Fractional Order Research Articles

An Exact Solution of the Fractional Transient Electromagnetic Field Inside an Atmospheric Duct

The objective of this paper is to find the exact solution of the fractional electromagnetic (EM) equation within an atmospheric duct (dielectric medium) of the transient EM field caused by a vertical magnetic dipole and discuss the results in both fractional D-dimensional space and integer space. The novelty of this work is the exact solution of the fractional EM equation in terms of Bessel and Mittage-Leffler functions based on Caputo fractional derivative order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and the Laplacian operator in D -dimensional fractional space. Using the separation of variables method, the resulting magnetic field can be controlled by D=α1+α2+α3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha _1+\\alpha _2+\\alpha _3$$\\end{document} and alpha as spatial and time fractional parameters, respectively. The transient magnetic field behaviour inside the duct is plotted through some of figures depending on fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}. The classical integer results in usual space are recovered from fractional solution at D=α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha =1$$\\end{document}. The main results highlighted are that spatial frequencies grow at a faster rate during EM wave propagation; thus, the amplitude of the propagated wave in integer space is greater than that in fractional space. Increasing the fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} simultaneously increases the amplitude of the EM wave and can be used to control the power and energy of the EM wave.

A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems

A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems

Fractional Wiener chaos: Part 1

AbstractIn this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $${\mathcal {H}}_\alpha (x,y)$$ H α ( x , y ) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, $${\mathcal {H}}_\alpha (W_t,t)$$ H α ( W t , t ) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately.

FRACTIONAL ANALYSIS OF POPULATION DYNAMICAL MODEL IN (3+1)-DIMENSIONS WITH FRACTAL-FRACTIONAL DERIVATIVES

The fractal-fractional derivative is a powerful tool that is generally used for the mathematical analysis of complex and unpredictable structures. In this paper, we study a population dynamical model of (3+1)-dimensional form using the fractal derivative involving fractional order with power law kernel. The proposed scheme is known as the Sumudu Homotopy Transform Method ([Formula: see text]HTM), which depends on the association between the Sumudu Transform ([Formula: see text]T) and the Homotopy Perturbation Method (HPM). The convergence of the derived results is verified by comparing the errors in consecutive iterations with the [Formula: see text]HTM results. We display the behavior of the obtained results in three-dimensional shape across the various orders of fractal and fractional derivatives. We present three numerical tests to validate the accuracy of [Formula: see text]HTM and compare the acquired findings to the exact outcomes of the suggested model. This analysis confirms that the [Formula: see text]HTM results align perfectly with the accurate results. As a result, the [Formula: see text]HTM is widely recognized as a leading computational method for obtaining approximate solutions to various nonlinear complex fractal-fractional problems.

A self-adaptive fractional-order PID controller for the particle velocity regulation in a pneumatic conveying system

This paper presents the methodical development of a state-error-driven self-adaptive fractional-order proportional–integral–derivative (AFOPID) control algorithm to efficiently regulate the velocity of pneumatically conveyed particles at the desired set point and to prevent the blockage of particles in a pipe due to an imbalanced combination of their velocity and corresponding mass flow rate. The proposed fractional control law is constituted by adaptively modulating fractional orders of the integral and differential operators in the control based on the state-error variations in the velocity of solid particles. The particle’s velocity is measured and updated via electrostatic sensors in conjunction with cross-correlation signal processing algorithms. All the other fixed hyper-parameters associated with the proportional–integral–derivative (PID) control scheme are meta-heuristically optimized by using a genetic algorithm. The proposed AFOPID is benchmarked against conventional integer-order PID and the fractional-order proportional–integral–derivative (FOPID) controllers. Experiments are performed on a laboratory-scale test rig to comparatively analyze the aforesaid control schemes, where each controller is examined for three velocity set points and three disturbance levels. The experimental results validate the superior time optimality and robustness of the proposed AFOPID controller against bounded disturbances and abrupt velocity set-point variations by manifesting relatively faster settling time, low overshoots (and undershoots), and smaller steady-state fluctuations.

Synchronization of Chaotic Satellite Systems with Fractional Derivatives Analysis Using Feedback Active Control Techniques

This research article introduces a novel chaotic satellite system based on fractional derivatives. The study explores the characteristics of various fractional derivative satellite systems through detailed phase portrait analysis and computational simulations, employing fractional calculus. We provide illustrations and tabulate the phase portraits of these satellite systems, highlighting the influence of different fractional derivative orders and parameter values. Notably, our findings reveal that chaos can occur even in systems with fewer than three dimensions. To validate our results, we utilize a range of analytical tools, including equilibrium point analysis, dissipative measures, Lyapunov exponents, and bifurcation diagrams. These methods confirm the presence of chaos and offer insights into the system’s dynamic behavior. Additionally, we demonstrate effective control of chaotic dynamics using feedback active control techniques, providing practical solutions for managing chaos in satellite systems.

A new method for converting impulsive Riemann–Liouville fractional order system into the integral equation

A new method for converting impulsive Riemann–Liouville fractional order system into the integral equation

A fractional order SIR model describing hesitancy to the COVID-19 vaccination

This study introduces a SIR (Susceptible-Infectious-Recovered) model using fractional derivatives to assess the population's hesitancy to the COVID-19 vaccination campaign in Portugal. Leveraging the framework developed by Angstmann [1], our approach incorporates fractional derivatives to best describe the nuanced dynamics of the vaccination process. We begin by examining the qualitative properties of the proposed model. To substantiate the inclusion of fractional derivatives, empirical data along with statistical criteria are applied. Numerical simulations are performed to compare both integer and fractional order models. An epidemiological interpretation for the fractional order of the model is provided, in the context of a vaccination campaign.

Analysis of Caputo fractional variable order multi-point initial value problems: existence, uniqueness, and stability

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski’s and Banach’s fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example.

Fractional order modeling of ecological and epidemiological systems: ambiguities and challenges

Fractional order modeling of ecological and epidemiological systems: ambiguities and challenges

A new approach to the directional derivative of fractional order

The fractional derivative approximations offer many approaches to understanding real‐world problems. The conformable fractional derivative operator that is one of the fractional derivative operators has recently attracted a lot of interesting. In this paper, a new approach to the directional derivative obtained with the help of the conformal fractional derivative is presented. Considering this approach, the definition of the fractional partial derivative is reformulated. In addition, a new definition of fractional gradient, fractional curl, and fractional divergence is given, and the properties of these new concepts are examined.

Continuous selections of solution sets of semilinear differential inclusions of fractional order

continuous selections of solution sets of semilinear differential inclusions of fractional order

Influence of Fractional Order on the Behavior of a Normalized Time-Fractional SIR Model

In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized by the property that the sum of its weight function equals one. This unity property is crucial because it helps with evaluating how the fractional order influences the behavior of time-fractional differential equations over time. The normalized time-fractional derivative, with its unity property, provides an intuitive understanding of how fractional orders influence the SIR model’s dynamics and enables systematic exploration of how changes in the fractional order affect the model’s behavior. We numerically investigate how these variations impact the epidemiological dynamics of our normalized time-fractional SIR model and highlight the role of fractional order in improving the accuracy of infectious disease predictions. The appendix provides the program code for the model.

Existence and Uniqueness Solution for a Mathematical Model with Mittag-Leffler Kernel

In this work, we analyse the fractional order West Nile Virus model involving the Atangana-Baleanu derivatives. Existence and uniqueness solutions were obtained by the fixed-point theorem. Another impressive aspect of the work is illustrated by simulations of different fractional orders by calculating the numerical solutions of the mathematical model.

Hamilton energy, competitive modes and ultimate bound estimation of a new 3D chaotic system, and its application in chaos synchronization

Abstract This paper discusses the dynamical behavior of a new 3D chaotic system of integer and fractional order. To get a comprehensive knowledge of the dynamics of the proposed system, we have studied competitive modes and Hamilton energy for different parameter values. In order to get the ultimate bound set for the proposed system, we employed the Lagrange coefficient approach to solve the optimization problem. We have also explored the use of the bound set in synchronization. Furthermore, we have examined the Hamilton energy, time series, bifurcation diagrams, and Lyapunov exponents for the fractional version of the proposed chaotic system. Finally, we looked at the Mittage-Leffler positive invariant sets and global attractive sets by merging the Lyapunov function approach with the Mittage-Leffler function. Numerical simulations have shown the obtained bound sets and other analytical outcomes.

A Galerkin finite element technique with Iweobodo-Mamadu-Njoseh wavelet (IMNW) basis function for the solution of time-fractional advection-diffusion problems

In this paper, the authors used wavelet-based Galerkin finite element technique constructed with Iweobodo-Mamadu-Njoseh wavelet as the basis function, for the numerical solution of time-fractional advection-diffusion equations. To achieve this, the authors used the Iweobodo-Mamadu-Njoseh wavelet as well as fractional calculus, wavelet and wavelet transform, and the Galerkin finite element technique. Also, time and space discretization in relation to the finite element technique were considered, followed by the steps in implementing numerical solutions to TFADE with the new technique. The new technique was considered in seeking numerical solutions of some Caputo type TFADE test problems, and the resulting numerical evidence displayed the effectiveness and accuracy of the method as the results obtained with the new method converged at a good pace to the exact solutions. The results obtained at different fractional order were also compared and the resulting evidence showed that at certain fractional value the convergence behavior displayed slight differences. Every numerical computation was done with the use of MAPLE 18 software.

An optimal nonlinear fractional order controller for passive/active base isolation building equipped with friction-tuned mass dampers

This paper presents an optimal nonlinear fractional-order controller (ONFOC) designed to reduce the seismic responses of tall buildings equipped with a base-isolation (BI) system and friction-tuned mass dampers (FTMDs). The parameters for the BI and FTMD systems, as well as their combinations (BI-FTMD and active BI-FTMD or ABI-FTMD), were optimized separately using a multi-objective quantum-inspired seagull optimization algorithm (MOQSOA). The seismic performances of the BI, FTMD, BI-FTMD, and ABI-FTMD systems for a 15-storey building subjected to two far-field (Loma Prieta and Landers) and two near-fields (Tabas and Northridge) earthquakes were evaluated. The results indicated that structures with BI, FTMD, BI-FTMD, and ABI-FTMD systems outperformed the uncontrolled structure in reducing structural responses during the design earthquakes (Loma Prieta and Tabas). However, under validation earthquakes (Landers and Northridge), the peak acceleration of the building with the FTMD system was worse than that of the uncontrolled structure during the near-field Tabas earthquake. To address this issue, we proposed a combination of the active BI system and the FTMD system. Time history analysis results demonstrated that for the building equipped with the ABI-FTMD system, the peak displacement, peak acceleration, and peak inter-storey drift were reduced by approximately 60%, 64%, and 78%, respectively, as compared to the uncontrolled structure.

Fractional Order Feedforward Odd-Harmonic Repetitive Controller for Grid-Tied Inverters in Microgrids

Fractional Order Feedforward Odd-Harmonic Repetitive Controller for Grid-Tied Inverters in Microgrids

A new mMSA-FOFPID controller for AGC of multi-area power system with multi-type generations

The exceptional growth in the penetration of renewable sources as well as complex and variable operating conditions of load demand in power system may jeopardize its operation without an appropriate automatic generation control (AGC) methodology. Hence, an intelligent resilient fractional order fuzzy PID (FOFPID) controlled AGC system is required. The parameters of controller are tuned utilizing a new modified moth swarm algorithm (mMSA) inspired by the movement of moth towards moon light. At first, the effectiveness of the controller is verified on a nonlinear 5-area thermal power system. The simulation outcomes bring out that suggested controllers provide the best performance over the lately published strategies. In the subsequent step, the methodology is extended to a 5-area system having 10-units of power generations, namely thermal, hydro, wind, diesel, gas turbine with2-units in each area. It is observed that mMSA based FOFPID is more effective related to other approaches. In order to establish the robustness of the controller, the sensitivity examination is executed. Then, experiments are conducted on OPAL-RT based real-time simulation to confirm the feasibility of the method. Finally, mMSA based FOFPID controller is observed superior than the recently published approaches for standard 2-area thermal and IEEE 10 generator 39 bus systems.