Fractional Order Derivative Research Articles

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.

Convergence analysis of a fully discrete scheme for diffusion-wave equation forced by tempered fractional Brownian motion

In this paper, we investigate a fully discrete approximation of stochastic diffusion-wave equation driven by additive tempered fractional Gaussian noise. This additive noise exhibits semi-long range dependence. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative and a tempered fractional Brownian motion. The space discretization is achieved via the spectral Galerkin method. The Caputo fractional derivative of order α∈(1,2) is approximated by using the Grünwald–Letnikov formula. Combining Mittag-Leffler function, Laplace transform and z-transform, we establish the error estimates of the full discretization in the sense of mean-squared L2-norm. Numerical experiments are presented to confirm the strong convergence rates.

Wave behaviors for fractional generalized nonlinear Schrödinger equation via Riemann–Hilbert method

This paper aims to study the explicit fractional generalized nonlinear Schrödinger (fGNLS) equation by the Riemann–Hilbert (RH) method and to explore the impact of the order of fractional derivatives ϵ on solitons. Firstly, utilizing the recursion operator of the generalized nonlinear Schrödinger (GNLS) equation, the anomalous dispersion relation is constructed. Secondly, the explicit form of the fGNLS equation is obtained by the anomalous dispersion relation and the completeness. Then, the N-soliton solutions are acquired through RH problems. We found that the energy of the solitons decreases with the increase of the order of fractional derivatives ϵ. Specifically, we demonstrate that the fractional one-soliton solution constitutes a valid solution of the fGNLS equation by the Darboux transform.

Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions.

Analysis of Caputo Fractional-Order Co-Infection COVID-19 and Influenza SEIR Epidemiology by Laplace Adomian Decomposition Method

Around the world, the people are simultaneously susceptible to or infected with several infections. This work aims at the analysis of the dynamics of transmission of two deadly viruses, COVID-19 and Influenza, using a co-infection epidemiological model by applying the Caputo fractional derivative. Fractional differential equations are currently used worldwide to model physical and biological phenomena. Our comprehension of complicated phenomena is improved when fractional-order derivatives are used to model systems with memory effects and long-range interactions. Mathematical depictions of infectious disease dynamics and dissemination across communities are provided by epidemiological models, which are essential resources for understanding and controlling infectious diseases. These models support informed decision making to prevent outbreaks, evaluate intervention measures, and help researchers and policymakers understand how diseases spread. A subclass of epidemiological models called co-infection models focuses on studying the dynamics of several infectious illnesses that occur in the same population at the same time. They are especially useful in situations where people are simultaneously susceptible to or infected with several infections. Co-infection models provide information on the development of effective control techniques, the progression of disease, and the interactions between several pathogens. The qualitative study via stability analysis is discussed at equilibrium, the reproduction number is computed, and the results are simulated using the Laplace Adomian Decomposition Method (LADM) for Fractional Differential Equations. We employ MATLAB R2023a for graphical presentations and numerical simulations.

A novel method for assessing water quality status using MODIS images: A case study of large lakes and reservoirs in China

The changing climate and economic development have exerted significant pressure on the water quality of global lakes and reservoirs (hereinafter referred to as lakes). Existing sporadic in-situ monitoring limits a comprehensive understanding of water quality changes at large spatiotemporal scales. Although remote sensing techniques offer efficient water quality observations, they have primarily focused on monitoring single optical active substances, making it difficult to assess water quality changes caused by chemical indicators such as total phosphorus and total nitrogen. However, changes in chemical indicators within a certain range can cause responses to physical indicators, which are comprehensively reflected in the water-leaving reflectance. Based on this physical mechanism, this study obtained the water quality status categories (from excellent to severe pollution) by calculating the water quality index (WQI) and developed a new remote sensing assessment model of water quality status using moderate resolution imaging spectroradiometer (MODIS) imagery. This model, combining fractional-order derivatives and various-dimensional spectral indices, significantly enhanced the sensitivity of spectral reflectance to water quality states, achieving a maximum correlation of 0.72. To address the applicability of the model at large spatiotemporal scales, this study proposed a classification modeling scheme based on the optimal optical water categories, resulting in a 20.69%–34.48% improvement in model accuracy after classification. Additionally, this study proposed an improved ensemble learning model combining the concepts of Stacking and Bagging, achieving an average accuracy of 82% and enhancing accuracy by 5%–15% compared to the original single model. Subsequently, the model was used for the first time to assess the water quality status of 180 large lakes in China from 2000 to 2022. The results revealed that 76.11% of the lakes exhibited excellent and good water quality, with a spatial distribution pattern showing a “better in the west, worse in the east” pattern. Over the 23-year period, 28.33% and 58.89% of the lakes showed improvement and stability trends in water quality status, with stability and improvement predominating in the western and eastern regions, respectively. The research results provide technical support for the rapid assessment of water quality status and sustainable resource management, highlighting the potential of remote sensing technology in water quality monitoring.

A novel modified Cheetah Optimizer for designing fractional-order PID-LFC placed in multi-interconnected system with renewable generation units

Establishing robust electrical interconnections between nations is a pivotal foundation for significant investments and addressing energy shortfalls in regions grappling with generational challenges. However, developing electrical load disruptions within interconnected systems can lead to substantial variations in frequencies and energy transmission. Load Frequency Control (LFC) is a crucial mechanism to mitigate these disruptions and ensure stable operations in interconnected regions. While meta-heuristics have been employed for LFC design, some techniques face challenges like early convergence and poor accuracy due to a lack of population diversity. In this study, a novel Modified Cheetah Optimizer (mCO) is proposed to optimize the parameters of LFC system, incorporating fractional-order proportional integral derivative (FOPID) controllers within multi-interconnected system with renewable energy integration. The mCO integrates learning-from-experience and random contraction strategies to enhance convergence accuracy and overcome local optima, demonstrating superior efficiency in solving optimization problems. The proposed mCO is evaluated by solving twelve functions from the CEC2022 test suite, showcasing its effectiveness. The optimization problem involves minimizing the Integral Time Absolute Error (ITAE) of the area control error, considering changes in frequencies and exchanged power, with controller parameters λd, kd, ki, kp, and μ to be identified. Two interconnected systems, photovoltaic (PV)-thermal and thermal-wind turbine (WT)-thermal-PV, are assessed under various load disturbances. The mCO is compared with other methods, including Modified Hunger Games Search Optimizer (MHGS), Driving Training-Based Optimizer (DTBO), Grey Wolf Optimizer (GWO), Aquila Optimal Search (AOS), and Cheetah Optimizer (CO). In the case of PV-thermal linked system, the proposed mCO succeeded in mitigating the ITAE by 19.21% compared to the reported MHGS and 8.63% compared to the conventional CO. In the four interconnected systems, the suggested approach reduced the ITAE by 89.21% and 15.26% compared to the reported MHGS and conventional CO, respectively. This confirmed the efficacy of FOPID-LFC, which was designed using the proposed mCO in all examined scenarios.

Higher-ordered hybrid fractional differential equations with fractional boundary conditions: Stability analysis and existence theory

In the present article, the p-Laplacian operator is applied for examining hybrid fractional differential equations (HFDEs). Establishing the existence and uniqueness (EU) results and analyzing the Hyers–Ulam (HU) stability for HFDEs incorporating fractional derivatives of different orders with the p-Laplacian operator are the primary objectives of this research. With an understanding of the Green function, we will transform the provided HFDE into a corresponding integral form of hybrid FDEs for EU results. A fixed point theorem is employed to examine the existence of solution (ES), and the Banach contraction mapping principle technique is employed to determine the uniqueness of solution. The result of the HU stability study is examined using dynamical systems and functional analysis approaches. Additionally, an application is presented to demonstrate the findings.

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the extended (3 + 1)-dimensional Zakharov-Kuznetsov equation, which describes the propagation of ion-acoustic waves in a magnetic environment, is investigated. Due to the exposure of the propagation to unpredictable factors, the stochastic model is assessed including the Brownian process, in addition to including the recent concept of truncated M-fractional derivative. A fractional stochastic transformation is applied to transform the model into an integer-order ordinary differential equation which in turn is equivalent to a conservative Hamiltonian model. Novel solutions, such as hyperbolic, trigonometric, and Jacobian elliptic functions, are established by employing both of the qualitative analysis of dynamical systems and the first integral of the Hamiltonian model. We explore and graphically display the effects of the fractional derivative order and noise intensity on the solutions structures. In the deterministic instance, i.e. in the absence of noise, solitary and cnoidal solutions among other traveling wave solutions of the Zakharov-Kuznetsov equation, are derived. Further, it is found that the curvature of the wave disturbs and the surface turns substantially flat by increasing the value of noise. While the curve in all cases loses its characteristic shape and degenerates into another deterministic shape by changing the fractional derivative order.

Transmission Dynamics of Malware in Networks Using Caputo Fractional Order Derivative

Transmission Dynamics of Malware in Networks Using Caputo Fractional Order Derivative

Fundamental solution of the time-space bi-fractional diffusion equation with a kinetic source term for anomalous transport

The purpose of this paper is to study the fundamental solution of the time-space bi-fractional diffusion equation incorporating an additional kinetic source term in semi-infinite space. The equation is a generalization of the integer-order model ∂tρ(x,t)=∂x2ρ(x,t)-ρ(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _{{t}} {\\rho (x,t)}} = {\\partial ^2_{{x}} {\\rho (x,t)}} - { \\rho (x,t)}$$\\end{document} (also known as the Debye–Falkenhagen equation) by replacing the first-order time derivative with the Caputo fractional derivative of order 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\alpha < 1$$\\end{document}, and the second-order space derivative with the Riesz-Feller fractional derivative of order 0<β<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\beta <2$$\\end{document}. Using the Laplace-Fourier transforms method, it is shown that the parametric solutions are expressed in terms of the Fox’s H-function that we evaluate for different values of 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 β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}.

Existence Results for Tempered-Hilfer Fractional Differential Problems on Hölder Spaces

This paper considers a nonlinear fractional-order boundary value problem HDa,gα1,β,μx(t)+f(t,x(t),HDa,gα2,β,μx(t))=0, for t∈[a,b], α1∈(1,2], α2∈(0,1], β∈[0,1] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and HDa,gα,β,μ represents the tempered-Hilfer fractional derivative of order α&gt;0 with parameter μ∈R+ and type β∈[0,1]. The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice.

Growth and addition in a herding model with fractional orders of derivatives

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order α where 0<α⩽1 . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as nk=p(2p−1)(2−α)p(1−α)+1Γ(α+2−α1−p)Γ(k)Γ(k−1+α+2−α1−p)t for α∈(0,1] , where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for α = 1. The results corresponding to various values of α and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p, thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.