Modeling of bio-heat transfers in lungs with fractional models

LPV continuous fractional modeling applied to ultracapacitor impedance identification

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

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

In this manuscript, a new approach to study the fractionalized Oldroyd-B fluid flow based on the fundamental symmetry is described by critically examining the Prabhakar fractional derivative near an infinitely vertical plate, wall slip condition on temperature along with Newtonian heating effects and constant concentration. The phenomenon has been described in forms of partial differential equations along with heat and mass transportation effect taken into account. The Prabhakar fractional operator which was recently introduced is used in this work together with generalized Fick’s and Fourier’s law. The fractional model is transfromed into a non-dimentional form by using some suitable quantities and the symmetry of fluid flow is analyzed. The non-dimensional developed fractional model for momentum, thermal and diffusion equations based on Prabhakar fractional operator has been solved analytically via Laplace transformation method and calculated solutions expressed in terms of Mittag-Leffler special functions. Graphical demonstrations are made to characterize the physical behavior of different parameters and significance of such system parameters over the momentum, concentration and energy profiles. Moreover, to validate our current results, some limiting models such as fractional and classical fluid models for Maxwell and Newtonian are recovered, in the presence of with/without slip boundary wall conditions. Further, it is observed from the graphs the velocity curves for classical fluid models are relatively higher than fractional fluid models. A comparative analysis between fractional and classical models depicts that the Prabhakar fractional model explains the memory effects more adequately.

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.

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.

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.

As recently shown, a fractional model can be viewed as a doubly infinite model: its “real state” is of infinite dimension as it is distributed, but it is distributed on an infinite domain. It is shown in the paper, that this last feature induces a physically inconsistent property: the model real state has the ability to store an infinite amount of energy. This property demonstration is based on an electrical interpretation of fractional models. As a consequence, even if fractional models permit to capture accurately the input-output dynamical behavior of many physical systems, such a property highlights a physical inconsistence of fractional models: they do not reflect the reality of macroscopic physical systems in terms of energy storage ability. This property is shown for implicit fractional models and extends previous result of the authors for explicit fractional models.

Many constitutive relationships have been derived to model the viscoelastic behaviors of materials, but limited works are done to point out which model is the most suitable one in certain conditions. In this article, we present the detailed comparisons of the classic rheological and fractional derivative models to make up for the deficiency. First, creep and stress relaxation tests of carbon black (CB)–filled rubber are carried out at the room temperature for 12 h, and frequency-sweep tests are conducted at a temperature ranging from Tg − 10°C to Tg + 100°C under tension loading. The master curve of storage modulus over a wide range of frequency covering about 21 decades is constructed according to the time/frequency–temperature superposition (TTS/fTS). And then the classic models (Maxwell, Kelvin, and their generalized models) and fractional derivative models (4, 5, 6, and 10 parameters Zener models) are used to fit the test data. The fitting results show that the classic generalized models and fractional Zener models are quantitatively equivalent. The classic generalized models with several parameters are much simpler and are able to predict the viscoelastic behaviors for short-term or narrow frequency conditions, while the fractional derivative models are suitable for wide frequency conditions.

A Leslie population model is an interesting mathematical discrete-time system because of its significant and wide applications in biology and ecology. In this paper, we extensively studied a fractional Leslie population model in the fractional μ-Caputo sense. For the different fractional order value and system parameters, the dynamics of the fractional population model are studied. It is verified that the new fractional population model undergoes doubling route to chaos and Neimark-Sacker bifurcation. Moreover, The dynamic of this model is experimentally investigated via bifurcation diagrams, phase portraits, largest Lyapunov exponent. Furthermore, the chaotic dynamic of the proposed population model is confirmed using a 0-1 test method. Simulation results reveal that chaos can be observed in such fractional model and its dynamic behavior depends on the fractional order value.

This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment is a significant research area with many mathematical models developed by mathematicians to represent the cancer treatment processes like hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives and the use of fractional derivatives is still new to many mathematicians. Therefore, it is imperative to review fractional cancer treatment models. The review was done by first presenting 22 various definitions of fractional derivative. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the used fractional derivative models were analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivative for cancer treatment modeling. Moreover, the fractional derivative is a general definition of all derivatives. Also, the fractional models can be applied to different cancer treatment procedures and the most used fractional derivative is the Caputo as well as its non-singular kernel versions. Finally, it was concluded that the future direction for cancer treatment modeling is the adoption of fractional derivative models corroborated with experimental or clinical data.

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.

مطالعه حاضر با هدف بررسی نحوه اثرگذاری متغیرهای آب و هوایی شامل دما، بارش، سرعت باد و رطوبت بر سهم سطح زیرکشت انواع محصولات سالانه زراعی شامل غلات، حبوبات، سبزیجات، محصولات جالیزی، محصولات علوفه ای و محصولات صنعتی در ایران صورت گرفت. در این راستا با استفاده از اطلاعات زراعی و هواشناسی 336 شهرستان کشور در دوره زمانی 92-1391 اقدام به برآورد مدل لاجیت چندگانه کسری گردید. نتایج مطالعه نشان داد افزایش دما سهم سطح زیرکشت غلات و محصولات جالیزی را افزایش و سهم سطح زیرکشت حبوبات را کاهش می-دهد. لذا با توجه به پیش بینی های صورت گرفته در مورد افزایش دما در سال های آتی، انتظار بر این است که میزان کشت غلات افزایش و میزان کشت حبوبات کاهش یابد. بارش متغیر دیگری است که با افزایش آن سهم سطح زیرکشت غلات افزایش و سهم سایر انواع محصولات کاهش می-یابد. درصد رطوبت بر سهم سطح زیرکشت سبزیجات و محصولات صنعتی و سرعت باد نیز بر سهم سطح زیرکشت محصولات صنعتی و غلات موثر می باشد. از این رو توصیه می گردد نحوه واکنش تولیدکنندگان محصولات زراعی سالانه به تغییرات آب و هوایی تحت سناریوهای گوناگون پیش بینی و با مقایسه مقدار تولید بالقوه با نیازهای غذایی جامعه در آینده و تعیین شکاف های موجود، مبنای سیاست گذاری های لازم در این زمینه فراهم شود. همچنین با توجه به اینکه مطالعه حاضر تنها تخصیص زمین بین انواع محصولات سالانه زراعی را مدنظر قرار داده است، توصیه می-گردد مطالعات دیگری نیز در زمینه بررسی نحوه اثرگذاری تغییرات آب و هوایی بر تولیدات سایر بخش های کشاورزی از قبیل محصولات باغی و دامی صورت گیرد.

Analysis of novel fractional COVID-19 model with real-life data application

Very recently, researchers dealing with constitutive law pertinent viscoelastic materials put forward the successful idea to introduce viscoelastic laws embedded with fractional calculus, relating the stress function to a real order derivative of the strain function. The latter consideration leads to represent both, relaxation and creep functions, through a power law function. In literature there are many papers in which the best fitting of the peculiar viscoelastic functions using a fractional model is performed. However there are not present studies about best fitting of relaxation function and/or creep function of materials that exhibit a non-linear viscoelastic behavior, as polymer melts, using a fractional model. In this paper the authors propose an advanced model for capturing the non-linear trend of the shear viscosity of polymer melts as function of the shear rate. Results obtained with the fractional model are compared with those obtained using a classical model which involves classical Maxwell elements. The comparison between experimental data and the theoretical model shows a good agreement, emphasizing that fractional model is proper for studying viscoelasticity, even if the material exhibits a non-linear behavior.