Fractional Order Derivative Research Articles

Optimization of the different controller parameters via OBL approaches based artificial ecosystem optimization involving fitness distance balance guiding mechanism for efficient motor speed regulation of DC motor

This study proposes a new optimization approach, which is called as artificial ecosystem optimization algorithm with fitness-distance balance guiding mechanism by using opposite based learning methods (FDBAEO_OBLs) for the speed regulation of direct current (DC) motor. The performance of the proposed FDBAEO_OBL algorithm is tested in two different experimental studies. In the first experimental study, the proposed approach is tested in the CEC2020 benchmark test functions and the FDBAEO algorithm, which included the best OBL approach, is determined using non-parametric Wilcoxon and Friedman statistical analysis methods. Second, the parameters of proportional integral derivative (PID), tilt integral derivative (TID), proportional integral derivative with filter (PIDF), tilt integral derivative with filter (TIDF), fractional-order proportional integral derivative (FOPID), fractional-order proportional integral derivative with filter (FOPIDF), proportional integral derivative with fractional-order filter (PIDFF) and fractional-order proportional integral derivative with fractional-order filter (FOPIDFF) controller structures to be used in DC motor closed loop speed control are determined with FDBAEO_OBL, and the performances of the controllers are investigated. Integral absolute error (IAE), integral time absolute error (ITAE), integral time squared error (ITSE) and integral squared error (ISE) performance indices are used as the objective function of the operation process in which the control parameters are determined. According to the comparative step response results of the controller structures, the four best controller structures for DC motor speed regulation are determined. The performances of these controllers are examined under different simulation conditions and according to the results obtained, it is seen that the best controller structure is FOPIDFF. The FDBAEO_OBL algorithm, which is used in both benchmark test functions and DC motor speed regulation, shows an effective, durable and superior performance in finding the optimal solution values during the optimization.

Hyperspectral Indices Developed from Fractional-Order Derivative Spectra Improved Estimation of Leaf Chlorophyll Fluorescence Parameters.

Chlorophyll fluorescence (ChlF) parameters offer valuable insights into quantifying energy transfer and allocation at the photosystem level. However, tracking their variation based on reflectance spectral information remains challenging for large-scale remote sensing applications and ecological modeling. Spectral preprocessing methods, such as fractional-order derivatives (FODs), have been demonstrated to have advantages in highlighting spectral features. In this study, we developed and assessed the ability of novel spectral indices derived from FOD spectra and other spectral transformations to retrieve the ChlF parameters of various species and leaf groups. The results obtained showed that the empirical spectral indices were of low reliability in estimating the ChlF parameters. In contrast, the indices developed from low-order FOD spectra demonstrated a significant improvement in estimation. Furthermore, the incorporation of species specificity enhanced the tracking of the non-photochemical quenching (NPQ) of sunlit leaves (R2 = 0.61, r = 0.79, RMSE = 0.15, MAE = 0.13), the fraction of PSII open centers (qL) of shaded leaves (R2 = 0.50, r = 0.71, RMSE = 0.09, MAE = 0.08), and the fluorescence quantum yield (ΦF) of shaded leaves (R2 = 0.71, r = 0.85, RMSE = 0.002, MAE = 0.001). Our study demonstrates the potential of FOD spectra in capturing variations in ChlF parameters. Nevertheless, given the complexity and sensitivity of ChlF parameters, it is prudent to exercise caution when utilizing spectral indices for tracking them.

Application of Newton’s polynomial interpolation scheme for variable order fractional derivative with power-law kernel

This paper, offers a new method for simulating variable-order fractional differential operators with numerous types of fractional derivatives, such as the Caputo derivative, the Caputo–Fabrizio derivative, the Atangana–Baleanu fractal and fractional derivative, and the Atangana–Baleanu Caputo derivative via power-law kernels. Modeling chaotical systems and nonlinear fractional differential equations can be accomplished with the utilization of variable-order differential operators. The computational structures are based on the fractional calculus and Newton’s polynomial interpolation. These methods are applied to different variable-order fractional derivatives for Wang–Sun, Rucklidge, and Rikitake systems. We illustrate this novel approach’s significance and effectiveness through numerical examples.

Effects of external magnetic field on the reflection and transmission of thermoelastic coupled waves with consideration of fractional order thermoelasticity

Effects of external magnetic field on the dispersion and attenuation feature as well as the reflection and transmission behavior of thermoelastic coupled waves are studied in the paper. The heat conduction behavior at instant heating is modeled by non-Fourier law with the fractional order derivatives. The effects of external magnetic field are incorporated into the motion equation by consideration of the Lorentz force. The dispersion and attenuation feature of the two longitudinal waves and one transversal wave and their reflection and transmission behaviors at interface are studied. The effects of external magnetic field on the dispersion and attenuation and on the reflection and transmission behavior are discussed based on the theoretic formulation and the numerical examples. It is found that the external magnetic field have influences on three kinds of coupled wave while the fractional order heat conduction only has evident influences on the thermal wave.

Fractional dynamics of entropy generation in unsteady mixed convection of a reacting nanofluid over a slippery permeable plate in Darcy–Forchheimer porous medium

AbstractThis paper presents a theoretical investigation of the inherent irreversibility in unsteady fractional time derivative mixed convection of a reacting nanofluid with heat and mass transfer mechanism over a slippery permeable plate embedded in a Darcy–Forchheimer porous medium. The model fractional partial differential equations are obtained based on conservation laws and numerically solved using the implicit finite difference scheme. The study displays and discusses the effects of various emerging parameters on the overall flow structure, such as velocity profiles, temperature distribution, nanoparticles concentration profiles, skin friction, Nusselt number, Sherwood number, entropy generation rate, and Bejan number. It was found that an increase in dimensionless time and fractional parameters leads to a decrease in both the entropy generation rate and the Bejan number. The study revealed that fractional order derivatives can capture intrinsic memory effects, non‐local behaviour, and anomalous diffusion in the nanofluid flow process. This can ultimately lead to better engineering system design and control.

Exact exploration of time fractional-based magnetized flow of a generalized second grade fluid through an oscillating rectangular duct

To model several engineering and physical models, the approach of the fractional derivative is highly anticipated. As compared to the ordinary derivatives, the fractional derivatives with more flexibility can estimate the data due to the involvement of the fractional-order derivatives. Due to these advantages of the fractional approach, this study communicates with the determination of the fractional-based exact outcomes of an oscillatory rectangular duct problem of a generalized second-grade fluid. The approach of the fractional operator is involved in the relationship of the constitutive equations. For cosine oscillation of the rectangular duct, exact results of the magnetized unsteady flow problem are evaluated through the technique of Laplace transform with double finite Fourier sine transform. This study concludes that the velocity field exhibits escalating behavior relative to the improved fractional parameter. Moreover, the magnetic parameter with increasing values declines the flow field while the accelerating values of the fluid parameter enhance the velocity field.

Sliding mode tracking control of a class of fractional-order nonstrict-feedback nonlinear systems

Since the Leibniz rule for integer-order derivatives of the product of functions, which includes a finite number of terms, is not true for fractional-order (FO) derivatives of that, all sliding mode control (SMC) methods introduced in the literature involved a very limited class of FO nonlinear systems. This article presents a solution for the unsolved problem of SMC of a class of FO nonstrict-feedback nonlinear systems with uncertainties. Using the Leibniz rule for the FO derivative of the product of two functions, which includes an infinite number of terms, it is shown that only one of these terms is needed to design a SMC law. Using this point, an algorithm is given to design the controller for reference tracking, that significantly reduces the number of design parameters, compared to the literature. Then, it is proved that the algorithm has a closed-form solution which presents a straightforward tool to the designer to obtain the controller. The solution is applicable to the systems with a mixture of integer-order and FO dynamics. Stability and finite-time convergence of the offered control law are also demonstrated. In the end, the availability of the suggested SMC is illustrated through a numerical example arising from a real system.

Analytical Scheme for Time Fractional Kawahara and Modified Kawahara Problems in Shallow Water Waves

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems.

Enhancing Dynamic Performance of Islanded Microgrids by Fractional-Order Derivative Droop

Enhancing Dynamic Performance of Islanded Microgrids by Fractional-Order Derivative Droop

Bernstein Operational Matrix for Solving Variable Order Fractional Differential Equations with Generalized Caputo-Type

In this research, we present novel generalized derivative founded on the newly constructed (NGCFVODs) novel generalized Caputo Fractional variable order derivatives. Utilizing these operators, a numerical methodology has been formulated to resolve the Fractional Variable Order Differential Equations (FVODEs). We estimate the solution for (FVODEs) by employing bernstein polynomials as foundational vectors. We have further expanded the derivative operational matrix of bernstein Polynomials (bPs) to generalized derivative an operational matrix in sense of (NGCFVODs). The efficiency of developed numerical methodology is tested by a taking a various test examples. We also a compare results of our suggested approach with the methodologies existing in academic papers. In this study the Fractional variable order differential operator of new generalized Caputo was described by three categories: (i) various value in ρ and Fractional variable order a parameter, (ii) various value in a fractional parameters while Fractional variable order and ρ parameter are fixed, and (iii) various value in Fractional variable order parameter controlling fractional and ρ parameter.

A Factory of Fractional Derivatives

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Quasi-synchronisation of conformable fractional-order complex dynamical networks with mixed delays

This paper investigates the exponential quasi-synchronisation of conformable fractional-order complex dynamical networks with mixed delays under periodically intermittent pinning control. First, by the definition of the conformable fractional-order derivative and fractional-order comparison principle, two generalised conformable fractional-order Halanay inequalities are established. Then, by these two inequalities, Lyapunov method and periodically intermittent controller, criteria for realising exponential quasi-synchronisation for the researched network are obtained. Finally, numerical examples are given to illustrate the feasibility of the derived results.

A simple approach for studying stability properties of an SEIRS epidemic model

Abstract In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.

Power and Mittag–Leffler laws for examining the dynamics of fractional unemployment model: A comparative analysis

Unemployment is a major problem worldwide and is one of the key factors determining a nation’s economic status. The issue of unemployment is made more difficult globally by the ongoing rise in labor force participation and the scarcity of job positions. In this work, we study the unemployment model with two distinct fractional-order derivatives: the Caputo operator and the Atangana–Baleanu operator in the sense of Caputo (ABC). These derivatives under consideration are the operators widely utilized in modeling real-world phenomena in fractional dynamics. The existence and uniqueness of the solutions to the fractional model under consideration are ascertained using the fixed-point theory. The Hyers-Ulam analysis is employed to determine stability. For the numerical results, we present an Adams-type predictor–corrector (PC) technique for Caputo derivative and an extended Adams Bashforth (ABM) method for Atangana–Baleanu derivative. The outcomes achieved with the Atangana–Baleanu–Caputo and Caputo derivatives are identical to those of the regular case when fractional order ν=1.00. However, the results obtained change slightly as fractional order assumes values smaller than one, and this variation becomes most noticeable when the fractional order ν<0.72. This is because of the fractional derivative definitions’ underlying kernel. It is shown that the Mittag–Leffler kernel derivative provides better results for smaller fractional orders.

Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model

Abstract A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $\alpha \in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960\le \tau _{k}/\tau _{k-1}&amp;lt;r^{*}(\alpha )$, where $\tau _{k}$ is the $k$th time-step size and $r^{*}(\alpha )\ge 4.660$ for $\alpha \in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.

A fractional-order generalized Richards growth model and its implementation to COVID-19 data

In this article, we include the memory effect in the generalized Richards model (GRM), which is proposed in the form of a fractional-order GRM. The fractional-order GRM is developed by replacing the first-order derivative in the GRM with a fractional-order derivative and then taking care of model homogeneity of dimension. We consider two fractional-order differential operators: Caputo and Atangana–Baleanu in the Caputo sense (ABC). The proposed fractional-order models are then implemented to fit the COVID-19 data in East Java, Indonesia from March 25 until October 31, 2020. The fitting is performed by minimizing the sum of the squared residual between the numerical solutions of each fractional-order model and the daily data of a cumulative number of COVID-19 cases. The numerical solutions of both fractional-order models are determined by the predictor-corrector method. The performance of the two fractional models is measured by two performance metrics: coefficient of determination (R 2) and root mean square error (RMSE). By considering that the order of fractional derivative as an extra degree of freedom, we perform data fitting for several orders of fractional derivative and evaluate the two performance metrics. It is observed that the fractional-order model with the ABC operator generally has the best performance in calibrating and forecasting both the cumulative number of COVID-19 cases and daily new cases of COVID-19.

An innovative fractional-order evolutionary game theoretical study of personal protection, quarantine, and isolation policies for combating epidemic diseases

This study uses imposed control techniques and vaccination game theory to study disease dynamics with transitory or diminishing immunity. Our model uses the ABC fractional-order derivative mechanism to show the effect of non-pharmaceutical interventions such as personal protection or awareness, quarantine, and isolation to simulate the essential control strategies against an infectious disease spread in an infinite and uniformly distributed population. A comprehensive evolutionary game theory study quantified the significant influence of people’s vaccination choices, with government forces participating in vaccination programs to improve obligatory control measures to reduce epidemic spread. This model uses the intervention options described above as a control strategy to reduce disease prevalence in human societies. Again, our simulated results show that a combined control strategy works exquisitely when the disease spreads even faster. A sluggish dissemination rate slows an epidemic outbreak, but modest control techniques can reestablish a disease-free equilibrium. Preventive vaccination regulates the border between the three phases, while personal protection, quarantine, and isolation methods reduce disease transmission in existing places. Thus, successfully combining these three intervention measures reduces epidemic or pandemic size, as represented by line graphs and 3D surface diagrams. For the first time, we use a fractional-order derivate to display the phase-portrayed trajectory graph to show the model’s dynamics if immunity wanes at a specific pace, considering various vaccination cost and effectiveness settings.

Combining the fractional order derivative and machine learning for leaf water content estimation of spring wheat using hyper-spectral indices

Leaf water content (LWC) is a vital indicator of crop growth and development. While visible and near-infrared (VIS–NIR) spectroscopy makes it possible to estimate crop leaf moisture, spectral preprocessing and multiband spectral indices have important significance in the quantitative analysis of LWC. In this work, the fractional order derivative (FOD) was used for leaf spectral processing, and multiband spectral indices were constructed based on the band-optimization algorithm. Eventually, an integrated index, namely, the multiband spectral index (MBSI) and moisture index (MI), is proposed to estimate the LWC in spring wheat around Fu-Kang City, Xinjiang, China. The MBSIs for LWC were calculated from two types of spectral data: raw reflectance (RR) and the spectrum based on FOD. The LWC was estimated by combining machine learning (K-nearest neighbor, KNN; support vector machine, SVM; and artificial neural network, ANN). The results showed that the fractional derivative pretreatment of spectral data enhances the implied information of the spectrum (the maximum correlation coefficient appeared using a 0.8-order differential) and increases the number of sensitive bands, especially in the near-infrared bands (700–1100 nm). The correlations between LWC and the two-band index (RVI1156, 1628 nm), three-band indices (3BI-3(766, 478, 1042 nm), 3BI-4(1129, 1175, 471 nm), 3BI-5(814, 929, 525 nm), 3BI-6(1156, 1214, 802 nm), 3BI-7(929, 851, 446 nm)) based on FOD were higher than that of moisture indices and single-band spectrum, with r of − 0.71**, 0.74**, 0.73**, − 0.72**, 0.75** and − 0.76** for the correlation. The prediction accuracy of the two-band spectral indices (DVI(698, 1274 nm) DVI(698, 1274 nm) DVI(698, 1274 nm)) was higher than that of the moisture spectral index, with R2 of 0.81 and R2 of 0.79 for the calibration and validation, respectively. Due to a large amount of spectral indices, the correlation coefficient method was used to select the characteristic spectral index from full three-band indices. Among twenty seven models, the FWBI-3BI− 0.8 order model performed the best predictive ability (with an R2 of 0.86, RMSE of 2.11%, and RPD of 2.65). These findings confirm that combining spectral index optimization with machine learning is a highly effective method for inverting the leaf water content in spring wheat.

Dust acoustic nonlinearity of nonlinear mode in plasma to compute temporal and spatial results

Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

The sine and cosine diffusive representations for the Caputo fractional derivative

In recent years, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order 0<α<1. An error analysis of the newly presented methods together with some numerical examples is provided at the end.