Measuring memory with the order of fractional derivative.
Ontology highlight
ABSTRACT: Fractional derivative has a history as long as that of classical calculus, but it is much less popular than it should be. What is the physical meaning of fractional derivative? This is still an open problem. In modeling various memory phenomena, we observe that a memory process usually consists of two stages. One is short with permanent retention, and the other is governed by a simple model of fractional derivative. With the numerical least square method, we show that the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics, but also in biology and psychology. Based on this model, we find that a physical meaning of the fractional order is an index of memory.
Project description:We provide a SEIR epidemic model for the spread of COVID-19 using the Caputo fractional derivative. The feasibility region of the system and equilibrium points are calculated and the stability of the equilibrium points is investigated. We prove the existence of a unique solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. To predict the transmission of COVID-19 in Iran and in the world, we provide a numerical simulation based on real data.
Project description:Accurate monitoring and estimation of heavy metal concentrations is an important process in the prevention and treatment of soil pollution. However, the weak correlation between spectra and heavy metals in soil makes it difficult to use spectroscopy in predicting areas with a risk of heavy metal pollution. In this paper, a method for detection of Ni in soil in eastern China using the fractional-order derivative (FOD) and spectral indices was proposed. The visible-near-infrared (Vis-NIR) spectra were preprocessed using the FOD (range: 0 to 2, interval: 0.1) to solve the problems of baseline drift and overlapping peaks in the original spectra. The product index (PI), ratio index (RI), sum index (SI), difference index (DI), normalized difference index (NDI), and brightness index (BI) were applied and compared. The results showed that the spectral detail increased as the FOD increased, and the interference of the baseline drift and overlapping peaks was eliminated as the spectral reflectance decreased. Furthermore, the FOD extracted the spectral sensitivity information more effectively and improved the correlation between the Vis-NIR spectra and the Ni concentration, and the NDI had a maximum correlation coefficient (r) of 0.803 for order 1.9. The estimation model based on the NDI dataset constructed after FOD processing had the best performance, with a validation accuracy [Formula: see text] of 0.735, RMSE of 3.848, and RPD of 2.423. In addition, this method is easy to carry out and suitable for estimating other heavy metal elements in soil.
Project description:In this work, we establish a fractional-order neural field mathematical model with Caputo's fractional derivative temporal order α considering 0 < α < 2, to analyze the effect of fractional-order on cortical wave features observed preceding seizure termination. The importance of this incorporation relies on the theoretical framework established by fractional-order derivatives in which memory and hereditary properties of a system are considered. Employing Mittag-Leffler functions, we first obtain approximate fractional-order solutions that provide information about the initial wave dynamics in a fractional-order frame. We then consider the Adomian decomposition method to approximate pulse solutions in a wider range of orders and longer times. The former approach establishes a direct way to investigate the initial relationships between fractional-order and wave features, such as wave speed and wave width. In contrast, the latter approach displays wave propagation dynamics in different fractional orders for longer times. Using the previous two approaches, we establish approximate wave solutions with characteristics consistent with in vivo cortical waves preceding seizure termination. In our analysis, we find consistent differences in the initial effect of the fractional-order on the features of wave speed and wave width, depending on whether α <1 or α>1. Both cases can model the shape of cortical wave propagation for different fractional-orders at the cost of modifying the wave speed. Our results also show that the effect of fractional-order on wave width depends on the synaptic threshold and the synaptic connectivity extent. Fractional-order derivatives have been interpreted as the memory trace of the system. This property and the results of our analysis suggest that fractional-order derivatives and neuronal collective memory modify cortical wave features.
Project description:Chlorophyll density (ChD) can reflect the photosynthetic capacity of the winter wheat population, therefore achieving real-time non-destructive monitoring of ChD in winter wheat is of great significance for evaluating the growth status of winter wheat. Derivative preprocessing has a wide range of applications in the hyperspectral monitoring of winter wheat chlorophyll. In order to research the role of fractional-order derivative (FOD) in the hyperspectral monitoring model of ChD, this study based on an irrigation experiment of winter wheat to obtain ChD and canopy hyperspectral reflectance. The original spectral reflectance curves were preprocessed using 3 FOD methods: Grünwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo. Hyperspectral monitoring models for winter wheat ChD were constructed using 8 machine learning algorithms, including partial least squares regression, support vector regression, multi-layer perceptron regression, random forest regression, extra-trees regression (ETsR), decision tree regression, K-nearest neighbors regression, and gaussian process regression, based on the full spectrum band and the band selected by competitive adaptive reweighted sampling (CARS). The main results were as follows: For the 3 types of FOD, GL-FOD was suitable for analyzing the change process of the original spectral curve towards the integer-order derivative spectral curve. RL-FOD was suitable for constructing the hyperspectral monitoring model of winter wheat ChD. Caputo-FOD was not suitable for hyperspectral research due to its insensitivity to changes in order. The 3 FOD calculation methods could all improve the correlation between the original spectral curve and Log(ChD) to varying degrees, but only the GL method and RL method could observe the change process of correlation with order changes, and the shorter the wavelength, the smaller the order, and the higher the correlation. The bands screened by CARS were distributed throughout the entire spectral range, but there was a relatively concentrated distribution in the visible light region. Among all models, CARS was used to screen bands based on the 0.3-order RL-FOD spectrum, and the model constructed using ETsR reached the best accuracy and stability. Its R2c, RMSEc, R2v, RMSEv, and RPD were 1.0000, 0.0000, 0.8667, 0.1732, and 2.6660, respectively. In conclusion, based on the winter wheat ChD data set and the corresponding canopy hyperspectral data set, combined with 3 FOD calculation methods, 1 band screening method, and 8 modeling algorithms, this study constructed hyperspectral monitoring models for winter wheat ChD. The results showed that based on the 0.3-order RL-FOD, combined with the CARS screening band, ETsR modeling has the highest accuracy, and hyperspectral estimation of winter wheat ChD can be realized. The results of this study can provide some reference for the rapid and nondestructive estimation of ChD in winter wheat.
Project description:In this study, we show that the discharge voltage pattern of a supercapacitor exhibiting fractional-order behavior from the same initial steady-state voltage into a constant resistor is dependent on the past charging voltage profile. The charging voltage was designed to follow a power-law function, i.e. [Formula: see text], in which [Formula: see text] (charging time duration between zero voltage to the terminal voltage [Formula: see text]) and p ([Formula: see text]) act as two variable parameters. We used this history-dependence of the dynamic behavior of the device to uniquely retrieve information pre-coded in the charging waveform pattern. Furthermore, we provide an analytical model based on fractional calculus that explains phenomenologically the information storage mechanism. The use of this intrinsic material memory effect may lead to new types of methods for information storage and retrieval.
Project description:Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.
Project description:Fractional derivative has a memory and non-localization features that make it very useful in modelling epidemics’ transition. The kernel of Caputo-Fabrizio fractional derivative has many features such as non-singularity, non-locality and an exponential form. Therefore, it is preferred for modeling disease spreading systems. In this work, we suggest to formulate COVID-19 epidemic transmission via
Project description:Control and synchronization of fractional-order chaotic systems have attracted wide attention due to their numerous potential applications. To get suitable control method and parameters for fractional-order chaotic systems, the stability analysis of time-varying fractional-order systems should be discussed in the first place. Therefore, this paper analyzes the stability of the time-varying fractional-order systems and presents a stability theorem for the system with the order 0<α<1. This theorem is a sufficient condition which can discriminate the stability of time-varying systems conveniently. Feedback controllers are designed for control and synchronization of the fractional-order Lü chaotic system. The simulation results demonstrate the effectiveness of the proposed theorem.
Project description:Isotropic linear and nonlinear fractional derivative constitutive relations are formulated and examined in terms of many parameter generalized Kelvin models and are analytically extended to cover general anisotropic homogeneous or non-homogeneous as well as functionally graded viscoelastic material behavior. Equivalent integral constitutive relations, which are computationally more powerful, are derived from fractional differential ones and the associated anisotropic temperature-moisture-degree-of-cure shift functions and reduced times are established. Approximate Fourier transform inversions for fractional derivative relations are formulated and their accuracy is evaluated. The efficacy of integer and fractional derivative constitutive relations is compared and the preferential use of either characterization in analyzing isotropic and anisotropic real materials must be examined on a case-by-case basis. Approximate protocols for curve fitting analytical fractional derivative results to experimental data are formulated and evaluated.
Project description:Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.