Fractional Differential Model Research Articles

Empirical analysis of fractional differential equations model for relationship between enterprise management and financial performance

Abstract With the development of science and technology, people also find that fractional-order models are more accurate than integer-order models in describing some phenomena and reflecting some properties of objects when studying practical problems. Since the fractional derivative is a quasi-differential operator, its memory-preserving non-locality, while beautifully portraying the real problem, also brings considerable difficulties to theoretical analysis and numerical calculation. This paper analyses the relationship between enterprise management and financial performance, analyses the mean and heterogeneity of the characteristics of enterprise management team, mathematically models its relationship, constructs fractional differential equations, and tests it through empirical research. The influence of the characteristics of enterprise management age, international experience, education level, team size and government background on the financial performance of the company.

Battery aging assessment and parametric study of lithium-ion batteries by means of a fractional differential model

An extensive parametric study of an alternative battery model, the fractional differential model (FDM), is presented for the first time over the entire lifetime of 20Ah NMC cells. The evolution of the FDM's model parameters, identified in the time-domain, is compared to electrochemical impedance spectroscopy results to assess their physico-chemical significance. This comprehensive parametric analysis shows that the FDM is able to capture relevant physico-chemical information with regards to the ohmic resistance over the entire SoC range and battery lifetime. Furthermore, the FDM's electrical performance is compared to the conventional first order RC battery model over the entire battery lifetime. Thanks to the FDM's non-integer derivatives for system states, it is able to capture intrinsic fractional derivative properties such as diffusion dynamics, charge transfer and memory hysteresis, which results in an improved simulation accuracy compared to the RC model. The comparative study showed an improved simulation accuracy of the FDM up to 85% in the lowest SoC range and a minimum improvement of 40% over the entire SoC range, considering the entire battery lifetime. Inherently proving that the FDM's ability to capture highly nonlinear battery behavior in the low SoC region persists over battery lifetime.

The Fractional Differential Model of HIV-1 Infection of CD4+ T-Cells with Description of the Effect of Antiviral Drug Treatment.

In this paper, the fractional-order differential model of HIV-1 infection of CD4+ T-cells with the effect of drug therapy has been introduced. There are three components: uninfected CD4+ T-cells, x, infected CD4+ T-cells, y, and density of virions in plasma, z. The aim is to gain numerical solution of this fractional-order HIV-1 model by Laplace Adomian decomposition method (LADM). The solution of the proposed model has been achieved in a series form. Moreover, to illustrate the ability and efficiency of the proposed approach, the solution will be compared with the solutions of some other numerical methods. The Caputo sense has been used for fractional derivatives.

A fractional differential equation model for bike share systems

A fractional differential equation model for bike share systems

Numerical Vibration Displacement Solutions of Fractional Drawing Self‐Excited Vibration Model Based on Fractional Legendre Functions

In practice, due to the fact that the phenomenon of drawing self‐excited vibration can be deemed as one of the hunting phenomena of the mechanical system, this study focuses on investigating the drawing self‐excited vibration process through proposing the fractional differential equation model of hunting phenomenon of the mechanical system. The fractional Legendre functions together with their fractional differential operational matrices are used to numerically solve the model. In this way, the numerical solutions of vibration displacement of the model are obtained. At the end, the proposed model and algorithm are proved to be effective via analyzing the numerical results and phase position.

Mathematical Epidemic Model of HIV/AIDS in Pakistan

In Pakistan the effect population mobility, specifically labor migration and refugees is also thought to have been important in explaining the rapid spread of HIV/AIDS. One of the effects labor migration is likely to have had increased the prevalence of the overlap of sexual partnership. A nonlinear fractional differential equation model is discussed for transmission and control of HIV/AIDS in Pakistan. We shall also discuss the disease free equilibrium and stability behavior of the model
 Bangladesh Journal of Medical Science Vol.18(1) 2019 p.14-23

Fractional Differential Model of Resonant Transfer of Excitations in Plasma

The most crucial feature of excitation transfer in low-temperature plasma forming the physical basis for operation of contemporary lighting devices is participation of resonant photons in the process. The widely used approximation of complete frequency redistribution assumes independence of the emitted photon frequency from frequency of the absorbed one. Averaged over frequency distribution in a resonance line, the natural transfer equation with the exponential kernel becomes the equation with an inverse power kernel, which in case of an infinite homogeneous medium is asymptotically equivalent to the Laplace equation of non-integer (fractional) order, considered in this article.

Response spectra generation using a fractional differential model

Over the past three decades, fractional calculus has been of increasingly interest to geotechnical researchers for studying time-dependent problems such as those that arise in Rheology. From previous investigations on Rheology (i.e., [1,2]) it has been shown that the accuracy requirements and the merits of hereditary phenomena with long memory make the fractional calculus an attractive mathematical tool for developing constitutive models. Although the advantages of this mathematical tool were mainly brought out by applying it to rheological problems, the authors are certain that fractional calculus can be used as alternative to integer models for evaluating design response spectra to carry out aseismic structural designs. Accordingly, this paper presents an alternative to generating response spectra of recorded or synthetic time histories using fractional differential equations (FDEs). As it is well known an acceleration response spectrum is the locus of the absolute value of maxima response accelerations of a single degree of freedom system (SDFS) for different natural periods and at a constant damping. A loci of response spectra is developed repeating the above mentioned computations for a set of different damping values. As broadly known, the equation of motion of the simple oscillator is obtained from the equilibrium of inertia, damping, elastic forces and dynamic excitation. This force equilibrium equation yields to a second order differential equation. In this paper only the first order derivative (velocity) of the second order differential equation of motion for the single degree of freedom system is considered fractional. Varying the order of the first order derivative (velocity) influences directly the damping capacity and tangentially the stiffness of the simple oscillator. Accordingly, its natural frequency and spectral ordinates of its response will be modified. As will be shown in the paper these modifications lead to more severe seismic conditions raising the question on the appropriateness of current design spectra specified in construction building codes and used in practice. Pondering this issue, the authors considered that it would be of great interest to use a fractional differential equation to model the simple damped oscillator and develop an alternative way to compute response spectra, particularly if these are more critical than those computed by currently used procedures. Thus, in this work a fractional differential equation (FDE) is used to generate response spectra of a number of recorded acceleration time histories. Furthermore, it is shown that the exponent α of the velocity term (dx/dt into dαx/dtα) of the FDE can be expressed in terms of the SDFS stiffness and damping parameters providing the exponent α physical meaning. Also, it is shown that the SDFS damping ratio is frequency dependent.

External force estimation of a piezo-actuated compliant mechanism based on a fractional order hysteresis model

Abstract Sensing of external forces applied on piezo-actuated compliant mechanisms plays a key role in state monitoring, process control and feedback strategy design in a variety of fields, including micro-/nanopositioning, micro-/nanomanipulation, and micro-/nanomanufacturing. In this paper, a force estimation strategy is proposed by comparing the practically measured displacement with the estimated free one without external forces, when subjecting to any given actuation voltages. To have an accurate prediction of the free displacement relating to the actuation voltage, an improved fractional order model is proposed. With this model, the system response is decomposed into a basic linear component and a nonlinear hysteresis component, and the extracted hysteresis is then described by an ordinary fractional order differential model using a modified voltage signal as the model input, which extracts a linear time-delay component from the original voltage. Comparative study with other two differential type hysteresis models are experimentally conducted on a piezo-actuated bridge-type compliant mechanism, demonstrating well the effectiveness and superiority of the proposed model for both system response modeling and external force estimation.

A study of the nanoindentation creep behavior of (La0.5Ce0.5)65Al10Co25 metallic glass based on fractional differential rheological model

A fractional differential rheological model was proposed to describe the nanoindentation creep behavior of metallic glasses (MGs) at room temperature. The model had been proved to not only include the conventional rheological model, but can be used to obtain the empirical formula that has been commonly used to characterize nanoindentation creep behavior of MGs. The reversible and irreversible parts of time-dependent deformations of (La0.5Ce0.5)65Al10Co25 metallic glass were obtained through nanoindentation creep experiments, and the results were used to investigate the effects of the creep load Pmax and the loading rate Ṗ on the creep behavior and the variations of the model parameters. The results indicate that the fractional differential rheological model can describe the creep deformation quite well. Parameters analysis shows that viscoelastic part of creep and β relaxation has many common points. The variation of irreversible viscoplastic part of creep with creep load is the main cause of indentation size effect. The load rate Ṗ has effect on all components of creep deformation, which is reflected by the variations of the rheological model parameters. The viscosity and retardation time are related to the loading history as indicators that characterize the system's instantaneous viscosity at the onset of creep. The fractional order and creep stress exponent varies with loading rate have similar pattern, which could reflect the flowing ability upon the completion of the creep experiment.

An integrated fractional partial differential equation and molecular dynamics model of anomalously diffusive transport in heterogeneous nano-pore structures

Fluid flows confined in nano-pore structures exhibit physical behaviors that are not observed in large-scale structures. Molecular dynamics simulation has been used for modeling nano-scale fluid flows at nanoscale, but is deemed to be computationally very expensive and is often limited to problems with simple geometry.We develop an integrated fractional partial differential equation and molecular dynamics upscaling modeling of anomalously diffusive transport in heterogeneous nano-pore structures, which has a significantly improved computational efficiency and memory requirement over the molecular dynamics simulation. Representative applications demonstrate the usage of the new model, which takes the model a few hours on a laptop that would otherwise require a molecular dynamics simulation in the matter of at least 100 years of CPU time on a parallel computer.

Features of Charging–Discharging of Supercapacitors

We present the results of experimental investigations of supercapacitors produced by Panasonic using several methods, namely, measurements of temporal dependences of charging–discharging currents, cyclic dc charging, and cyclic voltammetry. The values of the internal resistance, static and dynamic capacitance, as well as their dependences on the bias voltage of the capacitors, have been determined. Measurements have shown that the initial stage of relaxation of current is well approximated by the exponential time dependence, and then transition to a powerlike dependence occurs. This well-known effect is explained on the basis of the allowance for specific transport properties of a porous fractal-type medium. These processes are adequately described by the fractional-differential model of anomalous diffusion. A weak dependence of relaxation curves on the voltage at low values of the latter ( 3 V) is explained by the appearance of new percolation paths blocked at low charging voltages due to the presence of high-potential barriers. The internal resistance and the static capacitance have been determined by measuring the voltage across the supercapacitor in the mode of dc charging. These parameters have been shown to depend on the voltage applied to the capacitor. The dependence of the dynamic capacitance on the voltage has been determined using cyclic voltammetry. It has been shown that the capacitance depends not only on the voltage, but also on the prehistory of charging and discharging of the capacitor. Comparison of the experimental results and the published data on the models and equivalent circuits with passive R, L, and C elements allows one to conclude that such models and equivalent circuits can be applied only when explaining a limited number of phenomena, in particular, behavior at small relaxation times.

Spectrum behavior for the nonlinear fractional point reactor kinetics model

The nonlinear fractional point reactor kinetics equation in the presence of Newtonian temperature reactivity feedback with a multi-group of delayed neutrons, which describes the spectrum behavior of neutron density into the homogenous nuclear reactors, is developed. This system is one of the most important stiff coupled nonlinear fractional differentials for nuclear reactor dynamics. The generalization of Taylor’s formula that involves Caputo fractional derivatives is developed in an attempt to overcome the difficulty of the stiffness of the nonlinear fractional differential model. Moreover, the general fractional derivatives are calculated analytically throughout this work. Furthermore, the local and global estimated errors were analyzed, which suggest that the error quantification should take into account the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error (local truncation error). The neutron density response with time is analyzed for the anomalous diffusion, sub-diffusion, and super-diffusion processes.

The Adaptive Fractional Order Differential Model for Image Enhancement Based on Segmentation

For image enhancement method based on the fractional order differential, it is difficult to artificially give the optimal order of the fractional differential which can make the image enhancement effect better, and it is hard to ensure the enhancement of the target object while preserving the information of background pixels if the entire image is filtered by a fixed differential order. In order to solve this problem, the image is segmented into the object area and the background area according to the Otsu threshold algorithm based on Markov Random Field firstly. On the basis of the principle of the fractional differential for image enhancement, a piecewise function is established by combining with the different characteristics of pixels in each area, then the best order of fractional differential in the two areas can be determined adaptively. Thus, a novel adaptive fractional order differential algorithm for image enhancement on the basis of segmentation is put forward. Several fog-degraded traffic images are selected for experiments and processed by three other algorithms. The results of comparison exhibit the superiority of our algorithm.

Relation-theoretic metrical fixed-point results via $$\varvec{w}$$ w -distance with applications

In this article, utilizing the concept of w-distance, we prove the celebrated Banach’s fixed-point theorem in metric spaces equipped with an arbitrary binary relation. Necessarily, our findings unveil another direction of relation-theoretic metrical fixed-point theory. In addition, our paper consists of several non-trivial examples which signify the motivation of such investigations. Finally, our obtained results enable us to explore the existence of solutions of nonlinear fractional differential equations and fractional thermostat model involving the Caputo fractional derivative.

Identification of Fractional Differential Models for Lithium-ion Polymer Battery Dynamics

This paper presents a non-integer order (fractional) derivative model for modeling lithium-ion (Li-ion) battery dynamics in which direct continuous-time (CT) model identification methods are used to estimate the battery model parameters. In particular, the CT least squares-based state-variable filter (lssvf) identification method is extended to be used in fractional differential model identification of a battery system. The model performance and validation accuracy are evaluated on the basis of experimental data of a Li-ion polymer (LiPo) battery. It is shown how the proposed lssvf identification method can accurately capture the fractional order battery dynamics and exhibit better performance than integer order differential models.

A fractional differential fidelity-based PDE model for image denoising

In this paper, a new partial differential equation (PDE)-based model is proposed for image denoising. The new method is inspired by previous works in which the nonlinear diffusion approach obtained by using a coupling gradient fidelity term. Based on the long-term memory and nonlocality of fractional differential, we introduce a new fidelity term based on the combination of fractional-order fidelity term and global fidelity term to measure the similarity in the variation of images, which can prevent the staircase effect, and simultaneously enhance the noisy image, thus, the image becomes clearer and brighter. Numerical results are presented in the end to demonstrate that with respect to image denoising capability, our fractional fidelity-based model outperforms the gradient fidelity-based model.

Data-based fractional differential models for non-linear dynamic modeling of a lithium-ion battery

This paper presents a battery model with non-integer order derivatives for modeling the dynamics of a lithium-ion battery over a large operating range. The non-integer or fractional differential model includes a constant phase element term to approximate the non-linear dynamical behavior of the battery. The proposed fractional differential model is an amalgamation of electrochemical impedance spectroscopy experimental data and standard 1-resistor-capacitor electrical circuit model. The standard least squares-based state-variable filter identification method used for continuous-time system identification is used to estimate the model parameters and the fractional derivative coefficients of the proposed fractional differential model. For application of modeling fractional differential order battery dynamics, the continuous-time least squares-based state-variable filter parameter estimation approach is extended to an instrumental variable method to be robust to (non-white) noise perturbed output measurement. The model accuracy and model performance are validated on experimental data obtained from a lithium-ion battery and confirm that the proposed fractional differential model is able to accurately capture the battery dynamics over broad operating range. In comparison, the fractional differential model shows significant improvement on data prediction accuracy compared to a conventional integer model, making the fractional differential model suitable for monitoring battery dynamical behavior in a battery management system.

A fractional differential equation model for continuous glucose monitoring data

The main aim of this research was to test if fractional-order differential equation models could give better fits than integer-order models to continuous glucose monitoring (CGM) data from subjects with type 1 diabetes. In this research, real continuous glucose monitoring (CGM) data was analyzed by three mathematical models, namely, a deterministic first-order differential equation model, a stochastic first-order differential equation model with Brownian motion, and a deterministic fractional-order model. CGM data was analyzed to find optimal values of parameters by using ordinary least squares fitting or maximum likelihood estimation using a kernel-density approximation. Matlab and R programs have been developed for each model to find optimal values of the parameters to fit observed data and to test the usefulness of each model. The fractional-order model giving the best fit has been estimated for each subject. Although our results show that fractional-order models can give better fits to the data than integer-order models in some cases, it is clear that the models need further improvement before they can give satisfactory fits.

A variable-order fractional differential equation model of shape memory polymers

A shape-memory polymer (SMP) is capable of memorizing its original shape, and can acquire a temporary shape upon deformation and returns to its permanent shape in response to an external stimulus such as a temperature change. SMPs have been widely used industrial and medical applications. Previously, differential equation models were developed to describe SMPs and their applications. However, these models are often of very complicated form, which require accurate numerical simulations.In this paper we argue that a variable-order fractional differential equation model of the shape-memory behavior is more suitable than constant-order fractional differential equation models in terms of modeling the memory behavior of SMPs. We develop a numerical method to simulate the variable-order model and, in particular, to identify the unknown variable order of the model. Numerical experiments are presented to show the utility of the method.