Numerical Investigation of Fractional Order Buruli Ulcer Model

Nonlinear vibration analysis of a fractional dynamic model for the viscoelastic pipe conveying fluid

This paper proposes a state of charge (SOC) estimator of Lithium-ion battery based on a fractional order impedance spectra model. Firstly, a battery fractional order impedance model is derived on the grounds of the characteristics of Warburg element and constant phase element (CPE) over a wide range of frequency domain. Secondly, a frequency fitting method and parameter identification algorithm based on output error are presented to identify parameters of the fractional order model of Lithium-ion battery. Finally, the fractional order Kalman filter approach is introduced to estimate the SOC of the lithium-ion battery based on the fractional order model. The simulation results show that the fractional-order model can ensure an acceptable accuracy of the SOC estimation, and the error of estimation reaches maximally up to 0.5% SOC.

In this paper, we propose a nonlinear fractional order model in order to explain and understand the outbreaks of influenza A(H1N1). In the fractional model, the next state depends not only upon its current state but also upon all of its historical states. Thus, the fractional model is more general than the classical epidemic models. In order to deal with the fractional derivatives of the model, we rely on the Caputo operator and on the Grünwald–Letnikov method to numerically approximate the fractional derivatives. We conclude that the nonlinear fractional order epidemic model is well suited to provide numerical results that agree very well with real data of influenza A(H1N1) at the level population. In addition, the proposed model can provide useful information for the understanding, prediction, and control of the transmission of different epidemics worldwide. Copyright © 2013 John Wiley & Sons, Ltd.

On selection of improved fractional model and control of different systems with experimental validation

This paper deals with a fractional order state space model for the lithium-ion battery and its time domain system identification method. Currently the equivalent circuit models are the most popular model which was frequently used to simulate the performance of the battery. But as we know, the equivalent circuit model is based on the integer differential equations, and the accuracy is limited. And the real processes are usually of fractional order as opposed to the ideal integral order models. So here we propose a lithium-ion battery fractional order state space model, and compare it with the equivalent circuit models, to see which model fit with the experiment results best. Then the hybrid pulse power characterization (HPPC) test has been implemented in the lithium-ion battery during varied state-of-charge (SOC). Based on the Levenberg–Marquardt algorithm, the parameters for each model have been obtained using the time-domain test data. Experimental results show that the proposed lithium-ion fractional order state space model has a better fitness than the classical equivalent circuit models. Meanwhile, five other cycles are adopt here to validate the prediction error of the two models, and final results indicate that the fractional model has better generalization ability.

Fractional growth model of abalone length

The paper presents the application of fractional viscoelastic models to characterize viscoleastic properties of asphalt concrete. This implies the replacement of integer order derivatives in the constitutive equations with fractional derivatives. Integer order stress and strain derivatives lead to exponential relaxation and typically a large number of Maxwell or Kelvin elements are needed to characterize the full viscoelastic response range. In each case, the representation is not unique and the parameters cannot be linked to the composition of asphalt concrete. Fractional models lead to non-exponential relaxation making it possible to characterize the full viscoelastic response range with a small number of elements (typically 1 or 2). As such, the representation is unique and can be linked to the composition of asphalt concrete. Fractional models can also be used to construct the dynamic modulus master curve. As Witczak’s sigmoidal model is a simple curve fitting it has no real physical meaning. Fractional models on the other hand have physical meaning, uniquely define the creep compliance and relaxation modulus, and allow better analysis of the physics of the relaxation process by considering the storage modulus, the loss modulus, and the phase angle. Using fractional models to analyze experimental asphalt concrete dynamic modulus results suggested two distinct relaxation processes; one at low temperatures and another at high temperatures. A possible explanation for this behavior can be attributed to the composition of asphalt concrete; at low temperatures, the binder behaves as a viscoelastic solid with aggregate particles more or less securely tied to the binder. The relaxation process is therefore restricted to the binder. At high temperatures, the binder behaves more as a viscoelastic fluid allowing aggregate particles to slide past each other which introduces another aspect of the relaxation process.

COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained mathcal{R}_{0} approx 1.50. Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

Buruli ulcer is an increasingly common tropical disease that has been even ignored in advanced nations like Australia and Japan. Some mammals, including possums, have shown symptoms of the disease. Fractional derivatives are applied for a better understanding of biological processes and their crossover behavior. Infectious disease can be controlled by predicting the future spread while using Mathematical Models. Recently, various mathematical models based on classical integer-order and non-integer-order models have been proposed to predict the dynamics of infectious disease outbreaks. As a consequence, it is investigated the dynamics of the possum model using non-integer order derivatives in order to gain a better understanding and deeper insight into several biological models. The model’s important properties, such as the positivity of the model solution, equilibrium points, and the invariant property of the proposed model have also been discussed. The next-generation approach is used to compute the basic reproductive number 0 . Both the stability i.e. the local and global are obtained when 0 is less or greater than 1. The global stability of the fractional Possum model is achieved using the Lyapunov function in a fractional environment. Furthermore, The existence and uniqueness of the fractional order models are demonstrated. Numerical simulations of the model are done, along with their graphical representations, to examine the effects of arbitrary order derivatives and depict the implications of our theoretical results. It can be seen from the graphical findings that the fractional model provides more clarity and a better understanding.

A fractional order model that describes thermal behavior in the greenhouse was performed. Based on a good description of the soil model a fractional order greenhouse model is obtained, this model take account the change of the thermal conductivity in the soil. To make the fractional model, firstly we use the Laplace transform to remove the time terms from the soil heat conduction equation and from the energy equation for each greenhouse elements, then we resolve the obtained system of linear equations by the Gaussian elimination algorithm, to finally obtain the matrix transfer function that describes the dependency between the climatic conditions, the soil temperatures and the internal greenhouse temperature. The performances of this fractional model are illustrated by a base of experimental data.

New description of the mechanical creep response of rocks by fractional derivative theory

We develop rheological representations, i.e., discrete spectrum models, for the fractional derivative viscoelastic element (fractional dashpot or springpot). Our representations are generalized Maxwell models or series of Kelvin-Voigt units, which, however, maintain the number of parameters of the corresponding fractional order model. Accordingly, the number of parameters of the rheological representation is independent of the number of rheological units. We prove that the representations converge to the corresponding fractional model in the limit as the number of units tends to infinity. The representations extend to compound fractional derivative models such as the fractional Maxwell model, fractional Kelvin-Voigt model, and fractional standard linear solid. Computational experiments show that the rheological representations are accurate approximations of the fractional order models even for a small number of units.

We introduce a data-driven fractional modeling framework for complex materials, and particularly bio-tissues. From multi-step relaxation experiments of distinct anatomical locations of porcine urinary bladder, we identify an anomalous relaxation character, with two power-law-like behaviors for short/long long times, and nonlinearity for strains greater than 25%. The first component of our framework is an existence study, to determine admissible fractional viscoelastic models that qualitatively describe linear relaxation. After the linear viscoelastic model is selected, the second stage adds large-strain effects to the framework through a fractional quasi-linear viscoelastic approach for the nonlinear elastic response of the bio-tissue of interest. From single-step relaxation data of the urinary bladder, a fractional Maxwell model captures both short/long-term behaviors with two fractional orders, being the most suitable model for small strains at the first stage. For the second stage, multi-step relaxation data under large strains were employed to calibrate a four-parameter fractional quasi-linear viscoelastic model, that combines a Scott-Blair relaxation function and an exponential instantaneous stress response, to describe the elastin/collagen phases of bladder rheology. Our obtained results demonstrate that the employed fractional quasi-linear model, with a single fractional order in the range α = 0.25–0.30, is suitable for the porcine urinary bladder, producing errors below 2% without need for recalibration over subsequent applied strains. We conclude that fractional models are attractive tools to capture the bladder tissue behavior under small-to-large strains and multiple time scales, therefore being potential alternatives to describe multiple stages of bladder functionality.

International audience

Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point.