Fractional Order Derivative Research Articles

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

A new chemical networked system: spatial-temporal evolution and control

Abstract This paper constructs a scale-free chemical network based on the Gierer-Meinhardt (GM) system and investigates its Turing instability. We establish a fractional-order single-node GM system with delay and design a fractional-order proportional derivative (PD) control strategy for the issue of bifurcation control. Using delay as bifurcation parameter, the existence of Hopf bifurcation is proven, and control over bifurcation threshold points is achieved through a fractional-order PD control strategy. For the scale-free chemical network based on the GM system, we obtain the condition of how the Turing instability occurs. We derive how the number of edges for the new nodes changes the stability of the network-organized system and investigate the relationship between degrees of nodes and eigenvalues of the network matrix. We give the instability condition caused by diffusion in the network-organized system. Finally, the numerical simulations verify analytical results.

A mathematical modeling of patient-derived lung cancer stem cells with fractional-order derivative

The aim of this article is to help predict the course of lung cancer patients. To make this prediction as close to reality as possible, we used data from lung cancer patients receiving treatment at Erciyes University Hospitals in Kayseri, Turkey. First, we developed a mathematical model considering the cells in the microenvironment of lung cancer tumors with the assistance of Caputo fractional derivatives. Subsequently, we identified the equilibrium points of the proposed mathematical model and examined the coexistence equilibrium point. In addition, we demonstrated the existence and uniqueness of the solutions through the fixed-point theorem. We also investigated the positivity and boundedness of the model’s solutions to show whether they are biologically meaningful. Using laboratory experimental results from cancer stem cells isolated from resected tumor tissues of lung cancer patients, we determined the most biologically realistic parameter values through the least squares curve fitting approach. Then, using these parameter values, we performed numerical simulations with the Adams-Bashforth-Moulton predictor-corrector method to validate the theoretical results. We considered different values of fractional derivatives to investigate how the model is affected by fractional derivatives. As a result, we obtained the dynamics and expectations of lung cancer and made predictions specific to individual patients. In our simulations based on the parameter values obtained from actual patient data, it has been observed that after a certain period, both tumor cells and cancer stem cells have been eliminated. Consequently, an increase in normal tissue cells and immune cells has been observed. This implies that the patient in question, and similar behaving patients, will recover and overcome cancer. The findings from this study provide insights into the dynamics and prognosis of lung cancer, opening up the possibility for more personalized and effective approaches to treatment.

Comprehending symmetry in epidemiology: A review of analytical methods and insights from models of COVID-19, Ebola, Dengue, and Monkeypox.

The challenge of developing comprehensive mathematical models for guiding public health initiatives in disease control is varied. Creating complex models is essential to understanding the mechanics of the spread of infectious diseases. We reviewed papers that synthesized various mathematical models and analytical methods applied in epidemiological studies with a focus on infectious diseases such as Severe Acute Respiratory Syndrome Coronavirus-2, Ebola, Dengue, and Monkeypox. We address past shortcomings, including difficulties in simulating population growth, treatment efficacy and data collection dependability. We recently came up with highly specific and cost-effective diagnostic techniques for early virus detection. This research includes stability analysis, geographical modeling, fractional calculus, new techniques, and validated solvers such as validating solver for parametric ordinary differential equation. The study examines the consequences of different models, equilibrium points, and stability through a thorough qualitative analysis, highlighting the reliability of fractional order derivatives in representing the dynamics of infectious diseases. Unlike standard integer-order approaches, fractional calculus captures the memory and hereditary aspects of disease processes, resulting in a more complex and realistic representation of disease dynamics. This study underlines the impact of public health measures and the critical importance of spatial modeling in detecting transmission zones and informing targeted interventions. The results highlight the need for ongoing financing for research, especially beyond the coronavirus, and address the difficulties in converting analytically complicated findings into practical public health recommendations. Overall, this review emphasizes that further research and innovation in these areas are crucial for addressing ongoing and future public health challenges.

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

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.

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.

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.

Efficient results on fractional Langevin-Sturm-Liouville problem via generalized Caputo-Atangana-Baleanu derivatives.

In this paper, we investigate the generalized Langevin-Sturm-Liouville differential problems involving Caputo-Atangana-Baleanu fractional derivatives of higher orders with respect to another positive, increasing function denoted by ρ. The fixed point theorems in the framework of Kransnoselskii and Banach are utilized to discuss the existence and uniqueness of the results. In addition, the stability criteria of Ulam-Hyers, generalize Ulam-Hyers, Ulam-Hyers-Rassias, and generalize Ulam-Hyers-Rassias are investigated by non-linear analysis besides fractional calculus. Finally, illustrative examples are reinforced by tables and graphics to describe the main achievements.

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.

Random fractional kinematic wave equations of overland flow: The HPM solutions and applications

This paper presents new findings from analyses of a random fractional kinematic wave equation (rfKWE) for overland flow. The rfKWE is featured with orders of temporal and spatial fractional derivatives and with the roughness parameter, the effective rainfall intensity and infiltration rate as random variables. The new solutions are derived with the aid of a numerical method named the homotopy perturbation method (HPM) and approximate solutions are presented for different situations. The solutions are evaluated with data from overland flow flumes with simulated rainfall in the laboratory. The results suggest that on an infiltrating surface the temporal nonlocality of overland flow represented by the temporal order of fractional derivatives diminishes over time while the spatial nonlocality manifested by the spatial order of fractional derivatives continue if there is overland flow. It shows that the widely used unit discharge-height relationship is a special case of the solution of the rfKWE. Procedures are demonstrated for determining the fractional roughness coefficient, nf, the order of spatial fractional derivative, ρ, and the steady-state infiltration rate during the overland flow, As. The analyses of the data show that the mean spatial order of fractional derivative is ρ=1.25, the mean flow pattern parameter m=1.50, and the mean fractional roughness coefficient is nf=0.002 which is smaller than the conventional roughness coefficient, n=0.108. With these average values of the parameters and their standard deviations, simulations were performed to demonstrate the use of the methods, which is also a comparison of the classic KWE and rfKWE models.

Non-Destructive Identification of Virgin Cashmere and Chemically Modified Wool Fibers Based on Fractional Order Derivative and Improved Wavelength Extraction Algorithm Using NIR Spectroscopy and Chemometrics

ABSTRACT Virgin cashmere is highly prized for its superior quality, and chemically modified wool is often used to fake it. However, traditional spectral analysis and image processing methods struggle to identify the two fibers. To address this, combining chemometrics with near-infrared (NIR) spectroscopy is proposed to identify virgin cashmere and chemically modified wool based on fractional order derivatives and improved wavelength extraction algorithm. The original spectra of the two fibers are very close, making identification difficult. Therefore, fractional order derivatives are used to amplify tiny spectral differences, which are often overlooked in conventional analysis. Meanwhile, common wavelength extraction algorithms few consider inherent relationship between spectral features and chemical properties, so an improved wavelength extraction algorithm using Shuffled Frog Leaping Algorithm (SFLA) and Beluga Whale Optimization (BWO) is employed to explore the connection of the two. The proposed method with Partial Least Squares Discriminant Analysis (PLS-DA) achieves an accuracy of 94.4%. Experimental results demonstrate that fractional order derivatives enhance tiny spectral features and differences, while extracted feature wavelengths correspond to main chemical characteristic bands of the fibers. This study shows the feasibility of identifying virgin cashmere and chemically modified wool and offers a novel idea for identifying fibers in the textile industry.

Revised logarithmic Sobolev inequalities of fractional order

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on Rn with an explicit expression for the constant. Namely, we show that for all 0<s<n2 and a>0 we have the inequality∫Rn|f(x)|2log⁡(|f(x)|2‖f‖L2(Rn)2)dx+ns(1+log⁡a)‖f‖L2(Rn)2≤C(n,s,a)‖(−Δ)s/2f‖L2(Rn)2 with an explicit C(n,s,a) depending on a, the order s, and the dimension n, and investigate the behaviour of C(n,s,a) for large n. Notably, for large n and when s=1, the constant C(n,1,a) is asymptotically the same as the sharp constant of Lieb and Loss that was computed in [13]. Moreover, we prove a similar type inequality for functions f∈Lq(Rn)∩W1,p(Rn) whenever 1<p<n and p<q≤p(n−1)n−p.

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

On general tempered fractional calculus with Luchko kernels

In this paper, we construct the n-fold ψ-fractional integrals and derivatives and study their properties. This construction is purely based on the approach proposed by Luchko (2021). The fundamental theorems of fractional calculus are formulated and proved for the proposed n-fold ψ-fractional integrals and derivatives. On the other hand, a suitable generalization of the Luchko condition is presented to discuss the ψ-tempered fractional calculus of arbitrary order. We introduce an important class of kernels that satisfy this condition. For the ψ-tempered fractional integrals and derivatives of arbitrary order, two fundamental theorems are proven, along with a relation between Riemann–Liouville and Caputo derivatives. Finally, Cauchy problems for the fractional differential equations with the ψ-tempered fractional derivatives are solved.

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Edge currents for the time-fractional, half-plane, Schrödinger equation with constant magnetic field

We study the large-time asymptotics of the edge current for a family of time-fractional Schrödinger equations with a constant, transverse magnetic field on a half-plane (x,y)∈Rx+×Ry. The time-fractional Schrödinger equation is parameterized by two constants (α, β) in (0, 1], where α is the fractional order of the time derivative, and β is the power of i in the Schrödinger equation. We prove that for fixed α, there is a transition in the transport properties as β varies in (0, 1]: For 0 &amp;lt; β &amp;lt; α, the edge current grows exponentially in time, for α = β, the edge current is asymptotically constant, and for β &amp;gt; α, the edge current decays in time. We prove that the mean square displacement in the y∈R-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin [Phys. Rev. E 62, 3135 (2000)] that the latter two cases, α = β and α &amp;lt; β, are the physically relevant ones.

ON THE CHAOTIC NATURE OF A CAPUTO FRACTIONAL MATHEMATICAL MODEL OF CANCER AND ITS CROSSOVER BEHAVIORS

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts.

Review of approximation formulas for fractional derivatives of variable order

Fractional differential equations of variable order, depending on a time or space variable, are successfully used to study time and/or spatial dynamics. The purpose of this review is to consider the latest research and results related to the basic definitions and approximation formulas for fractional derivatives of variable order. The review begins with an examination of existing definitions based on various physical and practical knowledge. The following are formulas for discretizing fractional derivatives, since they play a key role in numerical modeling, providing an effective means of describing complex systems with long-term memory, heterogeneous parameters and non-local interactions. Their use not only improves the accuracy of models, but also facilitates the numerical solution of differential equations, which is essential for analyzing and predicting the behavior of real processes in various fields of science and engineering. This review is intended to help readers select appropriate definitions and discretization formulas to effectively solve specific physical and engineering problems.

On a Symmetry-Based Structural Deterministic Fractal Fractional Order Mathematical Model to Investigate Conjunctivitis Adenovirus Disease

In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.