Fractional Order Differential Operator Research Articles

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

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

Comparative analysis of classical and Caputo models for COVID-19 spread: vaccination and stability assessment

Several epidemiological models use the Caputo fractional-order differential operator without establishing its significance. This study verifies a Caputo operator-based fractional-order epidemiological model of the SAIVR type. COVID-19 kills. Infection weakens the immune system. The fractional Caputo operator describes COVID-19 immunization. Fundamental system characteristics are determined using fractional calculus. Our analysis included the fractional system’s Hyers–Ulam–Rassias stability and stable states. The uniqueness and existence of fractional Caputo system solutions are explored. The least-squares approach determines system parameters. The Caputo fractional-order α value is optimized to 6.757e−01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$6.757\ ext{e}{-}01$\\end{document}, indicating that the system best fits real-life medical data for infection. Caputo and classical systems were compared for absolute mean errors. The Box-Whisker chart case summaries show the Caputo operator superiority. When α→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha \\rightarrow 1$\\end{document}, the memory traces and hereditary traits are also observed. Finally, the Caputo fractional framework simulates COVID-19 using strong numerical methods.

Research on Application of Fractional Calculus Operator in Image Underlying Processing

Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing.

Spectroscopic analysis of residual radicals of gamma sterilized UHMWPE with fractional order differential operators

The purpose of this research was to evaluate the Grunwald-Letnikov definition of fractional order derivatives for determining the fractional derivative of ESR signals from UHMWPE free radicals using the fitted Gaussian distribution method. Specifically, the study focused on analyzing two long-lasting oxygen-induced residual radicals di- or tri-enyls with a carbon center radical (R1) and the oxygen-containing radical (R2). The impact of the derivative order on ESR spectral parameters, such as the Landé g-factor and peak-to-peak separation, was analyzed, and new spectral parameters were established for both radicals. The samples were measured in an ESR tube using ESR power saturation techniques at room temperature. Following the measurements, the fitted Gaussian distribution of the ESR signals was used to determine the fractional derivative using the Grunwald-Letnikov definition. Two estimators (I and II) were developed for both radicals, and their values were found to be 9.19 and 4.27 for Radical R1 and 11.51 and 2060.62 for Radical R2. Our results showed that the fractional derivative approach provided accurate and reliable estimations for the radicals. This method can be easily implemented for various ESR signals and can be useful in many applications, including materials science and biomedical research. The accuracy and reliability of our estimators were confirmed by comparing them with the results obtained from the conventional method. Our findings have important implications for materials science and biomedical research, where ESR signals are widely used. This method can help researchers to obtain more accurate and reliable estimations of radicals, which can ultimately lead to more accurate and reliable conclusions in their studies. Additional research is required to assess the effectiveness of this approach on various ESR signals and to investigate its potential use in other fields of study.

A fractional-order visual neural network for collision sensing in noisy and dynamic scenes

Autonomously sensing collision threats in intelligent mobile machines is a significant challenge. Inspired by the perfect collision sensing system of locusts, various neural networks based on lobular giant motion detectors (LGMD) have been proposed. These networks use integer-order differential operators to simulate the simple and efficient visual neural system of locusts. However, some biological studies have shown that the neurodynamics of neuronal response should be fractional-order, and fractional-order differential operators can accurately describe the biological logic brain neural response process. In view of this, firstly, this paper uses a fractional-order differential operator to simulate the membrane potential response of neurons, and a new fractional-order LGMD collision sensing visual neural network is proposed. This network is designed to accurately detect looming objects and generate reliable response spikes. Then, through experimental analysis, a range [0.36, 0.4] of order in the fractional-order system is given, which is considered to be a reasonable interval for modeling the behavior of biological nervous systems. This maybe contribute to the study of the locust’s fractional-order biological nervous system. Finally, the network was systematically tested in noisy and complex dynamic environments. The experimental results demonstrated that the fractional-order LGMD visual neural network generated responses with high adaptability and reliability, showcasing significantly improved collision detection performance.

Modeling and complexity analysis of a fractional-order memristor conservative chaotic system

Memristors are often utilized in circuit model analysis as one of the fundamental circuit components. In this paper, a five-dimensional conservative memristor chaotic system is built after the introduction of the memristor into a four-dimensional conservative chaotic system. The dynamic changes of the system are examined using phase diagram, mean value, and Lyapunov exponent spectrum. A line equilibrium point, symmetry and multi-stability are characteristics of the system; the phase trajectory can also produce shrinking and structure transformation behavior with the change of parameters. Furthermore, the system has initial offset boosting behaviors, conservative flows of it can be altered in position by changing two initial values, respectively. Most notably, we discover that the complexity of the system rises with the inclusion of memristor and again with the addition of fractional differential operators. It is shown that the complexity of chaotic systems may increase with the addition of memristors and fractional-order differential operators. At last, the NIST is used to test the randomness of the sequence, and the system's physical realizability is confirmed by the DSP platform.

A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images

High-quality image restoration is typically challenging due to low signal–to–background ratios (SBRs) and limited statistics frames. To address these challenges, this paper devised a method based on fractional-order total variation (FOTV) regularization for recovering Geiger-mode avalanche photodiode (GM-APD) light detection and ranging (lidar) depth images. First, the spatial differential peak-picking method was used to extract the target depth image from low SBR and limited frames. FOTV regularization was introduced based on the total variation regularization recovery model, which incorporates the fractional-order differential operator, in order to realize FOTV-regularization-based depth image recovery. These frameworks were used to establish an algorithm for GM-APD depth image recovery based on FOTV. The simulation and experimental results demonstrate that the devised FOTV-recovery algorithm improved the target reduction degree, peak signal–to–noise ratio, and structural similarity index measurement by 76.6%, 3.5%, and 6.9% more than the TV, respectively, in the same SBR and statistic frame conditions. Thus, the devised approach is able to effectively recover GM-APD lidar depth images in low SBR and limited statistic frame conditions.

Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators

Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability.

A directionally selective collision-sensing visual neural network based on fractional-order differential operator.

In this paper, we propose a directionally selective fractional-order lobular giant motion detector (LGMD) visual neural network. Unlike most collision-sensing network models based on LGMDs, our model can not only sense collision threats but also obtain the motion direction of the collision object. Firstly, this paper simulates the membrane potential response of neurons using the fractional-order differential operator to generate reliable collision response spikes. Then, a new correlation mechanism is proposed to obtain the motion direction of objects. Specifically, this paper performs correlation operation on the signals extracted from two pixels, utilizing the temporal delay of the signals to obtain their position relationship. In this way, the response characteristics of direction-selective neurons can be characterized. Finally, ON/OFF visual channels are introduced to encode increases and decreases in brightness, respectively, thereby modeling the bipolar response of special neurons. Extensive experimental results show that the proposed visual neural system conforms to the response characteristics of biological LGMD and direction-selective neurons, and that the performance of the system is stable and reliable.

The Problem for a Mixed Equation with Fractional Power of the Bessel Operator

В последнее время особый интерес представляют уравнения с частными производными, содержащими дифференциальный оператор дробного порядка. Подобные уравнения и задачи для них находят применение в теории вязкой упругости, электрохимии, теории управления, моделировании эпидемий и пандемий и в других различных областях. Настоящая работа посвящена решению дифференциальных уравнений, содержащих оператор Бесселя дробной степени. В статье рассматривается прямое и обратное преобразование Мейера, модифицированное для удобства работы с оператором Бесселя дробной степени. Для рассматриваемого преобразования Мейера получена свертка. Используя преобразования Лапласа и Пуассона получены факторизации прямого и обратного преобразований Мейера. С использованием рассмотренного модифицированного преобразования Мейера находится решение обыкновенного дифференциального уравнения с оператором Бесселя дробной степени. Рассматривается нелокальная краевая задача для смешанного параболо-гиперболческого уравнения, содержащего дробной степени оператор Бесселя. Доказывается, что, при выполнении определенных условий гладкости входных функций задачи и выполнения условия сопряжения на линии раздела областей гиперболичности и параболичности, регулярное решение нелокальной краевой задачи для смешанного параболо-гиперболического уравнения с оператором Бесселя дробной степени существует и единственно. Recently, of particular interest are partial differential equations containing a fractional order differential operator. Similar equations and problems for them find application in the theory of viscous elasticity, electrochemistry, control theory, modeling of epidemics and pandemics, and in various other areas. The present work is devoted to the solution of differential equations containing the Bessel operator of fractional degree. The article discusses the direct and inverse Meyer transforms, modified for the convenience of working with the Bessel operator of a fractional degree. For the considered Meyer transformation, a convolution is obtained. Using the Laplace and Poisson transformations, factorizations of the direct and inverse Meyer transformations are obtained. Using the considered modified Meyer transform, we find a solution to an ordinary differential equation with a Bessel operator of fractional degree. A nonlocal boundary value problem for a mixed parabolic-hyperbolic equation containing a fractional degree Bessel operator is considered. It is proved that, under certain conditions of smoothness of the input functions of the problem and the condition of conjugation on the dividing line of the regions of hyperbolicity and parabolicity, a regular solution of a nonlocal boundary value problem for a mixed parabolic-hyperbolic equation with a Bessel operator of fractional degree exists and is unique.

Data-driven discovery of Caputo fractional order systems

Due to the peculiar non-locality of fractional order differential and integral operators, Caputo fractional order systems are harder than integer order systems to be discoverd from data. To solve this open problem, we propose a framework of method capable of discovering Caputo fractional order (autonomous and non-autonomous) systems from measurable data. The interior point method and genetic algorithm are embedded respectively in the framework. The former is mainly presented in text, while the latter is implemented for comparison and validation. The framework is designed to dynamically and coordinately update the fractional order and vector field function for the system to be discovered till the difference between the measured and discovered systems is minimized. It is computationally efficient, robust and illustrated by discovering the Caputo fractional order Lorenz system, Chua’s circuit and Duffing’s oscillator hidden in measured data. As thus, this work provides one way to uncover underlying Caputo fractional order mathematical models (or physical laws and governing equations).

Research on Manufacturing Equipment Operation State Evaluation Technology Based on Fractional Calculus

The operational status of manufacturing equipment is directly related to the reliability of the operation of manufacturing equipment and the continuity of operation of the production system. Based on the analysis of the operation status of manufacturing equipment and its characteristics, it is proposed that the concept of assessing the operation status of manufacturing equipment can be realized by applying the real-time acquisition of accurate inspection data of important parts of weak-motion units and comparing them with their motion status evaluation criteria. A differential data fusion model based on the fractional-order differential operator is established through the study of the application characteristics of fractional-order calculus theory. The advantages of Internet of Things (IoT) technology and a fractional order differential fusion algorithm are integrated to obtain real-time high-precision data of the operating parameters of manufacturing equipment, and the research objective of the operating condition assessment of manufacturing equipment is realized. The feasibility and effectiveness of the method are verified by applying the method to the machining center operation status assessment.

Solvability of Mixed Problems for a Fourth-Order Equation with Involution and Fractional Derivative

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved.

Dynamic analysis of variable fractional order cantilever beam based on shifted Legendre polynomials algorithm

The shifted Legendre algorithm is used to solve the governing differential equation of variable order cantilever beam in time domain. According to the variable fractional model, the governing equations of viscoelastic polymer cantilever beam are built. The integer order and variable fractional order differential operator matrices are obtained based on the properties of shifted Legendre polynomials. The variable fractional differential equations are converted into algebraic equations and solved in time domain by operator matrix. The convergence analysis confirmed the high efficiency and accuracy of the proposed algorithm for solving differential equations. The deflection and stress of the polymer cantilever beams (PET and HDPE) under various external load (uniformly distributed and harmonic) are studied.

A Study of Adaptive Fractional-Order Total Variational Medical Image Denoising

Following the traditional total variational denoising model in removing medical image noise with blurred image texture details, among other problems, an adaptive medical image fractional-order total variational denoising model with an improved sparrow search algorithm is proposed in this study. This algorithm combines the characteristics of fractional-order differential operators and total variational models. The model preserves the weak texture region of the image improvement based on the unique amplitude-frequency characteristics of the fractional-order differential operator. The order of the fractional-order differential operator is adaptively determined by the improved sparrow search algorithm using both the sine search strategy and the diversity variation processing strategy, which can greatly improve the denoising ability of the fractional-order differential operator. The experimental results reveal that the model not only achieves the adaptivity of fractional-order total variable differential order, but also can effectively remove noise, preserve the texture structure of the image to the maximum extent, and improve the peak signal-to-noise ratio of the image; it also displays favorable prospects for applications in medical image denoising.

Separation method of semi-fixed variables together with dynamical system method for solving nonlinear time-fractional PDEs with higher-order terms

It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach combined with the separation method of semi-fixed variables and dynamical system method is introduced. As an example, a time-fractional reaction-diffusion equation with higher-order terms is studied under two different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional model are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed, and the graphs of them are illustrated.

A Fractional-Order Compartmental Model of Vaccination for COVID-19 with the Fear Factor

During the past several years, the deadly COVID-19 pandemic has dramatically affected the world; the death toll exceeds 4.8 million across the world according to current statistics. Mathematical modeling is one of the critical tools being used to fight against this deadly infectious disease. It has been observed that the transmission of COVID-19 follows a fading memory process. We have used the fractional order differential operator to identify this kind of disease transmission, considering both fear effects and vaccination in our proposed mathematical model. Our COVID-19 disease model was analyzed by considering the Caputo fractional operator. A brief description of this operator and a mathematical analysis of the proposed model involving this operator are presented. In addition, a numerical simulation of the proposed model is presented along with the resulting analytical findings. We show that fear effects play a pivotal role in reducing infections in the population as well as in encouraging the vaccination campaign. Furthermore, decreasing the fractional-order parameter α value minimizes the number of infected individuals. The analysis presented here reveals that the system switches its stability for the critical value of the basic reproduction number R0=1.

Numerical Simulation of Elastic Wave Field in Viscoelastic Two-Phasic Porous Materials Based on Constant Q Fractional-Order BISQ Model.

The fractional-order differential operator describes history dependence and global correlation. In this paper, we use this trait to describe the viscoelastic characteristics of the solid skeleton of a viscoelastic two-phasic porous material. Combining Kjartansson constant Q fractional order theory with the BISQ theory, a new BISQ model is proposed to simulate elastic wave propagation in a viscoelastic two-phasic porous material. The corresponding time-domain wave propagation equations are derived, and then the elastic waves are numerically simulated in different cases. The integer-order derivatives are discretised using higher-order staggered-grid finite differences, and the fractional-order time derivatives are discretised using short-time memory central differences. Numerical simulations and analysis of the wave field characterisation in different phase boundaries, different quality factor groups, and multilayered materials containing buried bodies are carried out. The simulation results show that it is feasible to combine the constant Q fractional-order derivative theory with the BISQ theory to simulate elastic waves in viscoelastic two-phasic porous materials. The combination can better describe the viscoelastic characteristics of the viscoelastic two-phasic porous materials, which is of great significance for further understanding the propagation mechanism of elastic waves in viscoelastic two-phasic porous materials and viscoelastic two-phasic porous materials containing buried bodies. This paper provides a theoretical forward simulation for fine inversion and reconstruction of layer information and buried body structure in viscoelastic two-phasic porous materials.

Fractional-order coronavirus models with vaccination strategies impacted on Saudi Arabia's infections

<abstract><p>Several newly nonlinear models for describing dynamics of COVID-19 pandemic have been proposed and investigated in literature recently. In light of these models, we attempt to reveal the role of fractional calculus in describing the growth of COVID-19 dynamics implemented on Saudi Arabia's society over 107 days; from 17 Dec 2020 to 31 March 2021. Above is achieved by operating two fractional-order differential operators, Caputo and the Caputo-Fabrizio operators, instead of the classical one. One of expanded SEIR models is utilized for achieving our purpose. With the help of using the Generalized Euler Method (GEM) and Adams-Bashforth Method (ABM), the numerical simulations are performed respectively in view of the Caputo and Caputo-Fabrizio operators. Accordance with said, the stability analysis of the two proposed fractional-order models is discussed and explored in view of obtaining the equilibrium points, determining the reproductive number ($ R_0 $) and computing the elasticity indices of $ R_0 $. Several numerical comparisons reveal that the fractional-order COVID-19 models proposed in this work are better than that of classical one when such comparisons are performed between them and some real data collected from Saudi Arabia's society. This inference together with the cases predictions that could easily deduced from the proposed fractional-order models can allow primary decision makers and influencers to set the right plans and logic strategies that should be followed to face this pandemic.</p></abstract>

The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator

<abstract><p>The theory of convex function has a lot of applications in the field of applied mathematics and engineering. The Caputo-Fabrizio non-singular operator is the most significant operator of fractional theory which permits to generalize the classical theory of differentiation. This study consider the well known Hermite-Hadamard type and associated inequalities to generalize further. To full fill this mileage, we use the exponential convexity and fractional-order differential operator and also apply some existing inequalities like Holder, power mean, and Holder-Iscan type inequalities for further extension. The generalized exponential type fractional integral Hermite-Hadamard type inequalities establish involving the global integral. The applications of the developed results are displayed to verify the applicability. The establish results of this paper can be considered an extension and generalization of the existing results of convex function and inequality in literature and we hope that will be more helpful for the researcher in future work.</p></abstract>