Fractional Dynamics Research Articles

Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces

A dynamical system is a particle or set of particles whose state changes over time. The dynamics of the system is described by a set of differential equations. If the derivatives involved are of non-integer order, we obtain a fractional dynamical system. In this paper, we considered a fractional dynamical system with the Caputo fractional derivative. We collocated the fractional differential problem in dyadic nodes and used refinable functions as approximation functions to achieve a good degree of freedom in the choice of the regularity. The collocation method stands out as a particularly useful and attractive tool for solving fractional differential problems of various forms. A numerical result is presented to show that the numerical solution fits the analytical one very well. We collocated the fractional differential problem in dyadic nodes using refinable functions as approximation functions to achieve a good degree of freedom in the choice of regularity.

Finite Length Triple Estimation Algorithm and its Application to Gyroscope MEMS Noise Identification

Abstract The noises associated with MEMS measurements can significantly impact their accuracy. The noises characterised by random walk and bias instability errors strictly depend on temperature effects that are difficult to specify during direct measurements. Therefore, the paper aims to estimate the fractional noise dynamics of the stationary MEMS gyroscope based on finite length triple estimation algorithm (FLTEA). The paper deals with the state, order and parameter estimation of fractional order noises originating from the MEMS gyroscope, being part of the popular Inertial Measurement Unit denoted as SparkFun MPU9250. The noise measurements from x, y and z gyroscope axes are identified using a modified triple estimation algorithm (TEA) with finite approximation length. The TEA allows a simultaneous estimation of the state, order and parameter of fractional order systems. Moreover, as it is well-known that the number of samples in fractional difference approximations plays a key role, we try to show the influence of applying the TEA with various approximation length constraints on final estimation results. The validation of finite length TEA in the noise estimation process coming from MEMS gyroscope has been conducted for implementation length reduction achieving 50% of samples needed to estimate the noise with no implementation losses. Additionally, the capabilities of modified TEA in the analysis of fractional constant and variable order systems are confirmed in several numerical examples.

Distributed Solving Linear Algebraic Equations with Switched Fractional Order Dynamics

Distributed Solving Linear Algebraic Equations with Switched Fractional Order Dynamics

Fractional Dynamics Foster Deep Learning of COPD Stage Prediction (Adv. Sci. 12/2023)

Deep Learning Analyzing physiological signals with fractional dynamics reduces the learning complexity for automated diagnosis with deep learning. In article number 2203485, Mihai Udrescu, Paul Bogdan, and co-workers show that fractional-order dynamical modeling can extract distinguishing signatures from the physiological signals recorded in COPD patients, then use fractional signatures to develop and train a deep neural network that accurately predicts COPD stages—a robust alternative to spirometry.

Investigation of Fractional Order Dynamics of Tuberculosis under Caputo Operator

In this article, a new deterministic disease system is constructed to study the influence of treatment adherence as well as awareness on the spread of tuberculosis (TB). The suggested model is composed of six various classes, whose dynamics are discussed in the sense of the Caputo fractional operator. Firstly the model existence of a solution along with a unique solution is checked to determine whether the system has a solution or not. The stability of a solution is also important, so we use the Ulam–Hyers concept of stability. The approximate solution analysis is checked by the technique of Laplace transformation using the Adomian decomposition concept. Such a solution is in series form which is decomposed into smaller terms and the next term is obtained from the previous one. The numerical simulation is established for the obtained schemes using different fractional orders along with a comparison of classical derivatives. Such an analysis will be helpful for testing more dynamics instead of only one type of integer order discussion.

Response of soil organic carbon fractions to cover cropping: A meta-analysis of agroecosystems

Agroecosystems have a large potential for soil carbon (C) sequestration, and cover cropping is one management practice that can enhance soil C while providing other ecological and agronomic benefits. Still, we lack an understanding of the distribution of this added C across soil organic C (SOC) fractions. Two major operational pools make up SOC: easily decomposed, faster-turnover particulate organic C (POC) and slowly decomposed, slower-turnover mineral-associated organic C (MAOC). Though POC is generally more sensitive to environmental change in the short term, recent projections hold that MAOC stocks could face considerable loss in croplands where it is the dominant form of SOC. We used a systematic meta-analysis to determine responses of POC and MAOC concentration and stock to cover cropping and how these responses are influenced by soil, agronomic, and climatic variables. Across 49 studies concentrated in the Americas and Europe, total POC increased by 14.94% and total MAOC increased by 5.56% on average in cover-cropped soils vs. non-cover cropped or fallow control soils, which would effectively reverse or offset projected losses of SOC with climate warming. The effect of cover cropping on POC was greater in soils sampled from 0 to 10 cm vs. 10–30 cm depths and under higher mean annual temperature, but the effect of cover cropping on MAOC was not influenced by either of these factors. The response of POC to cover cropping was significantly positive for certain cover crop growing seasons, cover crop types, and tillage practices in which the MAOC response was not significant, indicating that changes in POC rather than MAOC are responsible for driving the positive response of total SOC to cover cropping. One commonality between POC and MAOC responses to cover cropping was that both dramatically increased with the amount of aboveground C produced by cover crops. Still, POC and MAOC responses to cover cropping were not strongly altered by nitrogen fertilization rate, cover crop diversity, experimental duration, annual precipitation, or baseline soil properties (texture, total organic C, and pH). Our results show that POC is more responsive to cover cropping than MAOC in an average of five years. However, more experimental data are needed to determine how these SOC fractions respond in deeper soil as well as in different geographic regions, climates, and soil orders. Based on our results, we discuss drivers of total and fractional SOC dynamics in agroecosystems and lessons learned from our synthesis of available fractional SOC data.

Neural adaptation and fractional dynamics as a window to underlying neural excitability.

The relationship between macroscale electrophysiological recordings and the dynamics of underlying neural activity remains unclear. We have previously shown that low frequency EEG activity (<1 Hz) is decreased at the seizure onset zone (SOZ), while higher frequency activity (1-50 Hz) is increased. These changes result in power spectral densities (PSDs) with flattened slopes near the SOZ, which are assumed to be areas of increased excitability. We wanted to understand possible mechanisms underlying PSD changes in brain regions of increased excitability. We hypothesized that these observations are consistent with changes in adaptation within the neural circuit. We developed a theoretical framework and tested the effect of adaptation mechanisms, such as spike frequency adaptation and synaptic depression, on excitability and PSDs using filter-based neural mass models and conductance-based models. We compared the contribution of single timescale adaptation and multiple timescale adaptation. We found that adaptation with multiple timescales alters the PSDs. Multiple timescales of adaptation can approximate fractional dynamics, a form of calculus related to power laws, history dependence, and non-integer order derivatives. Coupled with input changes, these dynamics changed circuit responses in unexpected ways. Increased input without synaptic depression increases broadband power. However, increased input with synaptic depression may decrease power. The effects of adaptation were most pronounced for low frequency activity (< 1Hz). Increased input combined with a loss of adaptation yielded reduced low frequency activity and increased higher frequency activity, consistent with clinical EEG observations from SOZs. Spike frequency adaptation and synaptic depression, two forms of multiple timescale adaptation, affect low frequency EEG and the slope of PSDs. These neural mechanisms may underlie changes in EEG activity near the SOZ and relate to neural hyperexcitability. Neural adaptation may be evident in macroscale electrophysiological recordings and provide a window to understanding neural circuit excitability.

Fractional Dynamics Foster Deep Learning of COPD Stage Prediction.

Chronic obstructive pulmonary disease (COPD) is one of the leading causes of death worldwide. Current COPD diagnosis (i.e., spirometry) could be unreliable because the test depends on an adequate effort from the tester and testee. Moreover, the early diagnosis of COPD is challenging. The authors address COPD detection by constructing two novel physiological signals datasets (4432 records from 54 patients in the WestRo COPD dataset and 13824 medical records from 534 patients in the WestRo Porti COPD dataset). The authors demonstrate their complex coupled fractal dynamical characteristics and perform a fractional-order dynamics deep learning analysis to diagnose COPD. The authors found that the fractional-order dynamical modeling can extract distinguishing signatures from the physiological signals across patients with all COPD stages-from stage 0 (healthy) to stage 4 (very severe). They use the fractional signatures to develop and train a deep neural network that predicts COPD stages based on the input features (such as thorax breathing effort, respiratory rate, or oxygen saturation). The authors show that the fractional dynamic deep learning model (FDDLM) achieves a COPD prediction accuracy of 98.66% and can serve as a robust alternative to spirometry. The FDDLM also has high accuracy when validated on a dataset with different physiological signals.

Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

PurposeThis paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.Design/methodology/approachThis paper considered theoretical and numerical methodologies.FindingsThis paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.Originality/valueThe novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

Analytical Study of the Complexities in a Three Species Food Web Model with Modified Caputo–Fabrizio Operator

This article presents the analytical study of the three species fractional food web model in the framework of the Modified Caputo–Fabrizio operator. With the help of fixed point theory, the existence and uniqueness results are investigated for the fractional order model. To obtain the approximate solution for the suggested model, the well-known Laplace–Adomian decomposition method is used. The solutions are validated through simulations with a variety of fractional orders and initial values, where the complex nature of the system can be observed. The technique used here can be easily used to study a range of complex problems in different branches of science. From the figures, it can be observed that, at integer higher fractional order, there are a number of oscillations in the system and the system behaves chaotically, while, at lower fractional orders, the oscillation amplitudes decrease, resulting in the faster converging towards the equilibrium point. According to the results, the Modified Caputo–Fabrizio fractional-order derivative may be used in a variety of future fractional dynamics scenarios.

Water-Use Strategies and Habitat Adaptation of Four Tree Species in Karstic Climax Forest in Maolan

The technique of stable hydrogen and oxygen isotope tracing has become an important means to study the mechanism of water movement due to its high sensitivity and traceability. In this study, four dominant tree species in the karst forest of Maolan, Guizhou Province, were selected, and their water-use strategies and the mechanism of maintenance of tree species diversity were investigated using the stable hydrogen and oxygen isotope tracing technique. The results show that: (1) The regional precipitation varied evidently with the alternation of seasons, i.e., the values of δD and δ18O in precipitation had a positive bias in spring and a negative bias in summer and autumn. The value of deuterium excess (d-excess) was between 11.67‰ and 31.02‰, with a mean value of 22.98‰. (2) The soil temperature (ST), soil water content (SWC) and precipitation, which have a significant positive correlation, imposed a joint impact on the dynamics of the soil evaporative fractionation. (3) The line-conditioned excess (LC-excess) varied seasonally in different water bodies, i.e., the relative evaporative fractionation of the rhizosphere soil of deciduous tree species was stronger than that of evergreen tree species, and the evaporative fractionation of hydrogen and oxygen isotopes in the leaf water of evergreen tree species was stronger than that of deciduous tree species in spring and summer. However, that of the latter was stronger than that of the former in autumn. (4) The soil water was the most important potential water source for dominant tree species in karst terrain (71%), followed by epikarstic water, which made up an effective supplement (29%). (5) Finally, trees of different life forms and species varied in capacity and proportion in terms of using the potential water sources in different seasons, i.e., deciduous tree species had a greater capacity for using water from potential sources and variable water-use strategies. This may be a major water-limiting mechanism that maintains photosynthesis in the leaves of evergreen tree species (leaves are evergreen and plants continue to grow via photosynthesis) and constrains photosynthesis in deciduous tree species (leaves fall and plants become dormant and stop growing). These results lead to the conclusion that the dominant tree species in karstic forests resist water stress and adjust water-use strategies towards each potential water source to adapt to the harsh karstic habitat through root plasticity and leaf defoliation.

Application of trigonometric B-spline functions for solving Caputo time fractional gas dynamics equation

&lt;abstract&gt;&lt;p&gt;In this study, we present a numerical method that utilizes trigonometric cubic B-spline functions to solve the time fractional gas dynamics equation, which is a key component in the study of physical phenomena such as explosions, combustion, detonation and condensation in a moving flow. The Caputo formula is used to define the fractional time derivative, which generalizes the framework for both singular and non-singular kernels. To discretize the unknown function and its derivatives in the spatial direction, we employ trigonometric cubic B-spline functions, while the usual finite difference formulation is used to approximate the Caputo time fractional derivative. A stability analysis of the scheme is provided to ensure that errors do not propagate over time, and a convergence analysis is conducted to measure the accuracy of the solution. To demonstrate the effectiveness of the proposed methodology, we solve various relevant examples and present graphical and tabular results to evaluate the outcomes of the strategy.&lt;/p&gt;&lt;/abstract&gt;

Fractional-order quantum kicked top map and its discrete dynamic behaviors.

A kind of top with a fractional operator is discussed in this paper. The top has a periodic nonlinear pulse kick sequence in the magnetic field and constant precessing around the magnetic field. Then, a fractional quantum kicked top map based on the Caputo derivative is proposed. The numerical solutions of the fractional difference equation are obtained, and the chaotic behavior is observed numerically in three aspects. Fractional quantum dynamics behaviors take place in a finite dimensional Hilbert space where the squared angular momentum is free precession. Finally, the dynamic behaviors of the fractional quantum kicked top map are systematically analyzed by using the bifurcation diagram, the phase diagram, and the maximum Lyapunov exponent.

Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations

&lt;abstract&gt;&lt;p&gt;The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.&lt;/p&gt;&lt;/abstract&gt;

Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations

&lt;abstract&gt;&lt;p&gt;The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.&lt;/p&gt;&lt;/abstract&gt;

Clinicopathological study of fractional exhaled nitric oxide dynamics and intratumoral inducible nitric oxide synthase expression in primary lung cancer patients.

Inducible nitric oxide synthase (iNOS) is expressed in non-small cell lung cancer (NSCLC) tumor cells and contributes to tumorigenesis. Nitric oxide, an indicator of airway inflammation, is concurrently produced in the airway epithelium. However, the interrelationships and predictive importance of iNOS remain unclear. This study aimed to investigate whether iNOS could serve as a novel biomarker for NSCLC. Immunohistochemical analysis of iNOS expression in the tumor cells of 101 consecutive patients with NSCLC undergoing lung resection was conducted. The fractional exhaled nitric oxide (FeNO) levels were evaluated pre- and postoperatively using a clinically applied respiratory function testing device. iNOS expression was assessed by immunochemical staining for expression within tumor cells. iNOS expression in the tumor cells was significantly associated with squamous cell carcinoma (P<0.01). No significant correlation between the FeNO levels and iNOS expression scores existed; however, the FeNO levels in positive cases of squamous cell carcinoma were significantly higher than those in negative cases (P<0.01). The FeNO levels did not decrease in the iNOS-negative cases after tumor resection in the squamous cell carcinoma group but were significantly lower in the positive cases (P=0.03). iNOS expression in tumor cells showed a characteristic tendency toward squamous cell carcinoma, suggesting its potential for FeNO-mediated localization and diagnosing lung cancer.

Fractional Order Nonlinear Bone Remodeling Dynamics Using the Supervised Neural Network

This study aims to solve the nonlinear fractional-order mathematical model (FOMM) by using the normal and dysregulated bone remodeling of the myeloma bone disease (MBD). For the more precise performance of the model, fractional-order derivatives have been used to solve the disease model numerically. The FOMM is preliminarily designed to focus on the critical interactions between bone resorption or osteoclasts (OC) and bone formation or osteoblasts (OB). The connections of OC and OB are represented by a nonlinear differential system based on the cellular components, which depict stable fluctuation in the usual bone case and unstable fluctuation through the MBD. Untreated myeloma causes by increasing the OC and reducing the osteoblasts, resulting in net bone waste the tumor growth. The solutions of the FOMM will be provided by using the stochastic framework based on the Levenberg-Marquardt backpropagation (LVMBP) neural networks (NN), i.e., LVMBPNN. The mathematical performances of three variations of the fractional-order derivative based on the nonlinear disease model using the LVMPNN. The static structural performances are 82% for investigation and 9% for both learning and certification. The performances of the LVMBPNN are authenticated by using the results of the Adams-Bashforth-Moulton mechanism. To accomplish the capability, steadiness, accuracy, and ability of the LVMBPNN, the performances of the error histograms (EHs), mean square error (MSE), recurrence, and state transitions (STs) will be provided.

Study of Fractional Order Dynamical System of Viral Infection Disease under Piecewise Derivative

This research aims to understand the fractional order dynamics of the deadly Nipah virus (NiV) disease. We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the Caputo sense to investigate the crossover behavior of the considered dynamical system. We establish some qualitative results about the existence and uniqueness of the solution to the proposed problem. By utilizing the Newtonian polynomials interpolation technique, we recall a powerful algorithm to interpret the numerical findings for the aforesaid model. Here, we remark that the said viral infection is caused by an RNA type virus which can transmit from animals and also from an infected person to person. Fruits bats which are also known as flying foxes are one of the sources of transmission of NiV disease. Here in this work, we investigate its transmission mechanism through some new concepts of fractional calculus for further analysis and prediction. We present the approximate results for different compartments using different fractional orders. By using the piecewise derivative concept, we detect the crossover or multi-steps behavior in the transmission dynamics of the mentioned disease. Therefore, the considered form of the derivative is used to deal with problems exhibiting crossover behaviors.

ANALYSIS OF TRITROPHIC INTERACTION WITH VOLATILE COMPOUNDS IN PLANTS WITH FRACTAL–FRACTIONAL CAPUTO OPERATOR

This paper is devoted to the study of the dynamical behavior of the tritrophic interaction amongst plants, herbivores and carnivores mathematical model, expressed by three nonlinear ordinary differential equations under fractal–fractional derivative in the Caputo sense. We use fixed point theory to ensure that one solution exists to the proposed model. In addition, Hyers–Ulam’s stability analysis is studied by using theorem of functional analysis. For the numerical solution, we apply the fractional Adams–Bashforth iterative technique. For arbitrary fractional order and fractal dimensions, we study the dynamical and chaotic behavior of the obtained results for the considered model. Using Matlab 16, the system is then solved to get the required numerical solution for the proposed system. From the numerical simulations, we observed that the decay in fractional order dynamics of the system is stabilized when the amplitude of the oscillations becomes smaller.

Fractional dynamics of the transmission phenomena of dengue infection with vaccination

&lt;p style='text-indent:20px;'&gt;Dengue infection brings unimaginable damage in low-income countries across the world to public health, social, and economical fields. Scientists rely on modeling to preferable understand the transmission phenomena of dengue in order to forecast some preventive measures and to give better data for the development of vaccines and medications. In present work, we construct a compartmental model for dengue infection via fractional derivatives to show the influence of memory. The suggested system's solution is investigated for existence as well as uniqueness with the help of fractional calculus fundamental characteristics. The model is analyzed and the reproduction number is obtained through Next-generation Matrix technique, moreover, the sensitivity of the reproduction number is determined using the partial rank correlation coefficient (PRCC) method. We have shown that if &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ R_0&amp;lt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the disease-free steady-state of the dengue model is locally asymptotically stable (LAS) and globally asymptotically stable (GAS) in the absence of vaccine. In particular case, we have shown that the infection of dengue persists uniformly in the system for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; greater than one. Furthermore, we quantitatively showed memory's impact on &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; by fluctuation of different factors. Our result predicted that the biting frequency, mosquito recruiting rate, and memory index are the most sensitive aspects in efficiently controlling new dengue illnesses, according to our findings. The effect of vaccination on the suggested system's threshold is also investigated. Novel numerical schemes are presented to conceptualize the time series of the system in different scenarios. The findings suggested that more accurate and precious results are obtained through non-integer derivative and that memory plays a beneficial role in the management of dengue disease. We demonstrated that manipulating the index of memory in the system may manage the system reproduction number and endemic level of dengue infection.&lt;/p&gt;