Fractional Order Differential Operator Research Articles

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type (ψ−α)-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones.

Об идентификации характеристик неоднородных вязкоупругих тел в рамках модели дробного порядка

Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation, the inverse problems for specific objects , namely, an inhomogeneous rod and a round plate, are considered. In both model problems, the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental results made it possible to formulate recommendations for the selection of optimal sensing modes.

A Compartmental Approach to Modeling the Measles Disease: A Fractional Order Optimal Control Model

Measles is the most infectious disease with a high basic reproduction number (R0). For measles, it is reported that R0 lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with R0=1.

Improving Influenza Epidemiological Models under Caputo Fractional-Order Calculus

The Caputo fractional-order differential operator is used in epidemiological models, but its accuracy benefits are typically ignored. We validated the suggested fractional epidemiological seasonal influenza model of the SVEIHR type to demonstrate the Caputo operator’s relevance. We analysed the model using fractional calculus, revealing its basic properties and enhancing our understanding of disease progression. Furthermore, the positivity, bounds, and symmetry of the numerical scheme were examined. Adjusting the Caputo fractional-order parameter α = 0.99 provided the best fit for epidemiological data on infection rates. We compared the suggested model with the Caputo fractional-order system and the integer-order equivalent model. The fractional-order model had lower absolute mean errors, suggesting that it could better represent sickness transmission and development. The results underline the relevance of using the Caputo fractional-order operator to improve epidemiological models’ precision and forecasting. Integrating fractional calculus within the framework of symmetry helps us build more reliable models that improve public health interventions and policies.

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.

Research on Viscoelastic Impact Damage Constitutive Model and Its Behavioral Characteristics

To investigate the damage characteristics of viscoelastic materials under various impact scenarios, this study presents a viscoelastic impact damage constitutive model that incorporates a material coefficient, a variable fractional-order differential operator, and a damage factor. Split Hopkinson Pressure Bar experiments are conducted under single-impact and repeated-impact conditions. The results indicate that nonlinear features in stress-strain curves at different stages, exhibiting prominent strain rate dependency near the stress peak point in the single-impact scenario. In the repeated-impact scenario, the stress peak points decrease as the impact times increase, resulting in a corresponding decline in the material's load-bearing capacity. Furthermore, strain rate escalates the strain rate increases with an increase in impact times, making the material more susceptible to deformation. Parameter fitting and Sobol sensitivity analysis utilizing experimental data from both scenarios have been employed to explore the effects and patterns of single-parameter and multi-parameter coupling on the model, accurately delineating their influence. The findings reveal a strong coupling relationship among various parameters of the viscoelastic impact damage constitutive model in multiple impact scenarios, showing higher sensitivity to fluctuations in the viscoelastic coefficient and damage factor. These analyses have demonstrated the accuracy of the viscoelastic impact damage constitutive model in capturing dynamic impact damage characteristics of viscoelastic materials across various scenarios.

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.

Fractional-Order Dynamics in Epidemic Disease Modeling with Advanced Perspectives of Fractional Calculus

Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing fractional calculus, our approach aims to capture the crossover dynamics of disease spread, considering heterogeneity and transitions between epidemic phases. This research seeks to develop a framework using specialized mathematical techniques, such as the Caputo fractional derivative, with the potential to investigate the crossover dynamical behaviors of the considered epidemic model. The anticipated contribution lies in bridging fractional calculus and epidemiology, offering insights for both theoretical advancements and practical public health interventions. In order to improve our understanding of epidemic dynamics and support, we used MATLAB to simulate numerical results for a visual representation of our findings. For this interpretation, we used various fractional-order values. In addition, we also compare our simulated results with some reported results for infected and death classes to demonstrate the efficiency of our numerical method.

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.

Fractional-Order Super-Resolution Reconstruction Algorithm for GM-APD Lidar Distance Images Based on Convex Set Projection

The main objective of this paper is to solve the problem of low resolution of target distance images obtained by GM-APD lidar. Towards overcoming this problem, this paper proposes a fractional-order super-resolution reconstruction algorithm for GM-APD lidar distance images based on convex set projection. Firstly, the first low-resolution image is upsampled three times to obtain a high-resolution reference image. Second, starting from the image degradation model, the transformation model between the low-resolution image and the high-resolution reference image is constructed. Then the high-resolution reference image is convolved using the G-L fractional order differential operator, and the estimated low-resolution image is obtained by combining the point spread function. Finally, the difference between the remaining low-resolution images and the estimated low-resolution image is calculated, and the high-resolution reference image is corrected to obtain the final high-resolution distance image. In addition to solving the edge blurring problem of the original convex set projection algorithm, the edge structure of the image is enhanced and the image quality is improved. Simulation and experimental results show that the proposed algorithm improves the information entropy by at least 24.190% and the average gradient by at least 10.812%, and achieves the super-resolution reconstruction of GM-APD LIDAR target distance images.

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.