Fractional Derivative Modeling Research Articles

Dynamic analysis and data-driven inference of a fractional-order SEIHDR epidemic model with variable parameters

To analyze and predict the evolution of contagion dynamics, fractional derivative modeling has emerged as an important technique. However, inferring the dynamical structure of fractional-order models with high degrees of freedom poses a challenge. In this paper, to elucidate the spreading mechanism and non-local properties of disease evolution, we propose a novel fractional-order SEIHDR epidemiological model with variable parameters, incorporating fractional derivatives in the Caputo sense. We compute the basic reproduction number by the next-generation matrix and establish local and global stability conditions based on this reproduction number. By using the fractional Adams–Bashforth method, we validate dynamical behaviors at different equilibrium points in both autonomous and non-autonomous scenarios, while qualitatively analyze the effects of fractional order on the dynamics. To effectively address the inverse problem of the proposed fractional SEIHDR model, we construct a fractional Physics-Informed Neural Network framework to simultaneously infer time-dependent parameters, fractional orders, and state components. Graphical results based on the COVID-19 pandemic data from Canada demonstrate the effectiveness of the proposed framework.

Fractional derivative modeling of heat transfer and fluid flow around a contracting permeable infinite cylinder: Computational study

Understanding the complex interplay between the contracting behavior of the cylinder and the fluid flow dynamics has implications for the design of porous structures for heat exchange and filtration systems. In this study, we investigate the dynamics and thermal behavior of fluid flow past a contracting permeable infinite cylinder. First, we developed a mathematical model based on the Navier–Stokes equations to describe the fluid dynamics around the contracting permeable infinite cylinder. A new simple well-behaved definition of fractional derivative called conformal fractional derivative introduced by authors Khalil et al., (2014) is employed to generalize the PDE’s of momentum and energy. The similarity transformation technique is utilized to transform the proposed mathematical problem into a non-linear ODE’s. Then, we utilized IPS technique to numerically solve the governing equations and obtain detailed insights into the flow patterns and thermal characteristics. We observed significant changes in the flow velocity and temperature distribution as the cylinder contracted, highlighting the intricate thermal behavior of the system. Additionally, the changes in the fraction order was found to have a notable impact on the overall flow patterns and heat transfer processes.

Fractional derivative modeling for dynamic response of layered viscoelastic media under moving time-harmonic loads

The dynamic response of layered viscoelastic media under moving time-harmonic loads is an important issue in transportation geotechnics. However, the existing research often has limitations in terms of the constant creep rate or time-independent relaxation modulus. To fill this gap, this study introduces a fractional Zener model to simulate the viscoelastic properties of media. First, the governing equations of the elastic medium are converted into ordinary differential equations using an integral transform. Subsequently, by introducing the fractional viscoelastic model and extended precise integration method, a solution to the dynamic response of layered viscoelastic media under moving time-harmonic loads is obtained. After comparing the results with those in existing literature, the reliability and accuracy of the proposed solution were verified. Finally, a parametric analysis was performed to further explore the influence of load velocity and fractional order on the dynamic response of the viscoelastic media.

Characterization of multi-step creep behavior and fractional derivative modeling for a water-saturated coal

Characterization of multi-step creep behavior and fractional derivative modeling for a water-saturated coal

Viscoelastic Characteristics in Mouse Model of Hepatic Steatosis With Inflammation by Kelvin-Voigt Fractional Derivative Modeling.

The aim of the work described here was to measure the characteristics of viscoelasticity and fluidity in a mouse model of hepatic steatosis and inflammation using a nano-indentation test and the Kelvin-Voigt fractional derivative (KVFD) model and to explore the viscoelasticity and fluidity characteristics in mice with different degrees of hepatic steatosis with inflammation. Twenty-five ApoE mice were randomly divided into an experimental high-fat diet group (n=15) and an ordinary-food control group (n=10), then subdivided into four subgroups based on pathological degree of hepatic steatosis: S0 (normal), S1 (mild), S2 (moderate) and S3 (severe). The 25 liver specimens from these mice were evaluated by a slope-keeping relaxation nano-indentation test. Elasticity (E0) was significantly higher in the S3 group than in the S1 and S2 groups, while fluidity (α) and viscosity (τ) were significantly lower in S3 than in S1 and S2 (all p values < 0.05). The following cutoff values for the diagnosis of hepatic steatosis >33% with inflammation were also determined: E0 > 85.01 Pa (area under the curve [AUC]: 0.917, 95% confidence interval [CI]: 0.735-0.989), α ≤ 0.38 (AUC: 0.885, 95% CI: 0.695-0.977),\ and τ ≤ 3.92 (AUC: 0.813, 95% CI: 0.607-0.939). Increases in the degree of hepatic steatosis with inflammation in mice paralleled gradual increases in the stiffness of the liver and gradual decreases in the fluidity and viscosity of the liver.

A fractional derivative modeling study for measles infection with double dose vaccination

This study proposes a novel mathematical framework to study the spread dynamics of diseases. A measles model is developed by dividing the vaccinated compartment into the vaccinated-with-the-first-dose and the-second-dose populations. The model parameters are estimated using the genetic algorithm for measles transmission based on monthly cumulative measles data in Indonesia. The threshold is determined to measure a population’s potential spread of measles. The stability of the equilibria is investigated, and the sensitivity analysis is then presented to find the most dominant parameter on the spread of measles. We extend the classical model into Atangana–Baleanu Caputo (ABC) derivative and consider the effects of the first and second vaccination doses on susceptible and exposed individuals. These outcomes are based on the different fractional parameter values and can be utilized to identify significant disease-control strategies. The yields are graphically exhibited to support our results. The overall finding suggests that taking preventive measures has a significant influence on limiting the spread of measles in the population. Increasing vaccination coverage, in particular, will reduce the number of infected individuals, thereby lowering the disease burden in the population.

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying.

The viscoelastic properties of food materials will change significantly while drying is in progress, which greatly influences the food deformation caused by drying. This study aims to predict the viscoelastic mechanical behavior of Hami melon during drying using the fractional derivative model. To characterize the relaxation characteristics, based on the finite difference method, an improved Grünwald-Letnikov fractional stress relaxation model is proposed to derive an approximate discrete numerical solution of the relaxation modulus by applying the time fractional calculus. The Laplace transform method is used to verify the obtained results, and the equivalence of the two methods is proved. In addition, the stress relaxation tests prove that the fractional derivative model has a better prediction effect on the stress relaxation behavior of viscoelastic food than classical Zener model. The significant correlations between the fractional order and the stiffness coefficient and the moisture content is also studied. Which is negative correlation and positive correlation respectively.

Study on viscoelastic materials at micro scale pondering supramolecular interaction impacts with DMA tests and fractional derivative modeling

AbstractViscoelastic materials are commonly utilized in seismic vibration suppression of building structures. Supramolecular modification can extend the effective damping temperature range and improve the damping performance of viscoelastic materials. In present work, the mechanical behaviors of the nitrile rubber viscoelastic material are investigated by dynamic thermomechanical analyzer (DMA) with different strain amplitudes, excitation frequencies and environmental temperatures. The supramolecular interaction and molecular structures modified fractional Kelvin (SMF Kelvin) model taking into account the temperature and frequency impacts is proposed and verified with the DMA test results of the nitrile rubber viscoelastic material. The test results of the viscoelastic damper with different temperatures are employed to further prove the effectiveness of the new proposed SMF Kelvin model.

Dynamics of a fractional order mathematical model for COVID-19 epidemic transmission

To achieve the aim of immediately halting spread of COVID-19 it is essential to know the dynamic behavior of the virus of intensive level of replication. Simply analyzing experimental data to learn about this disease consumes a lot of effort and cost. Mathematical models may be able to assist in this regard. Through integrating the mathematical frameworks with the accessible disease data it will be useful and outlay to comprehend the primary components involved in the spreading of COVID-19. There are so many techniques to formulate the impact of disease on the population mathematically, including deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional derivative modeling is one of the essential techniques for analyzing real-world issues and making accurate assessments of situations. In this paper, a fractional order epidemic model that represents the transmission of COVID-19 using seven compartments of population susceptible, exposed, infective, recovered, the quarantine population, recovered–exposed, and dead population is provided. The fractional order derivative is considered in the Caputo sense. In order to determine the epidemic forecast and persistence, we calculate the reproduction number R0. Applying fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied . Moreover, we implement the generalized Adams–Bashforth–Moulton method to get an approximate solution of the fractional-order COVID-19 model. Finally, numerical result and an outstanding graphic simulation are presented.

Non-Fickian transport of sodium chloride in inter-river land: experiment validation and fractional derivative modeling

Non-Fickian transport of sodium chloride in inter-river land: experiment validation and fractional derivative modeling

Fractional derivative modeling for sediment suspension in ice-covered channels.

Characterizing the anomalous diffusion behavior of sediment transport is a key factor in calculating sediment concentration. This study attempts to seek an equation that captures the nonlocal movement feature of the transport of an ensemble of sediment particles on ice-covered channels with a steady uniform flow field. Given that the fractional advection-dispersion equation with a noninteger order on the space term is nonlocal and able to describe the long-distance transport, a mathematical model with the Caputo fractional derivative is proposed to estimate the vertical diffusion of suspended sediment particles in the ice-covered channel. Results show that the fractional derivative model has a good predictive ability to the suspended sediment concentration as compared to the measurements. Especially in regions close to the undersurface of the ice cover, the proposed model matches better with experiments than the existing analytical model. Sensitivity analyses indicate that the strength of the turbulent diffusion effect dominates the uniformity of the sediment concentration profile. Besides, the sediment concentration is more sensitive to the variation of the boundary roughness than to the change of the sediment settling velocity. It should be noted that the sediment concentration reduces with the decrease of the order of the fractional derivative, which differs from the findings in previous studies.

A Nonlinear Triaxial Damage Creep Model for Granite Based on Atangana–Baleanu Fractional Derivative

Finding a fractional derivative that can more accurately describe the characteristics of a dynamic process is one of the objectives for fractional derivative modeling. In this paper, Atangana–Baleanu (AB) fractional order dashpots with constant and variable coefficients are constructed by using AB fractional derivatives with nonlocal and nonsingular kernel characteristics. Compared with Riemann–Liouville (RL) and conformable (KA) fractional order dashpot, AB fractional order dashpot can describe the viscoelastic properties of materials in full time. A new damage AB fractional derivative (DABFD) creep model is proposed by improving the Nishihara model with the help of the variable coefficient AB fractional dashpot. The parameters for the DABFD creep model are determined by fitting creep experimental data of Beishan granite. Compared with the Nishihara, damage RL fractional derivative (DRLFD) and damage KA fractional derivative (DKAFD) creep models, DABFD creep model has a better fitting and prediction accuracy. The parameter sensitivity for the DABFD creep model shows that different creep curves can be demonstrated with different DABFD creep model parameters. The model can better represent the viscoelastic properties and three stages of the creep process for rock.

Modeling the impact of the vaccine on the COVID-19 epidemic transmission via fractional derivative.

To achieve the goal of ceasing the spread of COVID-19 entirely it is essential to understand the dynamical behavior of the proliferation of the virus at an intense level. Studying this disease simply based on experimental analysis is very time consuming and expensive. Mathematical modeling might play a worthy role in this regard. By incorporating the mathematical frameworks with the available disease data it will be beneficial and economical to understand the key factors involved in the spread of COVID-19. As there are many vaccines available globally at present, henceforth, by including the effect of vaccination into the model will also support to understand the visible influence of the vaccine on the spread of COVID-19 virus. There are several ways to mathematically formulate the effect of disease on the population like deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional order derivative modeling is one of the fundamental methods to understand real-world problems and evaluate accurate situations. In this article, a fractional order epidemic model S_{p}E_{p}I_{p}Er_{p}R_{p}D_{p}Q_{p}V_p on the spread of COVID-19 is presented. S_{p}E_{p}I_{p}Er_{p}R_{p}D_{p}Q_{p}V_p consists of eight compartments of population namely susceptible, exposed, infective, recovered, the quarantine population, recovered-exposed, and dead population. The fractional order derivative is considered in the Caputo sense. For the prophecy and tenacity of the epidemic, we compute the reproduction number R_0. Using fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied. Furthermore, we are using the generalized Adams–Bashforth–Moulton method, to obtain the approximate solution of the fractional-order COVID-19 model. Finally, numerical results and illustrative graphic simulation are given. Our results suggest that to reduce the number of cases of COVID-19 we should reduce the contact rate of the people if the population is not fully vaccinated. However, to tackle the issue of reducing the social distancing and lock down, which have very negative impact on the economy as well as on the mental health of the people, it is much better to increase the vaccine rate and get the whole nation to be fully vaccinated.

The viscoelastic characteristics of in-vitro carotid plaque by Kelvin-Voigt fractional derivative modeling

Atherosclerotic plaque with a thin fibrous cap can be ruptured by shear force. Exploiting the mechanical properties of plaques within different histological regions can help to better understand the physical mechanisms of the plaque. The association between the plaque components and viscoelasticity was studied when mapping the viscoelasticity to histological features. Eleven in-vitro carotid plaques were tested with ramp-hold relaxation nanoindentation tests. Viscoelasticity (elastic modulus E0, fluidity α, and viscosity τ) was characterized by Kelvin-Voigt fractional derivative (KVFD) modeling. There is a significant difference (p < 0.001) on E0, α, and τ between the collagen-rich (CR) group and the non-collagen-rich (NCR) group. In the CR group, the elastic modulus E0 was higher but the fluidity α and viscosity τ were lower than those of the NCR group. Receiver operating characteristic (ROC) analysis revealed that combinations of E0 and α can be used as a CR indicator with an area under the curve (AUC) of 0.770. There was a negative correlation between E0 and the percentages of myxoid degeneration (r = −0.160, p < 0.001), necrosis (r = −0.229, p < 0.001) and inflammatory cells (r = −0.130, p < 0.001), and a positive correlation between elasticity E0 and the percentage of foam cells (r = 0.121, p < 0.001). There was a positive correlation between fluidity α and the percentage of necrosis (r = 0.308, p < 0.001). The results confirmed the clinical evidence that the CR group with higher elasticity and lower fluidity has higher resisting ability, whereas the NCR group with lower elasticity and higher fluidity has accompanied with more myxoid degeneration, extracellular lipids and necrosis.

Caputo fractional order derivative model of Zika virus transmission dynamics

The Zika Virus (ZIKV) is a highly contagious disease, and several outbreaks have occurred since it emerged. It is transmitted from one to another human via a mosquito Aedes aegypti. There is no vaccine or established medicine available for ZIKV to date. There is an urgent need to enhance an understanding of the progression mechanism of the disease when drugs or vaccines are not available. Mathematical modeling is a tool that might be helpful to understand the progression dynamics of ZIKV which can enable us to make control strategies for invading the progression dynamics of disease. SEIR-SEI is a famous compartmental deterministic modeling based on integer-order derivative calculus. Nowadays, conversion from integer to fractional order-based derivative modeling is in trend, and it is a very effective and high degree of accuracy. In this paper, we proposed a Caputo fractional-order based susceptible-exposed-infected-recovered (SEIR) structure for hosts and a susceptible-exposed-infected (SEI) structure for mosquitoes for transmission dynamics of ZIKV. For this purpose, we modified the classical compartmental model used in the study of progression dynamics of the Zika fever outbreak in El-Salvador during 2015-16. The modified model involves nonlinear differential equations of fractional (non-integer) order which has an advantage over the classical model due to its memory effect property. Our study includes eight regions across the globe where the Zika outbreak has occurred during the year 2013-2016 including six major archipelagos of French Polynesia, i.e., Tahiti, Sous-le-vent, Moorea, Tuamotu, Marquises, and Australes. The other two regions included Costa Rica and Colombia. The outbreak in selected regions was studied first using a classical model and then compared by a fractional-order model. The data of outbreaks are best fitted with the fractional-order model which enables us to estimate the best parameters values for the outbreaks. Using this modeling, the epidemic threshold parameter $R_0$ was computed which is more accurate than the classical one. Hence, the fractional-order model for ZIKV transmission dynamics is much better prediction, analysis, and disease parameters estimation than the classical model. This modeling enhances the knowledge in the field of fractional order and understanding the ZIKV transmission accurately.

Transcranial Ultrasound Estimation of Viscoelasticity and Fluidity of the Soft Matter

In the fields of geology, materials and biomedical engineering, the noninvasive and accurate estimation of the dynamic mechanical properties of the soft matter enclosed in a rigid shell induced by a low-frequency vibration is worthy of exploration. The addition of fluidity on the basis of viscoelasticity to describe the properties of the soft matter, especially when containing a large amount of water, may be of great value in the detection of crustal movement, material structure, and early cerebral diseases. In the biomedical field, since the influence of the skull on ultrasound is unknown for viewing the propagation of the transcranial shear wave, it is challenging to obtain the mechanical properties of the brain tissue with the skull using the transcranial ultrasound. In this study, the propagations of the transcranial shear wave within the brain tissue induced by an external low-frequency vibration are presented by finite-element-method simulation. This experiment achieved a transcranial ultrasound estimation and differentiation of viscoelasticity and fluidity of brain phantoms enclosed in the skull using the low-frequency vibration and Kelvin-Voigt fractional derivative modeling, and the estimation results are consistent with the nontranscranial ones. These results represent a great potential in the estimation of viscoelasticity and fluidity of the soft matter under different boundary constraints in various areas.

New version of optical spherical electric and magnetic flow phase with some fractional solutions in [formula omitted

In this paper, we introduce a new characterization of spherical electric and magnetic flow density of magnetic spherical Heisenberg SHα particle by spherical frame in spherical Heisenberg space SH32. Thus, we get some new optical qualifications of magnetic spherical Heisenberg SHα electric and magnetic phase by using spherical magnetic fields. Also, we demonstrate spherical electric and magnetic flow density for spherical vector fields. Then, their fractional solutions are examined by using magnetic and optical geometric quantities with the conformable fractional derivative modeling. Finally, magnetic spherical HeisenbergSHα electric and magnetic phase of spherical surface are demonstrated in uniform spherical magnetic surface by using analytical and fractional numerical developments.

Fractional derivative modeling of time-varying viscosity materials considering initial loading ramp in real experiments

The fractional derivative models with time-varying viscosity have been used in characterizing creep or relaxation properties of different viscoelastic material, and many combination models were presented using the Boltzmann superposition principle. However, those models defined as initial ones in this manuscript usually ignored the initial loading ramp, and the ideal-loading condition is commonly assumed as a step function in modeling. The real-loading conditions of tested samples are usually a ramp load followed by constant stress or strain. The difference in loading conditions between the theoretical modeling and experimental procedure strongly influences the models’ rheological property characterization and parameter determination. It is especially the case for the fractional derivative model due to its memory or history-dependent characters, even though the ramp time is short compared with the total experimental time. An application example of the Maxwell model with time-varying viscosity Scott–Blair model (TVSM) shows that the initial loading ramp has a strong influence. To solve this problem, the authors propose modified models of TVSM based on real-loading conditions. The relative errors between initial and modified models are presented. In addition, a history-dependent optimization algorithm for parameter determination is proposed. Three sets of polymer experimental data are employed to suggest that the fitting results of models disregarding initial ramp loads are unreliable. The modified model should be used for characterizing rheological behavior, as this leads to obtaining the best fitting results even for a short experimental time.

The dynamics of COVID-19 in the UAE based on fractional derivative modeling using Riesz wavelets simulation

The well-known novel virus (COVID-19) is a new strain of coronavirus family, declared by the World Health Organization (WHO) as a dangerous epidemic. More than 3.5 million positive cases and 250 thousand deaths (up to May 5, 2020) caused by COVID-19 and has affected more than 280 countries over the world. Therefore studying the prediction of this virus spreading in further attracts a major public attention. In the Arab Emirates (UAE), up to the same date, there are 14,730 positive cases and 137 deaths according to national authorities. In this work, we study a dynamical model based on the fractional derivatives of nonlinear equations that describe the outbreak of COVID-19 according to the available infection data announced and approved by the national committee in the press. We simulate the available total cases reported based on Riesz wavelets generated by some refinable functions, namely the smoothed pseudosplines of types I and II with high vanishing moments. Based on these data, we also consider the formulation of the pandemic model using the Caputo fractional derivative. Then we numerically solve the nonlinear system that describes the dynamics of COVID-19 with given resources based on the collocation Riesz wavelet system constructed. We present graphical illustrations of the numerical solutions with parameters of the model handled under different situations. We anticipate that these results will contribute to the ongoing research to reduce the spreading of the virus and infection cases.

Stochastic Response Analysis and Reliability-Based Design Optimization of Nonlinear Electromechanical Energy Harvesters With Fractional Derivative Elements

Abstract A methodology based on the Wiener path integral (WPI) technique is developed for stochastic response determination and reliability-based design optimization of a class of nonlinear electromechanical energy harvesters endowed with fractional derivative elements. In this regard, first, the WPI technique is appropriately adapted and enhanced to account both for the singular diffusion matrix and for the fractional derivative modeling of the capacitance in the coupled electromechanical governing equations. Next, a reliability-based design optimization problem is formulated and solved, in conjunction with the WPI technique, for determining the optimal parameters of the harvester. It is noted that the herein proposed definition of the failure probability constraint is particularly suitable for harvester configurations subject to space limitations. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the satisfactory performance of the methodology.