Fractional Order Parameter Research Articles

Time-Fractional Cattaneo-Type Thermoelastic Interior-Boundary Value Problem Within A Rigid Ball

The paper discusses the solution of an interior-boundary value problem of one-dimensional time-fractional Cattaneo-type heat conduction and its stress fields for a rigid ball. The interior value problem describes the dependence of the boundary conditions within the ball's inner plane at any instant with a prescribed temperature state, in contrast to the exterior value problem, which relates the known surface temperature to boundary conditions. A single-phase-lag equation with Caputo fractional derivatives is proposed to model the heat equation in a medium subjected to time-dependent physical boundary conditions. The application of the finite spherical Hankel and Laplace transform technique to heat conduction is discussed. The influence of the fractional-order parameter and the relaxation time is examined on the temperature fields and their related stresses. The findings show that the slower the thermal wave, the bigger the fractional-order setting, and the higher the period of relaxation, the slower the heat flux propagates.

Dynamical Analysis of Generalized Tumor Model with Caputo Fractional-Order Derivative

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative is a mathematical tool that is recently being applied to model biological systems, including tumor growth. We present a detailed mathematical analysis of the generalized tumor model with the Caputo fractional-order derivative and examine its dynamical behavior. Our results show that the Caputo fractional-order derivative provides a more accurate description of the tumor growth dynamics compared to classical integer-order derivatives. We also provide a comprehensive stability analysis of the tumor model and show that the fractional-order derivative allows for a more nuanced understanding of the stability of the system. The least-square curve fitting method fits several biological parameters, including the fractional-order parameter α. In conclusion, our study provides new insights into the dynamics of tumor growth and highlights the potential of the Caputo fractional-order derivative as a valuable tool in biomedical research. The results of this study shell have significant implications for the development of more effective treatments for tumor growth and the design of more accurate mathematical models of tumor development.

A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method

This research focuses on the analysis of the competitive model used in the banking sector based on the stochastic fractional differential equation. For the approximate solution, a pseudospectral technique is utilized for the proposed model based on the stochastic Lotka–Volterra equation using a wide range of fractional order parameters in simulations. Conditions for stable and unstable equilibrium points are provided using the Jacobian. The Lotka–Volterra equation is unstable in the long term and can produce highly fluctuating dynamics, which is also one of the reasons that this equation is used to model the problems arising in finance, where fluctuations are important. For this reason, the conventional analytical and numerical methods are not the best choices. To overcome this difficulty, an automatic procedure is used to solve the resultant algebraic equation after the discretization of the operator. In order to fully use the properties of orthogonal polynomials, the proposed scheme is applied to the equivalent integral form of stochastic fractional differential equations under consideration. This also helps in the analysis of fractional differential equations, which mostly fall in the framework of their integrated form. We demonstrate that this fractional approach may be considered as the best tool to model such real-world data situations with very reasonable accuracy. Our numerical simulations further demonstrate that the use of the fractional Atangana–Baleanu operator approach produces results that are more precise and flexible, allowing individuals or companies to use it with confidence to model such real-world situations. It is shown that our numerical simulation results have a very good agreement with the real data, further showing the efficiency and effectiveness of our numerical scheme for the proposed model.

Constitutive modeling of human cornea through fractional calculus approach

In this work, the fractional calculus approach is considered for modeling the viscoelastic behavior of human cornea. It is observed that the degree of both elasticity and viscosity is easy to describe in terms of the fractional order parameters in such an approach. Modeling of the human cornea when subjected to simple stress up to the level of 250 MPa by fractional order Maxwell model along with the Fractional Kelvin Voigt Viscoelastic Model is reported. For the Maxwell governing fractional equation, two fractional parameters α and β have been considered to model the stress–strain relationship of the human cornea. The analytical solution of the fractional equation has been obtained for different values of α and β using Laplace transform methods. The effect of the fractional parameter values on the stress-deformation nature has been studied. A comparison between experimental values and calculated values for different fractional order of the Maxwell model equation defines the parameters which depict the real-time stress–strain relationship of the human cornea. It has been observed that the fractional model converges to the classical Maxwell model as a special case for α = β = 1.

Asymptotic Stabilization Control of Fractional-Order Memristor-Based Neural Networks System via Combining Vector Lyapunov Function With M-Matrix

This article examines a new measure of combining the vector Lyapunov function with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> -matrix for settling the asymptotic stabilization control of fractional-order memristor-based neural networks system (FOMBNNS) has large delays in various dimensional forms. Some new stability and stabilization criteria are deduced. First, the vector Lyapunov function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> -matrix are imported for investigating stabilization control for the above system. Then, we solve the problem for a special type of situation that the activation functions no longer consider Lipschitz parameters via the new method. Finally, four numerical examples from different kinds of situations are simulated for expounding the validity of the novel asymptotic stability and stabilization criteria. Compared with the methods mentioned in the current references, the proposed asymptotic stability and stabilization criteria in this article have strong generality and universality. They can be applied not only to the most common feedback control, accordingly, the feedback control law based on which they are designed but also to all fractional-order parameters from 0 to 1. In addition, the new method has lower conservativeness and fewer constraints. Moreover, the new stability and stabilization criteria can also overcome the difficulty in dealing with the above system owning large delays.

The Two Stage Moisture Diffusion Model for Non-Fickian Behaviors of 3D Woven Composite Exposed Based on Time Fractional Diffusion Equation

The moisture diffusion behaviors of 3D woven composites exhibit non-Fickian properties when they are exposed to a hydrothermal environment. Although some experimental works have been undertaken to investigate this phenomenon, very few mathematical works on non-Fickian moisture diffusion predictions of 3D woven composites are available in the literature. To capture the non-Fickian behavior of moisture diffusion in 3D woven composites, this study first utilized a time fractional diffusion equation to derive the percentage of moisture content of a homogeneous material under hydrothermal conditions. A two-stage moisture diffusion model was subsequently developed based on the moisture diffusion mechanics of both neat resin and 3D woven composites, which describes the initial fast diffusion and the long-term slow diffusion stages. Notably, the model incorporated fractional order parameters to account for the nonlinear property of moisture diffusion in composites. Finally, the weight gain curves of neat resin and the 3D woven composite were calculated to verify the fractional diffusion model, and the predicted moisture uptake curves were all in good agreement with the experimental results. It is important to note that when the fractional order parameter α &lt; 1, the initial moisture uptake will become larger with a later slow down process. This phenomenon can better describe non-Fickian behavior caused by initial voids or complicated structures.

Dynamics and Stability of a Fractional-Order Tumor–Immune Interaction Model with B-D Functional Response and Immunotherapy

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and a unique positive equilibrium point are obtained. Finally, we use numerical simulation method to visualize and verify the theoretical conclusions. It is known that the fractional-order parameter β has a stabilization effect, and the tumor cells can be destroyed or controlled by using immunotherapy.

The Sensitive Visualization and Generalized Fractional Solitons’ Construction for Regularized Long-Wave Governing Model

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields.

On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.

The paradigm of quantum cosmology through Dunkl fractional Laplacian operators and fractal dimensions

The semi-classical regime of fractional quantum cosmology has been discussed based on the Wheeler-DeWitt equation and ADM-Hamiltonian formalism. Fractionality has been introduced through two different approaches: the first one is based on fractional Dunkl Laplacian operator and the second one based on fractal calculus. Our models generalize the recent approaches of fractional quantum cosmology introduced in literature. In the first approach, it was revealed that power-law inflation and large number of e-folds may be obtained for various values of the fractional parameter. However, the second approach is plausible only if the cosmological constant decays with the scale factor of the universe. We have confronted our model with observations based on the Bayesian hierarchical model that describes the full distance ladder from nearby geometric anchors through Cepheids to Hubble-Flow SNe we have estimated the numerical values of the fractional order parameter. In both scenarios, the number of e-folds is too large.Pacs numbers: 05.45.Df, 45.10.Hj, 98.80.-k

Casimir effect associated with fractional laplacian and fractal dimensions

Casimir effect predicts that two parallel flat neutral plates are attracted to each other due to quantum fluctuations of the electromagnetic field. In this communication, we study Casimir effect based on two different approaches: the first one is correlated to fractional Laplacian by means of fractional derivatives operators whereas the second one is associated to fractal dimensions. It was observed that, in both approaches, the Casmir zero-point energy of the fields is affected by the fractional order parameter and the numerical value of the fractal dimensions. The first model requires the use of the Hurwitz zeta function and no regularization technique is strongly required. Nevertheless, the second model necessitates the use of the regularization technique augmented by the Lifshitz fixed point and Schwinger proper-time representation mainly when a particular eigenvalue problem is used. • 1-Casimir effect has been studied based on fractional Laplacian and fractal dimensions. • 2-Casmir zero-point energy is affected by the fractional order parameter and fractal dimensions. • 3-The fractional model requires the use of the Hurwitz zeta function without the need of the regularization technique. • 4-The fractal model requires the use of regularization technique and the Lifshitz fixed point approach.

ON FRACTAL FRACTIONAL HEPATITIS B EPIDEMIC MODEL WITH MODIFIED VACCINATION EFFECTS

Fractional calculus is strongly tied to the dynamics of complicated real-world phenomena. The goal of this study is to employ a fractal fractional mathematical model with vaccine effects to solve HBV infection. We provide a novel numerical technique for fractal–fractional model with many graphical results. Moreover, we present by comparing these two operators for many values of the fractal and fractional order parameters [Formula: see text] and [Formula: see text]. The effects of vaccination have been examined more thoroughly because vaccination results are not cent percent in some epidemic circumstances, e.g. vaccinated people can also be deemed as susceptible people. The dynamical behavior of the fractal–fractional HBV model with modified vaccination effects is examined utilizing the Atangana–Baleanu approach.

Dynamical analysis of an anthrax disease model in animals with nonlinear transmission rate

&lt;abstract&gt;&lt;p&gt;Anthrax is a bacterial infection caused by &lt;italic&gt;Bacillus anthracis&lt;/italic&gt;, primarily affecting animals and occasionally affecting humans. This paper presents two compartmental deterministic models of anthrax transmission having vaccination compartments. In both models, a nonlinear ratio-dependent disease transmission function is employed, and the latter model distinguishes itself by incorporating fractional order derivatives, which adds a novel aspect to the study. The basic reproduction number $ \mathcal{R}_0 $ of the epidemic is determined, below which the disease is eradicated. It is observed that among the various parameters, the contact rate, disease-induced mortality rate, and rate of animal recovery have the potential to influence this basic reproduction number. The endemic equilibrium becomes disease-free via transcritical bifurcations for different threshold parameters of animal recovery rate, disease-induced mortality rate and disease transmission rate, which is validated by utilizing Sotomayor's theorem. Numerical simulations have revealed that a higher vaccination rate contributes to eradicating the disease within the ecosystem. This can be achieved by effectively controlling the disease-induced death rate and promoting animal recovery. The extended fractional model is analyzed numerically using the Adams-Bashforth-Moulton type predictor-corrector scheme. Finally, it is observed that an increase in the fractional order parameter has the potential to reduce the time duration required to eradicate the disease from the ecosystem.&lt;/p&gt;&lt;/abstract&gt;

Modeling and hardware implementation of universal interface-based floating fractional-order mem-elements.

Fractional-order systems generalize classical differential systems and have empirically shown to achieve fine-grain modeling of the temporal dynamics and frequency responses of certain real-world phenomena. Although the study of integer-order memory element (mem-element) emulators has persisted for several years, the study of fractional-order mem-elements has received little attention. To promote the study of the characteristics and applications of mem-element systems in fractional calculus and memory systems, a novel universal fractional-order mem-elements interface for constructing three types of fractional-order mem-element emulators is proposed in this paper. With the same circuit topology, the floating fractional-order memristor, the fractional-order memcapacitor, and fractional-order meminductor emulators can be implemented by simply combining the impedances of different passive elements. PSPICE circuit simulation and printed circuit board hardware experiments validate the dynamical behaviors and effectiveness of our proposed emulators. In addition, the dynamic relationship between fractional-order parameters and values of fractional-order impedance is explored in MATLAB simulation. The proposed fractional-order mem-element emulators built based on the universal interface are constructed with a small number of active and passive elements, which not only reduces the cost but also promotes the development of fractional-order mem-element emulators and application research for the future.

Health Indicators Identification of Lithium-Ion Battery From Electrochemical Impedance Spectroscopy Using Geometric Analysis

Electrochemical impedance spectrum of lithium-ion battery changes regularly with cycling, and is an effective tool for analyzing aging. However, due to the anomalous diffusions and non-exponential effects in battery, EIS-based model is generally identified in complex and time-consuming ways, which limits its online application. In this paper, a parameter identification method for EIS-based model is proposed using geometric analysis. By fitting the impedance spectrum at intermediate frequencies with two depressed semicircles, the parameters of EIS-based model are directly calculated, greatly reducing the computational complexity. Compared with the traditional optimization algorithms, the indexes of the proposed method are optimal, considering running time, file size and goodness of fit. Furthermore, six model parameters are identified based on EIS at different temperatures, states of charge and aging cycles, and their relationship with capacity decay is analyzed. The results show that the charge transfer resistance always has an excellent linear correlation with capacity decay, and the correlation coefficient exceeds 0.94 regardless of temperature and state of charge. It can be used as health indicator of battery. Moreover, one fractional-order parameter has a good correlation with the capacity only at high temperature.

An optimal fractional-order accumulative Grey Markov model with variable parameters and its application in total energy consumption

&lt;abstract&gt; &lt;p&gt;In this paper, we propose an optimal fractional-order accumulative Grey Markov model with variable parameters (FOGMKM (1, 1)) to predict the annual total energy consumption in China and improve the accuracy of energy consumption forecasting. The new model is built upon the traditional Grey model and utilized matrix perturbation theory to study the natural and response characteristics of a system when the structural parameters change slightly. The particle swarm optimization algorithm (PSO) is used to determine the number of optimal fractional order and nonlinear parameters. An experiment is conducted to validate the high prediction accuracy of the FOGMKM (1, 1) model, with mean absolute percentage error (MAPE) and root mean square error (RMSE) values of 0.51% and 1886.6, respectively, and corresponding fitting values of 0.92% and 6108.8. These results demonstrate the superior fitting performance of the FOGMKM (1, 1) model when compared to other six competitive models, including GM (1, 1), ARIMA, Linear, FAONGBM (1, 1), FGM (1, 1) and FOGM (1, 1). Our study provides a scientific basis and technical references for further research in the finance as well as energy fields and can serve well for energy market benchmark research.&lt;/p&gt; &lt;/abstract&gt;

Fractional HCV infection model with adaptive immunity and treatment

Fractional HCV infection model with adaptive immunity and treatment is suggested and studied in this paper. The adaptive immunity includes the CTL response and antibodies. This model contains five ordinary differential equations. We will start our study by proving the existence, uniqueness, and boundedness of the positive solutions. The model has free-equilibrium points and other endemic equilibria. By using Lyapunov functional and LaSalle's invariance principle, we have shown the global stability of these equilibrium points. Finally, some numerical simulations will be given to validate our theoretical results and show the effect of the fractional derivative order parameter and the other treatment parameters.

IMPACT OF PUBLIC HEALTH AWARENESS PROGRAMS ON COVID-19 DYNAMICS: A FRACTIONAL MODELING APPROACH

Public health awareness programs have been a crucial strategy in mitigating the spread of emerging and re-emerging infectious disease outbreaks of public health significance such as COVID-19. This study adopts an Susceptible–Exposed–Infected–Recovered (SEIR) based model to assess the impact of public health awareness programs in mitigating the extent of the COVID-19 pandemic. The proposed model, which incorporates public health awareness programs, uses ABC fractional operator approach to study and analyze the transmission dynamics of SARS-CoV-2. It is possible to completely understand the dynamics of the model’s features because of the memory effect and hereditary qualities that exist in the fractional version. The fixed point theorem has been used to prove the presence of a unique solution, as well as the stability analysis of the model. The nonlinear least-squares method is used to estimate the parameters of the model based on the daily cumulative cases of the COVID-19 pandemic in Nigeria from March 29 to June 12, 2020. Through the use of simulations, the model’s best-suited parameters and the optimal ABC fractional-order parameter [Formula: see text] may be determined and optimized. The suggested model is proved to understand the virus’s dynamical behavior better than the integer-order version. In addition, numerous numerical simulations are run using an efficient numerical approach to provide further insight into the model’s features.

FRACTIONAL ORDER PIλDμ FOR TRACKING CONTROL OF A NOVEL REHABILITATION ROBOT BASED ON IIMO-BP NEURAL NETWORK ALGORITHM

In this study, we develop a novel multi-posture lower limb rehabilitation robot with three postures, which can provide different amplitudes and frequencies of rehabilitation training for hip, knee and ankle joints, respectively. The kinematic and dynamic analyses of the robot are carried out to solve the kinematic forward and backward solutions and the Lagrangian dynamics equations of the lower limbs. The angle, angular velocity and angular acceleration ideal trajectory curves of the rehabilitation motion are derived by using a quintic polynomial trajectory planning scheme. An improved ions motion optimization (IIMO) algorithm is proposed and applied to optimize the initial weight of back propagation (BP) neural network, and algorithm is used to adjust five parameters of fractional order [Formula: see text] ([Formula: see text]) control in controller design. The passive training experiment results of prototype show that the designed controller has the largest average error of angle and angular velocity of hip, knee and ankle joints in high amplitude and high frequency movement mode, which are 1.091∘, 0.716∘, 0.412∘, 1.551∘/s, 1.394∘/s, 1.498∘/s, respectively. At low amplitude and low frequency, the maximum average errors are the smallest, which are 0.351∘, 0.341∘, 0.167∘; 0.833∘/s, 0.842∘/s, 0.398∘/s, respectively. The actual trajectory curve fits well with the designed one. The highest accuracy of angle and angular velocity can reach 99.165% and 99.116% through comprehensive comparison of all motion modes. Therefore, the overall error is small. The stability of rehabilitation training process is ensured, and the rationality and effectiveness of trajectory planning and control design are verified.

Nonrelativistic quark model for mass spectra and decay constants of heavy-light mesons using conformable fractional derivative and asymptotic iteration method

In this paper, we use the concept of conformable fractional derivative to study the nonrelativistic radial Schrödinger equation. We suggest an extended version of the Cornell potential as the quark–antiquark interaction of light and heavy mesons. We generalize the asymptotic iteration method to the fractional domain. The latter is used to calculate the energy eigenvalues, as well as the effect of the fractional order [Formula: see text] on energy spectra. To test the applicability of our model, we use the obtained results to reproduce the mass spectra of some light and heavy mesons such as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] quarks. The mass spectra are obtained at different values of the fractional order parameter [Formula: see text] and were compared with experimental results and other relevant theoretical works. Using the wave function, we calculated the decay constants for heavy-light [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] mesons. Our results are found to be in good agreement with the experimental data, and improved in comparison with other theoretical previsions.