Locally Optimal Preconditioned Conjugate Gradient Method for Computing Ground State of Space-Fractional Nonlinear Schrodinger ¨ Equation

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

In this study, the peridynamic Drucker‐Prager plastic model with fractional order derivative is proposed to investigate the plastic behavior of surrounding rocks around tunnels, in which the Caputo fractional derivative is employed due to its mathematical simplicity. Instead of utilizing just one parameter for the typical nonassociated flow rule, such as dilation angle, multiple parameters, such as fractional order and interval of fractional derivative, are used to specify the direction of plastic deformation, and the fractional derivative based‐typical flow rule is proposed. As a result, compared with the traditional peridynamic Drucker‐Prager plastic model, the proposed method is a more generalized model including the typical nonassociated flow rule. Besides, based on the PD force density, the strategy of exertion of initial stress is proposed to simulate in‐situ stress in rocks. Taking a block subjected to compression as an example, the impacts of various factors, such as fractional order and interval of fractional derivative, are investigated. A numerical simulation of tunnel excavation in rocks is carried out and the numerical results obtained by the proposed method are verified by comparing with FEM results.

Atrial fibrillation (AF) is the most common arrhythmia within the clinical context. Advanced stages of the AF involve several difficulties in its management and treatment. This occurs mostly because the initiation and perpetuation mechanisms of the AF are still not fully understood. Cardiac scientific computation has become an important tool in researching the underlying mechanisms of the AF. In this work, an equation of action potential propagation that implements fractional order derivatives is used to model the atrial dynamics. The fractional derivative order represents the structural heterogeneities of the atrial myocardium. Using such mathematical operator, the Courtemanche and Maleckar human atrial electrophysiological models, during healthy and AF conditions, are assessed. The results indicate that, through the fractional order variations, there are electrophysiological properties whose behavior do not depend on the cellular model or physiological conditions. On the other hand, there are properties whose behavior under distinct values of the fractional order, are specific to the cellular model and to the physiological condition and they can be characterized quantitatively and qualitatively. Therefore, the fractional atrial propagation model can be a useful tool for modeling a wide range of electrophysiological scenarios in both healthy and AF conditions.

To achieve the goal of ceasing the spread of COVID-19 entirely it is essential to understand the dynamical behavior of the proliferation of the virus at an intense level. Studying this disease simply based on experimental analysis is very time consuming and expensive. Mathematical modeling might play a worthy role in this regard. By incorporating the mathematical frameworks with the available disease data it will be beneficial and economical to understand the key factors involved in the spread of COVID-19. As there are many vaccines available globally at present, henceforth, by including the effect of vaccination into the model will also support to understand the visible influence of the vaccine on the spread of COVID-19 virus. There are several ways to mathematically formulate the effect of disease on the population like deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional order derivative modeling is one of the fundamental methods to understand real-world problems and evaluate accurate situations. In this article, a fractional order epidemic model S_{p}E_{p}I_{p}Er_{p}R_{p}D_{p}Q_{p}V_p on the spread of COVID-19 is presented. S_{p}E_{p}I_{p}Er_{p}R_{p}D_{p}Q_{p}V_p consists of eight compartments of population namely susceptible, exposed, infective, recovered, the quarantine population, recovered-exposed, and dead population. The fractional order derivative is considered in the Caputo sense. For the prophecy and tenacity of the epidemic, we compute the reproduction number R_0. Using fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied. Furthermore, we are using the generalized Adams–Bashforth–Moulton method, to obtain the approximate solution of the fractional-order COVID-19 model. Finally, numerical results and illustrative graphic simulation are given. Our results suggest that to reduce the number of cases of COVID-19 we should reduce the contact rate of the people if the population is not fully vaccinated. However, to tackle the issue of reducing the social distancing and lock down, which have very negative impact on the economy as well as on the mental health of the people, it is much better to increase the vaccine rate and get the whole nation to be fully vaccinated.

A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model

The flexural wave propagation in a microbeam is studied based upon the nonlocal strain gradient model with the spatial and time fractional order differentials in the present work. To capture the dispersive behaviour induced by the inherent nanoscale heterogeneity, the stress gradient elasticity and the strain gradient elasticity are often used to model the mechanical behaviour. The present model incorporates the two models and introduces the fractional order derivatives which can be understood as a generalization of integral order nonlocal strain gradient model. The Laplacian operator in the constitutive equation is replaced with the symmetric Caputo fractional differential in the present model. To illustrate the flexibility of the present model, the flexural wave propagation in a microbeam is studied. The fractional order in the present model as a new material parameter can be adjusted appropriately to describe the dispersive properties of the flexural waves. The numerical results based on the new nonlocal strain gradient elasticity with fractional order derivatives are provided for both Euler–Bernoulli beam and Timoshenko beam. The comparisons with the integer order nonlocal strain gradient model and the molecular dynamic simulation are performed to validate the flexibility of the fractional order nonlocal strain gradient model.

It is important to master the creep properties of artificial frozen soil in engineering construction during the freezing in the shaft sinking .Given the shortage that the creep behaviors of artificial frozen soil were deemed as the materials between perfect solid and fluid, and that the components of integer order calculus constitutive relation were in great demand,the fractional order constitutive relation was used to calculate the creep character of the artificial frozen soil .Through the substitution of a multipled dashpot by a fractional order derivative dashpot, a fractional order derivative Kelvin’s creep model is developed. The series accelerated element are added to the fractional order element in Kelvin model, and the acceleratemodel is built. Using fractional order derivative Kelvin model to simulate the creep properties of artificial frozen soil, the model parameters was obtained by the global optimization of simulated annealing algorithm. The fractional order derivative creep model of artificial frozen soil can be reflected preferably in the 3 stages creep process. Introduction It is important to master the creep properties of artificial frozen soil in engineering construction during the freezing in the shaft sinking. The mechanical properties of artificial frozen soil were deemed as the materials between perfect solid and fluid.Neither do the creep properties of artificial frozen soil abide by Hooke Law, nor do it bide by Newton’s Law of viscosity, but it abide by a relationship between these two laws instead. The study of fractional calculus is the arbitrary order differentiation, the properties of integral operator and its application. The rheological model theory of using fraction order derivative not only reserve advantages of the classical model theory, but also the combination of series and parallel of a few several elements can be anastomosed preferably with test results. The parameter identification of constitutive model is the important subject that studies on the constitutive relation of artificial frozen soil. The general idea of parameter identification of geo-material constitutive model is to select suitable combined model, according to the experimental data and then to determine the parameter of model by regression analysis with least square method etc. The error in calculation was caused in the process of hypothesis of constitutive model, therefore, there is a great error between the calculation results and the measured value. With the development of artificial intelligence, the algorithm technology of simulated annealing was applied to geo-material constitutive model and parameter identification by many scholars. And, they have obtained plentiful achievements. Given the shortage that the creep behaviors of artificial frozen soil were deemed as the materials between perfect solid and fluid, and that the components of integer order calculus constitutive relation were in great demand, the fractional order constitutive relation was used to calculate the creep character of the artificial frozen soil .Through the substitution of a dashpot by a fractional order derivative dashpot, a fractional order derivative Kelvin’s creep model is developed. The series accelerated elements are added to the fractional order element in Kelvin model, and the accelerated model is built. Using fractional order derivative Kelvin model to simulate the creep properties of artificial frozen soil, the model parameters was obtained by the global optimization of simulated International Conference on Applied Science and Engineering Innovation (ASEI 2015) © 2015. The authors Published by Atlantis Press 1479 annealing algorithm. The fractional order derivative creep model of artificial frozen soil can be reflected preferably in the 3 stages creep process. The fractional order derivative model is a new method in the calculation of the field of artificial frozen soil. Fractional order derivative Kelvin creep model of artificial frozen soil General Kelvin model. According to the analysis of the existing geo-material constitutive model, Kelvin model is a model which is relatively comprehensive and has a broad application. It is used to describe the relationship of stress and strain of geotechnical materials. This model can be used to consider the mechanical properties of material, such as elasticity, visco-elasticity and elastic-plastic etc. It has a wide filed of application and more mature theoretical derivation. General Kelvin model is shown in figure 1.

ABSTRACTThe periodic motions of the fractional order and/or delayed nonlinear systems are investigated in the frequency domain using a harmonic balance method with the analytical gradients of the nonlinear quality constraints and the sensitivity information of the Fourier coefficients can also obtained. The properties of fractional order derivatives and trigonometric functions are utilized to construct the fractional order derivatives, delayed and product operational matrices. The operational matrices are used to derive the analytical formulae of nonlinear systems of algebraic equations. The stability of periodic solutions for the delayed nonlinear systems is identified by an eigenvalue analysis of quasi-polynomials characteristic equations. Sensitivity analysis is performed to study the influence of the structural parameters on the system responses. Finally, three numerical examples are presented to illustrate the validity and feasibility of the developed method. It is concluded that the proposed methodology has the potential to facilitate highly efficient optimization, as well as sensitivity and uncertainty analysis of nonlinear systems with fractional derivatives and/or time delayed.

The flow analysis of fiuids in fractal reservoir with the fractional derivative

Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment

Chaos in the fractional order nonlinear Bloch equation with delay

Effective pretreatment of spectral reflectance is vital to model accuracy in soil parameter estimation. However, the classic integer derivative has some disadvantages, including spectral information loss and the introduction of high-frequency noise. In this paper, the fractional order derivative algorithm was applied to the pretreatment and partial least squares regression (PLSR) was used to assess the clay content of desert soils. Overall, 103 soil samples were collected from the Ebinur Lake basin in the Xinjiang Uighur Autonomous Region of China, and used as data sets for calibration and validation. Following laboratory measurements of spectral reflectance and clay content, the raw spectral reflectance and absorbance data were treated using the fractional derivative order from the 0.0 to the 2.0 order (order interval: 0.2). The ratio of performance to deviation (RPD), determinant coefficients of calibration (), root mean square errors of calibration (RMSEC), determinant coefficients of prediction (), and root mean square errors of prediction (RMSEP) were applied to assess the performance of predicting models. The results showed that models built on the fractional derivative order performed better than when using the classic integer derivative. Comparison of the predictive effects of 22 models for estimating clay content, calibrated by PLSR, showed that those models based on the fractional derivative 1.8 order of spectral reflectance ( = 0.907, RMSEC = 0.425%, = 0.916, RMSEP = 0.364%, and RPD = 2.484 ≥ 2.000) and absorbance ( = 0.888, RMSEC = 0.446%, = 0.918, RMSEP = 0.383% and RPD = 2.511 ≥ 2.000) were most effective. Furthermore, they performed well in quantitative estimations of the clay content of soils in the study area.

Diffuse reflectance spectroscopy of γ-irradiated UHMWPE: A novel fractional order based filters approach for accessing the radiation modification

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like Grunwald-Letnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like Grunwald-Letnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.