or rotational movements, using only the vehicle’s control systems to maintain proper balance in all spatial dimensions. Algorithms and control systems have been developed to continuously adjust motor speeds to counteract deviations from the desired position and achieve effective hovering flight. This paper proposes a set of PID controllers with an integral effect based on the Riemann–Liouville fractional-order approach to improve the hovering flight of a quadrotor UAV. This research innovates by introducing a set of fractional-order PID controllers for UAV hover stability, which offer better adaptability to non-linear dynamics and robustness than traditional PID controllers. Also presented is the development of new performance metrics (MSE, BQC-LR), which allow for more comprehensive control system evaluations. A thorough comparative analysis with conventional control methods demonstrates the superior performance of fractional-order control in real-world simulations. The numerical simulation results show the effectiveness of the proposed Fractional Integral Action PID Controller in the control of UAV hovering flight, while comparative analyses against a classical controller emphasize the benefits of the fractional-order approach in terms of control accuracy.

An existence theory for nonlinear superposition operators of mixed fractional order

An existence theory for nonlinear superposition operators of mixed fractional order

An interior penalty method for a parabolic complementarity problem involving a fractional Black-Scholes operator

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

Numerical Analysis of the Discrete, Fractional Order PID Controller Using FOBD Approximation

This paper proposes a methodology of the numerical testing of the discrete, approximated Fractional Order PID Controller (FOPID). The fractional parts of the controller are approximated using the Fractional Order Backward Difference (FOBD) operator. The goal of the analysis is to find the memory length optimum from point of view both accuracy and duration of computations. To do it new cost functions describing both accuracy and numerical complexity were proposed and applied. Results of tests indicate that the optimum memory length lies between 200 and 400. The proposed approach can be also useful to examine of another discrete implementations of a fractional order operator using FOBD.

On time‐fractional partial differential equations of time‐dependent piecewise constant order

This contribution considers the time‐fractional subdiffusion with a time‐dependent variable‐order fractional operator of order . It is assumed that is a piecewise constant function with a finite number of jumps. A proof technique based on the Fourier method and results from constant‐order fractional subdiffusion equations has been designed. This novel approach results in the well‐posedness of the problem.

Robust fractional-order PID controller assisted by active disturbance rejection control for the first-order plus time-delay systems

Time-delay characteristics of various industrial processes may degrade the stability and dynamic performance of the control systems. Aiming at the problems of the existing methods in dealing with the time delay plant, a modified fractional-order proportional–integral–derivative (FOPID) controller for the first-order plus time-delay (FOPTD) system is developed. Assisted by a modified active disturbance rejection control (ADRC) scheme with increased observer bandwidth, the proposed FOPID controller inherently obtains good robustness to time-delay uncertainties and external disturbances. In addition, taking advantage of the fractional-order operator, the proposed controller can provide larger stability margin over the proportional–integral–derivative (PID) controller. By suitably establishing the relation between ADRC and FOPID controller parameters, the proposed controller can be analytically tuned based on the common design indices. A practical tuning guideline is developed according to frequency-domain characteristic analysis, making the proposed controller more acceptable to industrial application. The performance of the ADRC-based FOPID controller is tested by the control simulation of some typical FOPTD systems and a diesel engine speed regulation system. The efficiency of the ADRC-based FOPID controller is demonstrated by the comparisons with some existing controllers.

Improving Influenza Epidemiological Models under Caputo Fractional-Order Calculus

The Caputo fractional-order differential operator is used in epidemiological models, but its accuracy benefits are typically ignored. We validated the suggested fractional epidemiological seasonal influenza model of the SVEIHR type to demonstrate the Caputo operator’s relevance. We analysed the model using fractional calculus, revealing its basic properties and enhancing our understanding of disease progression. Furthermore, the positivity, bounds, and symmetry of the numerical scheme were examined. Adjusting the Caputo fractional-order parameter α = 0.99 provided the best fit for epidemiological data on infection rates. We compared the suggested model with the Caputo fractional-order system and the integer-order equivalent model. The fractional-order model had lower absolute mean errors, suggesting that it could better represent sickness transmission and development. The results underline the relevance of using the Caputo fractional-order operator to improve epidemiological models’ precision and forecasting. Integrating fractional calculus within the framework of symmetry helps us build more reliable models that improve public health interventions and policies.

On a Nonlocal Problem for a Mixed-Type Equation with a Fractional Order Operator

On a Nonlocal Problem for a Mixed-Type Equation with a Fractional Order Operator

Stability analysis of a fractional-order [formula omitted] epidemic model for the COVID-19 pandemic

To explore the impact of the COVID-19 vaccination rate and immune loss rate on the pandemic, this paper proposes a fractional-order SIV1V2S epidemic model with vaccination ineffectiveness and infection differences. And we compare and analyze the dynamic differences between integer-order and Caputo fractional-order operators. First, we show the non-negativity and boundedness of solutions for the Caputo fractional-order model. Based on the basic reproduction number, we use the fractional-order matrix eigenvalue method and the Routh–Hurwitz criterion to determine the local asymptotic stability of disease-free equilibrium points and endemic equilibrium points, construct Lyapunov functions, and use the fractional-order Barbalat’s Lemma to prove the global asymptotic stability of the two equilibrium points. Second, real data from India were used to fit the model parameters and get the optimal fractional-order parameter, and then sensitivity analysis was performed on the basic reproduction number. Further, through numerical simulation of different fractional-order parameter epidemic models, we verify the global asymptotic stability of the equilibrium point of the fractional-order SIV1V2S COVID-19 model. The comparison shows that the fractional-order model is more closely related to actual virus transmission than the integer-order model. In the case of the SIV1V2S model, which only considers vaccination measures, simultaneously increasing the vaccination rate of susceptible individuals and reducing the rate of immunization loss can effectively control epidemic transmission.

Exact solution for the electrical response of a constant phase element with a series resistance to linear voltage sweep

A resistance Rs in series with a constant phase element (CPE) of frequency-dependent impedance given by the power law function Zc(s)=1/(Cαsα) is commonly used for the analysis of steady-state frequency response data exhibiting non-purely capacitive behavior. This is the case in most (bio)(electro)chemical systems including dielectrics, batteries, supercapacitors, capacitive deionization units, biological tissues and bioelectrodes. Passing to the time domain, the current, voltage and charge of the system are governed by differential equations with non-integer, fractional-order operators. The purpose of this study is to provide the exact analytical expressions for the electrical response of an Rs-CPE model under linear sweep voltammetry with the use of the Laplace transform method. The electrical variables are expressed in terms of special functions regularly encountered with fractional calculus such as the Fox’s H-function and the Mittag-Leffler function, and can be used for modeling non-ideal devices as well as extracting their characteristic parameters.

Frequency Loop-Shaping and IMC-Based Integer-Order Robust PID Controller Design for Fractional-Order Processes with Time-Delay

It is well accepted that fractional-order calculus yields a more accurate description of complex physical processes compared to integer-order one. So, fractional-order modelling is getting a good attention in the process industry. On the other hand, to meet design goals accurately, control designers prefer fractional-order controllers over integer-order ones due to the availability of more parameters. However, the design of such controllers are complex. More often, the design faces the challenge in real-time implementations, and the problem is circumvented by higher order approximations of fractional-order operators. In the presented work, three methods for integer-order Proportional–Integral–Derivative (PID) controller design are proposed based on Frequency Loop-Shaping (FLS) and Internal Model Control (IMC). To prove the efficacy of the presented control law, simulations are conducted and included along with a performance measure based on ISE (Integral of Square Error), ITAE (Integral of Time Absolute Error) and IAE (Integral of Absolute Error).

Temporal second-order difference schemes for the nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay

This paper investigates a nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay. The problem is particularly challenging due to its nonlinear nature, the presence of a time delay, and the incorporation of both fractional diffusion and fractional wave terms, introducing computational complexities for numerical analysis. To address this, we transform the model into a new generalized form involving Riemann–Liouville fractional integrals and a Caputo derivative of order α∈(0,1). Subsequently, a temporal second-order linearized difference scheme is presented to approximate the model, and its unconditional stability is rigorously proven based on discrete Gronwall’s inequality. To reduce computational and storage costs, we extend the discussion to a fast variant of the proposed difference scheme. The sum of exponentials approach is employed to approximate the kernel function in fractional-order operators, leading to fast difference analogs for the Riemann–Liouville fractional integral and the Caputo derivative. A fast variant of the developed direct method is introduced based on these analogs. Numerical results are provided to validate our theoretical findings and assess the accuracy and efficiency of the difference schemes.

Reconfigurable Fractional-Order Operator and Bandwidth Expansion Suitable for PIα Controller

Reconfigurable Fractional-Order Operator and Bandwidth Expansion Suitable for PI^α Controller

Field-Programmable Analog Array Implementation of Neuromorphic Silicon Neurons with Fractional Dynamics

Silicon neurons are bioinspired circuits with the capability to reproduce the modulation through pulse-frequency observed in real neurons. They are of particular interest in closed-loop schemes to encode the control signal into pulses. This paper proposes the analog realization of neuromorphic silicon neurons with fractional dynamics. In particular, the fractional-order (FO) operator is introduced into classical neurons with the intention of reproducing the adaptation that has been observed experimentally in real neurons, which is the variation in the firing frequency even when considering a constant or periodic incoming stimulus. For validation purposes, simulations using a field-programmable analog array (FPAA) are performed to verify the behavior of the circuits.

The fractional soliton solutions of dynamical system arising in plasma physics: The comparative analysis

In light of fractional theory, this paper presents several new effective solitonic formulations for the Langmuir and ion sound wave equations. Prior to this study, no previous research has presented the comparision and obtained the generalized fractional soliton solutions of this kind with power law kernel and Mittag-Leffler kernel. The ion sound and Langmuir wave equations are essential in plasma physics, offering insights into the collective behavior of charged particles in plasmas and enabling diagnostics and control of these complex, ionized gas systems. The two distinct fractional order differential operators are substituted for the traditional order derivative to reshape the examined model. The Atangana-Baleanu non-singular and non-local operator and conformable fractional operator are the fractional-order operators that are used to create the fractional complex system equations for Langmuir waves and ion sound. A constructive approach new auxiliary equation method utilizes to obtain the exact analytical soliton solutions for ion sound and Langmuir wave equation. A wide range of soliton solutions is obtained, including mixed complex solitary shock solutions, singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed singular solutions, exact solutions, mixed periodic solutions, and mixed hyperbolic solutions, dark soliton, bright soliton, trigonometric solutions, periodic results, and hyperbolic results. The solitons solution of the ion sound and Langmuir wave equations lies in their ability to maintain wave stability, their role in modeling wave propagation and nonlinear effects, their potential use as diagnostic tools, and their relevance in wave-particle interactions in plasma physics. The solitons provide a valuable framework for understanding the behavior of waves in plasmas and offer insights into the complex dynamics of these charged particle systems. A graphical comparison analysis of a few solutions is also shown here, taking into account appropriate parametric values through the use of the software package. Moreover, the results of this study have important implications for Hamilton's equations and generalized momentum, where solitons are employed in long-range interactions.

Dengue Transmission Dynamics: A Fractional-Order Approach with Compartmental Modeling

This work presents a quantitative analysis of the transmission dynamics of dengue using the Caputo–Fabrizio fractional-order derivative. It presents an extensive framework for modeling a dengue epidemic, including the various stages of infection and encompassing a wide range of transmission pathways. The proposed model is subjected to a rigorous qualitative study, including the determination of a non-negative solution, the assessment of the basic reproduction number, and an evaluation of local stability. Numerical solutions are obtained using the Newton method. The fractional-order operator, developed using the Caputo–Fabrizio approach, provides a refined perspective on the transmission dynamics of dengue. This study contributes to a deeper understanding of the disease’s transmission mechanisms, considering both fractional-order dynamics and diverse transmission routes, thus offering insights for enhanced disease management and control.

On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces

AbstractThis paper is concerned with a parabolic evolution equation of the form , settled in a smooth bounded domain of , , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the ‐Laplacian for suitable , the “variable‐exponent” ‐Laplacian, or even some fractional order operators. The operator is assumed to be in the form with being measurable in and maximal monotone in . The main results are devoted to proving existence of weak solutions for a wide class of functions that extends the setting considered in previous results related to the variable exponent case where . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called ‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators , ) to which the result can be applied.

Qualitative dynamical study of hybrid system of Pantograph equations with nonlinear p-Laplacian operator in Banach’s space

This study provides a comprehensive exploration of the qualitative analysis of a hybrid system of pantograph equations with fractional order and a p-Laplacian operator. The existence of the solution of the system is explicitly established within the context of Riemann–Liouville’s fractional order operator, employing the Arzelà–Ascoli theorem for validation. The establishment of uniqueness criteria is accomplished by the utilization of the Banach contractive technique. In addition, the examination of solution stability is conducted using the Hyers–Ulam (HU) stability technique. In order to enhance the credibility of our main conclusions, we have included a representative and illustrative example in the concluding section of the study. This work serve to offer a thorough and applicable comprehension of the mathematical framework that has been proposed.