Fractional-order Differential Model Research Articles

Transient response of rectangular plate on viscoelastic foundation under time-variable load based on fractional-order differential model

Transient response of rectangular plate on viscoelastic foundation under time-variable load based on fractional-order differential model

Extended Hybrid Controller Design of Bifurcation in a Delayed Chemostat Model

Fractional-order differential models plays a pivotal role in depicting the relationship among concentration changes of various chemical substances in chemistry. In this current study, we will explore the dynamics of a delayed chemostat model. First of all, we prove that the solution of the delayed chemostat model exists and is unique by virtue of fixed point theorem. Secondly, we demonstrate that the solution of the delayed chemostat model is non-negative by applying some suitable inequality strategies. Thirdly, the boundedness of the solution to the delayed chemostat model is explored via constructing a reasonable function. Fourthly, the Hopf bifurcation and stability of the delayed chemostat model are dealt with by exploiting the stability criterion and bifurcation theory on fractional dynamical system. Fifthly, the stability domain and Hopf bifurcation of the delayed chemostat model are resoundingly controlled by making use of an extended hybrid controller. Sixthly, the stability domain and Hopf bifurcation of the delayed chemostat model are effectively adjusted by making use of an another extended hybrid controller. The role of delay in this chemostat model is revealed. Seventhly, software experiments are given to illustrate the rightness of the gained key conclusions. The acquired outcomes of this work are perfectly innovative and have crucial theoretical value in controlling the concentrations of various chemical substances.

Numerical Investigations of the Fractional-Order Mathematical Model Underlying Immune-Chemotherapeutic Treatment for Breast Cancer Using the Neural Networks

The aim of this work is to design a stochastic framework to solve the fractional-order differential model based on the breast cancer progression during the immune-chemotherapeutic treatment phase, including certain control parameters such as anti-cancer medications, ketogenic diet and immune boosters. The developed model considers tumor density progression throughout chemotherapy treatment, as well as an immune response during normal cell–tumor cell interaction. This study’s subject seems to be to demonstrate the implications and significance of the fractional-order breast cancer mathematical model. The goal of these studies is to improve accuracy in the breast cancer model by employing fractional derivatives. This study also includes an integer, nonlinear mathematical system with immune-chemotherapeutic treatment impacts. The mathematical system divides the fractional-order breast cancer mathematical model among four manifestations: normal cell population (N), tumor cells (T), immune response class (I), and estrogen compartment (E), i.e., (NTIE). The fractional-order NTIE mathematical system is still not published previously, nor has it ever been addressed employing the stochastic solvers’ strength. To solve a fractional-order NTIE mathematical system, stochastic solvers based on the Levenberg–Marquardt backpropagation scheme (LMBS) and neural networks (NNs), namely, LMBNNs, are been constructed. To solve the fractional-order NTIE mathematical model, three cases with varying values for this same fractional order have been supplied. The statistics used to offer the numerical solutions of the fractional-order NTIE mathematical model are divided as follows: 75% in training, 15% in testing, and 10% in the authorization. The acquired numerical findings were compared using the reference solutions to determine the accuracy of the LMBNNs using Adams–Bashforth–Moulton. The numerical performances employing error histograms (EHs), state transitions (STs), regression, correlation, including mean square error (MSE) have been further supplied to authenticate overall capability, competence, validity, consistency, as well as exactness of such LMBNNs.

Exponential stability and synchronisation of fuzzy Mittag–Leffler discrete-time Cohen–Grossberg neural networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper.

Non-fractional and fractional mathematical analysis and simulations for Q fever

The purpose of analysing the transmission dynamism of Q fever (Coxiellosis) in livestock and incorporating ticks is to outline some management practices to minimise the spread of the disease in livestock. Ticks pass coxiellosis from an infected to a susceptible animal through a bite. The faecal matter can also contain coxiellosis, thus contaminating the environment and spreading the disease. First, a nonlinear integer order mathematical model is developed to represent the spread of this infectious disease in livestock. The proposed integer model investigates the positivity and boundedness, disease equilibria, basic reproduction number, bifurcation, and sensitivity analysis. Through mathematical analysis and numerical simulations, it shows that if the environmental transmission and the effective shedding rate of coxiella burnetii into the environment by both asymptomatic and symptomatic livestock are zero, then the usual threshold hold and it produces forward bifurcation. It is noticed that an increase in the recruitment rate of ticks produces backward bifurcation. And also, it is seen that an increase in the natural decay rate of the bacterial in the environment reduces the backward bifurcation point. Furthermore, to take care of the memory aspect of ticks on their host, we modified the initially proposed integer order model by introducing Caputo, Caputo-Fabrizio, Atangana-Baleanu fractional differential operators. The existence and uniqueness of these three newly developed fractional-order differential models are shown using the Banach fixed point theorem. Numerical trajectories are obtained for each of the fractional-order mathematical models. The trajectory of some fractional orders converges to the same endemic equilibrium point as the integer order. Finally, it is seen that the Atangana-Baleanu fractional differential operator captures more susceptibilities and fewer infections than the other operators.

Hyperspectral Prediction of Soil Total Salt Content by Different Disturbance Degree under a Fractional-Order Differential Model with Differing Spectral Transformations

Soil salinization is an ecological challenge across the world. Particularly in arid and semi-arid regions where evaporation is rapid and rainfall is scarce, both primary soil salinization and secondary salinization due to human activity pose serious concerns. Soil is subject to various human disturbances in Xinjiang in this area. Samples with a depth of 0–10 cm from 90 soils were taken from three areas: a slightly disturbed area (Area A), a moderately disturbed area (Area B), and a severely disturbed area (Area C). In this study, we first calculated the hyperspectral reflectance of five spectra (R, R, 1/R, lgR, 1/lgR, or original, root mean square, reciprocal, logarithm, and reciprocal logarithm, respectively) using different fractional-order differential (FOD) models, then extracted the bands that passed the 0.01 significance level between spectra and total salt content, and finally proposed a partial least squares regression (PLSR) model based on the FOD of the significance level band (SLB). This proposed model (FOD-SLB-PLSR) is compared with the other three PLSR models to predict with precision the total salt content. The other three models are All-PLSR, FOD-All-PLSR, and IOD-SLB-PLSR, which respectively represent PLSR models based on all bands, all fractional-order differential bands, and significance level bands of the integral differential. The simulations show that: (1) The optimal model for predicting total salt content in Area A was the FOD-SLB-PLSR based on a 1.6 order 1/lgR, which provided good predictability of total salt content with a RPD (ratio of the performance to deviation) between 1.8 and 2.0. The optimal model for predicting total salt content in Area B was a FOD-SLB-PLSR based on a 1.7 order 1/R, which showed good predictability for total salt content with RPDs between 2.0 and 2.5. The optimal model for predicting total salt content in Area C was a FOD-SLB-PLSR based on a 1.8 order lgR, which also showed good predictability for total salt content with RPDs between 2.0 and 2.5. (2) Soils subject to various disturbance levels had optimal FOD-SLB-PLSR models located in the higher fractional order between 1.6 and 1.8. This indicates that higher-order FODs have a stronger ability to extract feature data from complex information. (3) The optimal FOD-SLB-PLSR model for each area was superior to the corresponding All-PSLR, FOD-All-PLSR, and IOD-SLB-PLSR models in predicting total salt content. The RPD value for the optimal FOD-SLB-PLSR model in each area compared to the best integral differential model showed an improvement of 9%, 45%, and 22% for Areas A, B, and C, respectively. It further showed that the fractional-order differential model provides superior prediction over the integral differential. (4) The RPD values that provided an optimal FOD-SLB-PLSR model for each area were: Area A (1.9061) < Area B (2.0761) < Area C (2.2892). This indicates that the prediction effect of data processed by fractional-order differential increases with human disturbance increases and results in a higher-precision model.

Effect of basset history drag on the accurate determination of streaming flow with PIV measurement

In acoustofluidics, the conflation of acoustic streaming and acoustic pressure effects on suspended particles, especially for particle image velocimetry (PIV) with micron-sized particles, leads to inaccurate streaming measurements. Typical multiphase streaming models fail to account for the Basset history drag, which emerges due to a mismatch between the momenta of the particle and the local flow. The history drag is proportional to a fractional-order derivative of the mismatch. Since this force integrates over time, its absence leads to significant modeling errors while also augmenting errors from other sources (e.g., particle-fluid density mismatch and acoustic radiation forcing). We investigate the effects of particle size selection on accurate streaming flow determination from PIV microacoustofluidic system measurements. We achieve this through experimental observation ofparticle behaviors under varied ultrasonic forcing frequencies (30–500MHz) and particle sizes. We pair the empirical study with an analytical fractional-order differential model of a spherical particle under the influence of radiation forces. Rigorous comparative analyses between the model and observations are undertaken to more completely characterize the magnitude of various effects leading to spurious PIV data. We expect the results of this investigation to clearly delineate optimal PIV particle size selection for the continued study of microacoustofluidic systems.

Dynamical analysis of a fractional-order foot-and-mouth disease model

In this paper, a fractional-order model that describes transmission and control of foot-and-mouth disease is presented and analyzed. The proposed model incorporates two foot-and-mouth disease intervention strategies: vaccination of susceptible animals and quarantine of clinically infected animals. Firstly, the global existence, positivity and boundedness of solutions for the proposed model were proved. This was followed by computation of the basic reproduction number. The basic reproduction number was then used to demonstrate the global stability of the model steady states. Theoretical results were validated by solving the proposed model in MATLAB software using the modified Adams–Bashforth–Moulton predictor corrector scheme. Among several other important results, numerical results demonstrate that coupling vaccination and animal isolation could be essential to attain effective management of foot-and-mouth disease for certain disease transmission levels. However, if disease transmission exceeds a certain threshold value, then coupling vaccination and quarantine may not be sufficient to effectively manage the disease. Furthermore, simulation results also demonstrated that the qualitative nature of the solutions of the classical integer model is the same as that of the fractional-order differential model. In addition, it was shown numerically that in the presence of quarantine alone, disease outbreaks may occur.

Pollination enthused residual optimization of some realistic nonlinear fractional order differential models

This work proposes a new trend for determining the numerical solutions of complex real life models by the profitable implementation of a nature inspired optimization technique, which is enthused by the pollination strategy of the flowering plants. The design methodology optimizes the determined residual fitness function achieved by the utilization of the generalized Taylor series for the deliberated fractional order differential model. Results of numerical experimentation achieved by the proposed pollination enthused residual optimization (PERO) technique are compared with some former numerical and metaheuristic techniques. Moreover, a detailed performance analysis is performed via statistical inference based on hundred independent runs.

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.

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.

Model-order reduction of lumped parameter systems via fractional calculus

This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.

FPGA implementation of fractional-order discrete memristor chaotic system and its commensurate and incommensurate synchronisations

A new fourth-order memristor chaotic oscillator is taken to investigate its fractional-order discrete synchronisation. The fractional-order differential model memristor system is transformed to its discrete model and the dynamic properties of the fractional-order discrete system are investigated. A new method for synchronising commensurate and incommensurate fractional discrete chaotic maps are proposed and validated. Numerical results are established to support the proposed methodologies. This method of synchronisation can be applied for any fractional discrete maps. Finally the fractional-order memristor system is implemented in FPGA to show that the chaotic system is hardware realisable.

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.

Dynamics of Hepatitis C Virus Infection: Mathematical Modeling and Parameter Estimation

In this paper, we provide a differential mathematical model with non-integer order derivative (fractional-order) to investigate the dynamics of Hepatitis-C Virus (HCV) replication, in presence of interferon-α (IFN) treatment. The fractional-order is considered to represent the intermediate cellular interactions and intracellular delay of the viral life cycle. We mathematically analyze and characterize the steady states and dynamical behavior of the model in presence of interferon-α treatment. We deduce a threshold parameter ℜ0 (average number of newly infected cells produced by a single infected cell) in terms of the treatment efficacy parameter 0 ≤ e < 1 and other parameters. We also provide a numerical technique for solving the fractional-order model and fitting the model to real data during treatment. The numerical simulations confirm that the fractional-order differential models have the ability to provide accurate descriptions of nonlinear biological systems with memory. The analyses presented here give an insight to understand the dynamics of HCV infection.

Numerical behavior of a fractional order dynamical model of RNA silencing

A class of fractional-order differential models of RNA silencing with memory is presented in this paper. We also carry out a detailed analysis on the stability of equilibrium and we show that the model established in this paper possesses non-negative solutions. Numerical solutions are obtained using a predictor-corrector method to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. Numerical simulations are presented to illustrate the results. Also, the numerical simulations show that, modeling the phenomena of RNA silencing by fractional ordinary differential equations (FODE) has more advantages than classical integer-order modeling.

Application of Nonequidistant Fractional-Order Accumulation Model on Trajectory Prediction of Space Manipulator

To predict the trajectory of a motion target floating in an outer space, a nonequidistant fractional-order accumulation (NEFA) model of a visual-based space manipulator is presented in this paper. This model is a modified extrapolate method of trajectory prediction. First, the initial discrete point set of a motion trajectory is real-time accumulated to a fractional-order differential model. Second, a follow trajectory is predicted both by the fractional-order differential model and a dynamic data refresh strategy. A novel measuring machine of a space manipulator is designed to orbital simulation and trajectory predictive test. The NEFA model is applied to visual-based space manipulator to trajectory prediction in this measuring instrument. The experimental result shows that the smaller the r has, the higher predictive accuracy the NEFA model has. Besides, the accuracy of 0.1-order NEFA (0.1-NEFA) model is approximately 30 times higher than that both of the traditional first-order grey model and the least-squares estimation (LSE) method. In all, the 0.1-NEFA model is superior to both the traditional first-order gray model and the LSE method.

Numerical Modeling of Fractional-Order Biological Systems

We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4<sup >+</sup> T cells. Stability and nonstability conditions for disease-free equilibrium and positive equilibria are obtained in terms of a threshold parameter <svg style="vertical-align:-3.25793pt;width:22.5px;" id="M1" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.737)"><path id="x211B" d="M852 669l7 -19q-33 -13 -66 -38q83 -53 83 -139q0 -65 -49 -111q-49 -47 -131 -50v-2q54 -40 54 -84q0 -36 -38 -97t-38 -77q0 -34 32 -34q52 0 145 101l17 -16q-100 -115 -176 -115q-38 0 -65.5 23.5t-27.5 58.5q-1 28 20.5 61t43.5 63t20 51q0 48 -33 48q-30 0 -47 11&#xA;q-81 -166 -177 -242.5t-219 -76.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 152 52q60 53 145 203q75 133 131 214q54 75 121 124q-76 48 -192 48q-150 0 -250 -75&#xA;q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 245h20q-8 -107 -69 -191q-62 -84 -158 -84q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41zM814 486q0 64 -48 102q-7 -6 -14 -14t-14.5 -18l-10.5 -14q-4 -5 -35 -55l-6 -11&#xA;q-6 -11 -10 -20t-10 -20l-6 -11l-17 -36q-4 -7 -7.5 -14.5l-6 -13t-3.5 -7.5q34 0 60 -17q57 6 92 50q36 43 36 99z" /></g> <g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg> (minimum infection parameter) for each model. We provide unconditionally stable method, using the Caputo fractional derivative of order <svg style="vertical-align:-0.1254pt;width:9.7749996px;" id="M2" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g> </svg> and implicit Euler&#x2019;s approximation, to find a numerical solution of the resulting systems. The numerical simulations confirm the advantages of the numerical technique and using fractional-order differential models in biological systems over the differential equations with integer order. The results may give insight to infectious disease specialists.

FRACTIONAL RANDOM FIELDS ON DOMAINS WITH FRACTAL BOUNDARY

For random fields with fractional regularity order (respectively, fractional singularity order), an orthogonal decomposition of the associated reproducing kernel Hilbert space with respect to domains with fractal boundary is derived. The approach presented is based on the theory of generalized random fields on fractional Sobolev spaces. The orthogonal decomposition derived is equivalent to the weak-sense Markov condition, in the second-order moment sense, studied in Ref. 50, and based on the concept of splitting Hilbert spaces. A mean-square fractional order differential representation on bounded domains with fractal boundary is also obtained. In the Gaussian case, the random fields studied have fractal sample paths (see Ref. 1). Examples of fractional-order differential models in the class considered are provided.

Fractional Generalized Random Fields on Bounded Domains

Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.