Fractional Derivative Model Research Articles

Impact response of a viscoelastic plate made of a material with negative Poisson’s ratio1

In this article, the problem on impact response of a viscoelastic plate, made of viscoelastic auxetic with fractional viscosity, has been formulated for the case, when the shear operator is governed by the fractional derivative model, resulting in auxetic’s Poisson’s ratio as a time-dependent operator changing its values from negative to positive magnitudes with time. The proposed mathematical model and numerical calculations based on the obtained asymptotic solutions of the stated problem have verified existing experimental results, which demonstrate enhancing indentation and impact resistance in auxetic materials.

Study of Transmission Dynamics of Streptococcus suis Infection Mathematical Model between Pig and Human under ABC Fractional Order Derivative

In this paper, a mathematical model for Streptococcus suis infection is improved by using the fractional order derivative. The modified model also investigates the transmission between pigs and humans. The proposed model can classify the pig population density into four classes, which are pig susceptible class, pig infectious class, pig quarantine class, and pig recovery class. Moreover, the human population density has been separated into three classes, these are human susceptible class, human infectious class, and human recovery class. The spread of the infection is analyzed by considering the contact between humans and pigs. The basic reproduction number (R0), the infectious indicator, is carried out using the next generation matrix. The disease-free equilibrium is locally asymptotically stable if R0<1, and the endemic equilibrium is locally asymptotically stable if R0>1. The theoretical analyses of the fractional order derivative model, existence and uniqueness, have been proposed. The numerical examples were illustrated to support the proposed stability theorems. The results show that the fractional order derivative model provides the various possible solution trajectories with different fractional orders for the same parameters. In addition, transmission between pigs and humans resulted in the spread of Streptococcus suis infection.

Random forest-based modelling of parameters of fractional derivative concrete creep model with Bayesian optimization

Random forest-based modelling of parameters of fractional derivative concrete creep model with Bayesian optimization

Fractional derivative viscoelastic response of high-viscosity modified asphalt

To explain the dynamic viscoelastic characteristics of high-viscosity modified asphalt (HVMA) more comprehensively and accurately, one type of finished HVMA and two types of laboratory-produced HVMA were chosen as the research objects. Firstly, a dynamic shear rheometer was employed for the strain sweep test to define the linear viscoelastic range of three different types of HVMA. Then, the frequency sweep test was performed in the linear viscoelastic range under different high-temperature conditions, and the viscoelastic properties of three different types of HVMA were analysed. Finally, an HVMA fractional derivative viscoelastic model was built and compared to the traditional Sigmoid function model. The results demonstrated that the traditional Sigmoid function model has limitations and cannot effectively illustrate the changes in the HVMA over a wide frequency range. The viscoelastic properties of HVMA can be fully described by the 1S1A1D fractional derivative model, including the storage modulus, loss modulus, complex shear modulus, and phase angle variations. For the 1S1A1D fractional derivative model, one Abel dashpot can illustrate the viscoelastic properties of HVMA, and increasing the number of Abel dashpots no significance on the performance.

A novel viscoelastic modelling of railway track elements and experimental validation

This study presents a novel approach for the viscoelastic (VE) modelling of railway track elements that depends on the combination of strong sides of classical and fractional derivative models (FDM) of viscoelasticity. First, parameters of a recent 10 parameter fractional derivative model are identified from dynamic mechanical analysis (DMA) data for three different elastomeric pads that are used in railway system. The FDM gives accurate results in wide frequency ranges, however, the time domain representation of this model includes fractional derivatives that substantially increase the computation time. To overcome this drawback, five-armed Generalized Maxwell Model (GMM) parameters are fitted in 0.01–1000 Hz to be used in the time domain analyses. The mathematical models of two distinct types of railway superstructure are obtained and solved by the Galerkin’s method. The results of proposed framework are tested against measurements and the results of finite element model (FEM) analyses and very good agreement is observed.

Investigation of One-dimensional Consolidation of Fractional Derivative Model for Viscoelastic Saturated Soils Caused by the Groundwater Level Change

The Fractional calculus theory is introduced into the Merchant model to study the mechanical characteristics of saturated soils. Laplace transform is applied to the 1D consolidation and the fractional derivative Merchant model constitutive equations of saturated soils. In the transform domain, the analytical solutions of effective stress and settlement are obtained by solving the simultaneous equation. The Crump's method is used to the numerical inversion of Laplace transform, and the semi-analytical solution of the one-dimensional consolidation is obtained. A series of parameters, such as the fractional order, compression modulus, coefficient of viscosity, permeability coefficient and the variation models of groundwater level, are studied. Considering the continuity conditions, the 1D consolidation equation of single-layer saturated soil is generalized to multi-layer soils, and then the influence of different boundaries on several kinds of consolidation behavior was analyzed.

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 study on the thermoelastic interaction in two-dimension orthotropic materials under the fractional derivative model

The extended thermoelastic theory with fractional derivative is described in this article to estimate temperature variation, displacement components, and stress components in two-dimensional orthotropic materials. In the Laplace domain, exact solutions are found. The physical quantities are calculated analytically using Laplace and Fourier transforms and the eigenvalues approaches. The Laplace-Fourier transform inversion procedures are carried out using numerical approaches. The numerical outcome for all the physical quantities investigated are introduced and shown visually. In this case, if the fractional parameter is equal to one and the thermal relaxation time is equal to zero, then the Lord and Shulman (LS) theory comes into play and the classical dynamical coupled (CT) theory comes in. Also, the comparison between the orthotropic medium and isotropic medium are presented.

Vibration Analysis of a 1-DOF System Coupled with a Nonlinear Energy Sink with a Fractional Order Inerter.

The fluid inerter described by the fractional derivative model is introduced into the traditional nonlinear energy sink (NES), which is called fractional-order NES in this paper. The slowly varying dynamic equation (SVDE) of the system coupled with fractional-order NES is obtained by the complex averaging method, in which the fractional derivative term is treated using the fractional Leibniz theorem. Then, the discriminants (Δ, Δ1, and Δ2) of the number of equilibrium points are derived. By using the variable substitution method, the characteristic equation for judging the stability is established. The results show: (1) the approximate SVDE is sufficient to reflect the slowly varying characteristics of the system, which shows that the mathematical treatment of the fractional derivative term is reliable; (2) the discriminant conditions (Δ1, Δ2) can accurately reflect the number of equilibrium points, and the corresponding range of nonlinear parameter κ can be calculated when the system has three equilibrium points. The expressions of Δ1, Δ2 are simpler than Δ, which is suitable for analysis and design parameters; (3) the stability discrimination methods of schemes 1 and 2 are accurate. Compared with scheme 2, scheme 1 is more prone to various responses, especially various strongly and weakly modulated responses. In scheme 2, the inertia effect of mass can be completely replaced by integer order inerter. Compared with integer order inerter, the introduction of fractional order inerter, whether in series or in parallel, means that the amplitude of the equilibrium point on the NES vibrator is smaller, but it is also for this reason that it is not easy to produce a modulated response with scheme 2, and the vibration suppression effect of the main structure is not good.

Computer Virus Fractional Order Model with Effects of Internal and External Storage Media

In this work, we focus on the implementation of epidemic techniques on computer virus and study the dynamic transmission of several viruses to minimize the destruction of computers. We aim to make and analyze computer viruses through the Atangana-Baleanu sense and the Atangana-Taufik scheme, which is used for the fractional derivative model for the computer virusepidemic. It contained infected external computer effects and removable storage media on the computer viruses. For the validation of the model, we also discussed its positivity and boundedness. Fixed point theory and the iterative methods helped a lot to find out the existence and uniqueness of the model. In the case of numerical simulation, we used Atanagana-Taufik technique to illustratethe effects of varying the fractional order. The graphical results support our theoretical results from which, we analyze the infected external computer effects and removable storage media on the computer viruses.

Mechanical validation of viscoelastic parameters for different interface pressures using the Kelvin-Voigt fractional derivative model.

The knowledge of the biomechanical properties of tissues is useful for different applications such as disease diagnosis and treatment monitoring. Reverberant Shear Wave Elastography (RSWE) is an approach that has reduced the restrictions on wave generation to characterize the shear wave velocity over a range of frequencies. This approach is based on the generation of a reverberant field that is generated by the reflections of waves from inhomogeneities and tissue boundaries that exist in the tissue. The Kelvin-Voigt Fractional Derivative model is commonly used to characterize elasticity and viscosity of soft tissue when using shear wave ultrasound elatography. These viscoelastic characteristics can be then validated using mechanical measurements (MM) such as stress relaxation. During RSWE acquisition, the effect of interface pressure, induced by pushing the probe on the skin through the gel pad, on the viscous and elastic characteristics of tissue can be investigated. However, the effect of interface pressure on the validity of the extracted viscous and elastic characteristics was not investigated before. Therefore, the purpose of this study was to compare the estimation of the viscoelastic parameters at different thickness of gel pad against the viscoelastic characteristics obtained from MM. The experiments were conducted in a tissue-mimicking phantom. The results confirm that the relaxed elastic constant (μ0) can be depreciated. In addition, a higher congruence was found in the viscous parameter (ηα) estimated at 6 and 7 mm. On the other hand, a difference in the order of fractional derivative (α) was found.

Applicability of the Time Fractional Derivative Model on Fickian and Non-Fickian Transport in the Single Fracture: An Experimental Investigation

The understanding of solute transport in rock fractures is of great importance in many engineering fields. In this study, two groups of experiments on artificial and natural single fractures with different fracture apertures and roughness were conducted to investigate the changes of solute transport regimes. The time fractional advection–dispersion equation (tFADE) as a promising model was applied to describe the anomalous transport. The performance of the classical advection-dispersion equation (ADE) and tFADE was compared according to the fitting precision of breakthrough curves (BTCs). The responses of the fitting parameters in the tFADE to the experimental conditions were also discussed. The results indicated that the non-Fickian transport more likely occurred in the short distance transport, and the larger Peclet number ( Pe ) led to the increase of the exponent of the power-law function in the phase of concentration decline. The tFADE was superior to ADE in capturing the non-Fickian transport especially the tailing behavior. The fractional order of time α in the tFADE was the key parameter to describe the anomalous transport process, and its responding mechanisms of were revealed: the best-fit α decreased with the increase of flow velocity and the decrease of the fracture aperture. The roughness of the single fracture which leads to a complex flow field had a significant effect on the best-fit α . The findings of this study can help for better understanding the effectiveness and physical significance of the tFADE.

Vibration analysis of variable fractional viscoelastic plate based on shifted Chebyshev wavelets algorithm

In this paper, new and effective methods are provided for modeling and numerical simulation of viscoelastic plate, respectively. Viscoelastic plate is modeled by a variable fractional derivative model with better fitting effect and is numerically analyzed directly in time domain using shifted Chebyshev wavelets algorithm for the first time. A governing equation with three independent variables is established. The feasibility and accuracy of proposed algorithm are verified by convergence analysis and error estimations of mathematical example. Under different conditions, the ternary variable fractional differential equation of plate is solved numerically by shifted Chebyshev wavelets algorithm directly in time domain. The vibrations of plate are analyzed under different loads, loading times and temperatures. Numerical solutions for the stress of plate with two materials are also calculated and analyzed under the same load. The displacement of plate increases with increasing load, loading time and temperature. The bending resistance of PET plate is better than that of polyurea plate. These results are consistent with the existing references and actual practice. Therefore, the variable fractional model and shifted Chebyshev wavelets algorithm are accurate for the study of plate. They can be applied to complex problems in engineering. All obtained conclusions can provide theoretical basis for the protection and design of load-bearing structures in engineering.

Variable fractional modeling and vibration analysis of variable-thickness viscoelastic circular plate

In this paper, a variable fractional derivative model (VFDM) is used to model the viscoelastic circular plate whose thickness varies exponentially. Based on the VFDM, a ternary variable fractional differential equation of viscoelastic circular plate is established. The shifted Chebyshev polynomials (SCPs) algorithm is introduced to numerically solve the complex equation directly in time domain. The effectiveness of proposed SCPs algorithm is verified by convergence analysis. Then the dynamic responses of viscoelastic circular plate are researched under different loads, radius, thickness parameters, boundary conditions and viscoelastic materials by SCPs algorithm. The displacement of circular plate increases with the increasing load. The vibration can be reduced by increasing the thickness of plate, reducing plate area or strengthening the restraint on plate. The bending resistance of polyethylene terephthalate (PET) plate is better than polyurea. These results are consistent with the results of literatures. Therefore, the proposed model and algorithm are correct and sensitive in the numerical simulation of viscoelastic circular plate with different conditions.

A Core Damage Constitutive Model for the Time-Dependent Creep and Relaxation Behavior of Coal

The creep and stress relaxation behaviors of coal are common in coal mining. The unified constitutive model is suitable to describe and predict both the creep and relaxation evolution characteristics of rocks. The generalized Kelvin model is the core element for traditional and improved component models to reflect both the nonlinear creep and relaxation. In this paper, an improved core damage model, which could both reflect the creep and stress relaxation in relation to the damage evolution, was established based on a comparison of the traditional and improved component models, and the responding constitutive equations (creep and stress relaxation equation) at constant stress/strain were deduced. Then, the core damage model was validated to the uniaxial compressive multistage creep and stress relaxation test results of coal, showing that the model curves had great accordance with the experimental data. Moreover, the model comparisons on accuracy, parameter meaning, and popularization among the core damage model, hardening-damage model, and the fractional derivative model were further discussed. The results showed that the parameters in the core damage model had clear and brief physical significances. The core damage model was also popularized to depict the time-dependent behaviors of other rocks, showing great accuracy.

A discussion on nonlocality: From fractional derivative model to peridynamic model

Characterization of nonlocality is an open problem in physics and engineering. This paper conducts a detailed investigation on two nonlocal models, namely, the fractional derivative model and the peridynamic model for anomalous diffusion. A generalized nonlocal model combining the advantages of the fractional derivative model and the peridynamic model, is introduced. In this paper, analytical solutions and the mean squared displacements of the two models are provided and discussed. In addition, their intrinsic relations and notable differences are investigated. Preliminary applications indicate that the peridynamic model can well capture an unremarkable transition from normal-diffusion to super-diffusion, while the fractional derivative model presents super-diffusion behaviors in the whole process. At last, a generalized nonlocal operator is proposed as a more general strategy to solve nonlocal problems.

Anomalous transport for multispecies reactive system with first order decay: time-fractional model

The prediction of pollutant migration and its concentration variation in the subsurface hydrology is vitally requisite for the assessment and restorative treatment of polluted groundwater system. Pollutant migration for the multispecies reactive system cannot be reliably investigated by classical form of convection-dispersion equation (CDE), due to the presence of more than one reactive species. This paper establishes a time-fractional model for multispecies reactive system under the first order sequential reaction network to understand the anomalous or non-Fickian migration phenomenon for reactive pollutants. At present, most of the fractional models are presented for the classical CDE to investigate migration phenomenon for single species system, not for the multispecies reactive system due to the complexity of the modelled framework. The impact of fractional derivative model is analysed for variable dependent migration parameters and constant parameters, both for the multispecies reactive migration phenomenon. The fractional derivative is expressed in the Caputo sense and to handle the non-linearity of problem, Homotopy perturbation method (HPM) is adopted. The advantage of this method, to get the solutions, is that the HPM is independent of small parameters required for the deformation process as used in other existing perturbation techniques, which make it much more convenient to use for non-linear systems. The impact of the fractional derivative index and other migration parameters is graphically depicted for the reactive species and significant influence of fractional term is observed. The derived solutions are then validated by using the corresponding solutions obtained by other existing well-established methods to ensure the convergence of the HPM solutions. As there do not exist any solutions for such fractional models for multispecies reactive system, this study may be advantageous to convey better understanding for the anomalous or non-Fickian migration for reactive pollutants and their remediation planning in the groundwater resources.

Forced Vibration of Axially Accelerating three Parameter Beam Constituted by Fraction Alderivative Model

The forced vibration characteristics of axially variable speed viscoelastic beams are studied. The material is described by the three parameter model of Poynting-Thompson beam. According to Newton's second law, the fractional derivative is introduced to deduce the governing equation of the beam. The approximate analytical solution and amplitude frequency equation are obtained by multi-scale method. The amplitude frequency equation is divided into real part and imaginary part by separating variables. According to the Routh-Hurwitz criterion, the Jacobi matrix and characteristic equation are established to determine the stable region and unstable region obtained under different parameters. Numerical examples show that the instability region shows a downward trend with the increase of the order of fractional derivative.

Analysis and Simulation of Fractional Order Smoking Epidemic Model.

In recent years, there are many new definitions that were proposed related to fractional derivatives, and with the help of these definitions, mathematical models were established to overcome the various real-life problems. The true purpose of the current work is to develop and analyze Atangana-Baleanu (AB) with Mittag-Leffler kernel and Atangana-Toufik method (ATM) of fractional derivative model for the Smoking epidemic. Qualitative analysis has been made to `verify the steady state. Stability analysis has been made using self-mapping and Banach space as well as fractional system is analyzed locally and globally by using first derivative of Lyapunov. Also derive a unique solution for fractional-order model which is a new approach for such type of biological models. A few numerical simulations are done by using the given method of fractional order to explain and support the theoretical results.

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.