Fractional Order Calculus Research Articles

The fractional neural grey system model and its application

Recently, grey system, neural network, and fractional order calculus theory have become popular research areas, and an increasing number of scholars have joined these studies, conducted illuminating research, and produced a number of significant results. Numerous research studies have demonstrated that these three strategies are crucial to solving a wide range of practical problems. In this paper, we present a fractional order neural grey system model with a three-layer structure in which the input of the network is a fractional order cumulative sequence, and the output is a predicted value in order to maximize the benefits of each of the three elements. The purpose of this research is to present a strategy for reducing the number of conditions in order to improve the stability of parameter estimation by using QR decomposition. The order of the models is determined by an intelligent optimization algorithm. Finally, real-world examples are used to validate the model’s validity, and experimental results indicate that the newly presented model is more accurate than previous models.

A Fractional-Order Density-Dependent Mathematical Model to Find the Better Strain of Wolbachia

The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through predicting their dynamics. Wolbachia is an effective tool for controlling mosquito-borne diseases, and several strains have been tested in laboratories and released into outbreak locations. This study aimed to determine the symmetrical features of the most efficient strain from a mathematical perspective. This was accomplished by integrating a density-dependent death rate and the rate of cytoplasmic incompatibility (CI) into the model to examine the spread of Wolbachia and non-Wolbachia mosquitoes. The fractional-order mathematical model developed here is physically meaningful and was assessed for equilibrium points in the presence and absence of disease. Eight equilibrium points were determined, and their local and global stability were determined using the Routh–Hurwitz criterion and linear matrix inequality theory. The basic reproduction number was calculated using the next-generation matrix method. The research also involved conducting numerical simulations to evaluate the behavior of the basic reproduction number for different equilibrium points and identify the optimal CI value for reducing disease spread.

Computational Complexity-based Fractional-Order Neural Network Models for the Diagnostic Treatments and Predictive Transdifferentiability of Heterogeneous Cancer Cell Propensity

Neural networks and fractional order calculus are powerful tools for system identification through which there exists the capability of approximating nonlinear functions owing to the use of nonlinear activation functions and of processing diverse inputs and outputs as well as the automatic adaptation of synaptic elements through a specified learning algorithm. Fractional-order calculus, concerning the differentiation and integration of non-integer orders, is reliant on fractional-order thinking which allows better understanding of complex and dynamic systems, enhancing the processing and control of complex, chaotic and heterogeneous elements. One of the most characteristic features of biological systems is their different levels of complexity; thus, chaos theory seems to be one of the most
 applicable areas of life sciences along with nonlinear dynamic and complex systems of living and non-living environment. Biocomplexity, with multiple scales ranging from molecules to cells and organisms, addresses complex structures and behaviors which emerge from nonlinear interactions of active biological agents. This sort of emergent complexity is concerned with the organization of molecules
 into cellular machinery by that of cells into tissues as well as that of individuals to communities. Healthy systems sustain complexity in their lifetime and are chaotic, so complexity loss or chaos loss results in diseases. Within the mathematics-informed frameworks, fractional-order calculus based Artificial Neural Networks (ANNs) can be employed for accurate understanding of complex biological
 processes. This approach aims at achieving optimized solutions through the maximization of the model’s accuracy and minimization of computational burden and exhaustive methods. Relying on a transdifferentiable mathematics-informed framework and multifarious integrative methods concerning computational complexity, this study aims at establishing an accurate and robust model based upon
 integration of fractional-order derivative and ANN for the diagnosis and prediction purposes for cancer cell whose propensity exhibits various transient and dynamic biological properties. The other aim is concerned with showing the significance of computational complexity for obtaining the fractional-order derivative with the least complexity in order that optimized solution could be achieved. The multifarious
 scheme of the study, by applying fractional-order calculus to optimization methods, the advantageous aspect concerning model accuracy maximization has been demonstrated through the proposed method’s applicability and predictability aspect in various domains manifested by dynamic and nonlinear nature displaying different levels of chaos and complexity.

Optimal Sizing and Power System Control of Hybrid Solar PV-Biogas Generator with Energy Storage System Power Plant

In this paper, the electrical parameters of a hybrid power system made of hybrid renewable energy sources (HRES) generation are primarily discussed. The main components of HRES with energy storage (ES) systems are the resources coordinated with multiple photovoltaic (PV) cell units, a biogas generator, and multiple ES systems, including superconducting magnetic energy storage (SMES) and pumped hydro energy storage (PHES). The performance characteristics of the HRES are determined by the constant power generation from various sources, as well as the shifting load perturbations. Constant power generation from a variety of sources, as well as shifting load perturbations, dictate the HRES’s performance characteristics. As a result of the fluctuating load demand, there will be steady generation but also fluctuating frequency and power. A suitable control strategy is therefore needed to overcome the frequency and power deviations under the aforementioned load demand and generation conditions. An integration in the environment of fractional order (FO) calculus for proportion-al-integral-derivative (PID) controllers and fuzzy controllers, referred to as FO-Fuzzy-PID controllers, tuned with the opposition-based whale optimization algorithm (OWOA), and compared with QOHSA, TBLOA, and PSO has been proposed to control the frequency deviation and power deviations in each power generation unites. The results of the frequency deviation obtained by using FO-fuzzy-PID controllers with OWOA tuned are 1.05%, 2.01%, and 2.73% lower than when QOHSA, TBLOA, and PSO have been used to tune, respectively. Through this analysis, the algorithm’s efficiency is determined. Sensitivity studies are also carried out to demonstrate the robustness of the technique under consideration in relation to changes in the sizes of the HRES and ES system parameters.

Research on Manufacturing Equipment Operation State Evaluation Technology Based on Fractional Calculus

The operational status of manufacturing equipment is directly related to the reliability of the operation of manufacturing equipment and the continuity of operation of the production system. Based on the analysis of the operation status of manufacturing equipment and its characteristics, it is proposed that the concept of assessing the operation status of manufacturing equipment can be realized by applying the real-time acquisition of accurate inspection data of important parts of weak-motion units and comparing them with their motion status evaluation criteria. A differential data fusion model based on the fractional-order differential operator is established through the study of the application characteristics of fractional-order calculus theory. The advantages of Internet of Things (IoT) technology and a fractional order differential fusion algorithm are integrated to obtain real-time high-precision data of the operating parameters of manufacturing equipment, and the research objective of the operating condition assessment of manufacturing equipment is realized. The feasibility and effectiveness of the method are verified by applying the method to the machining center operation status assessment.

Analysis and achievement for fractional optimal control of Hepatitis B with Caputo operator

Deciphering the effect of long memory on dynamic behavior is important for controlling disease transmission. Employing fractional order calculus, here, we propose a model with Caputo operator to emphasize the regulation effect of the long memory on the optimal control of Hepatitis B. Considering four different health classes of populations, susceptible, acute infections, chronic infections, and recovered, we try to elucidate the existence of the optimal solution of the fractional optimal control problems (FOCP) by applying a strategy of isolation, treatment, and vaccination. Initially, we discuss the locally asymptotically stability of the equilibrium solutions of the fractional differential model. Sensitivity analysis and numerical simulations are also carried out to verify how parameter changes affect the system dynamics, that is, the threshold effect depending on the basic reproduction number, which provides inspiration for constructing the controlled system. After that, the necessary conditions for the optimality of the Hepatitis B system are derived to focus on the Caputo fractional operator with a singular kernel. An effective numerical simulation algorithm based on trapezoidal approximation is used to validate the control effect in the sense of the transient response by comparing the difference between the fractional-order and integer-order derivatives, and the numeric results can vividly show the attenuation effect of memory on control. This study may provide a powerful rationale for uncovering the importance of the long-memory effects in disease control.

Dual-Loop Voltage–Current Control of a Fractional-Order Buck-Boost Converter Using a Fractional-Order PIλ Controller

Based on the fact that the inductor and capacitor are of a non-integer order by nature, to provide a more accurate theoretical basis for the optimal control of the converter, the fractional-order model of the Buck-Boost converter in the continuous mode of current is established according to the fractional-order calculus theory. The fractional-order PIλ control system of the fractional-order Buck-Boost converter is designed to compare the performance of the integer-order PI controller with the fractional-order controller. Secondly, the sparrow search algorithm is applied to the optimal design of the fractional-order PIλ control system of the fractional-order Buck-Boost converter to improve the system’s phase margin, stability, and robustness. Finally, the simulation is verified on the Matlab/Simulink simulation platform and compared with the integer-order PI controller.

A Hybrid Approximation Method for Integer-Order Approximate Realization of Fractional-Order Derivative Operators

The use of fractional-order (FO) calculus for the solution of different problems in many fields has increased recently. However, the usage of FO system models in practice brings some difficulties. The FO operator, fractance device, is usually realized via several integer-order approximation methods, which have pros and cons in the aspect of operation frequency, time response and stability region. These methods may not meet all performance expectations. In this regard, author proposes an efficient hybrid integer-order approximation method for FO derivative operator without causing any additional difficulty in realization. The proposed method combines Matsuda and modified stability boundary locus (M-SBL) approximation methods. The advantage of each method is combined in a single hybrid function by considering root mean square error (RMSE) rates for step response. The performance of hybrid transfer function is analyzed in comparison with Matsuda, Oustaloup, continued fraction expansion (CFE) and M-SBL transfer functions for both frequency and time response. Analog realization of the proposed model is performed experimentally via partial fraction expansion method. Analog design is verified via both Multisim simulations and experimental results. The improvements due to the hybrid behavior and the consistency of experimental results with theoretical and simulation results demonstrate the practicality and usefulness of the hybrid model.

Adaptive fuzzy variable fractional-order sliding mode vibration control of uncertain building structures

A novel robust adaptive fuzzy variable fractional-order sliding mode control (FVFOSMC) strategy is proposed to suppress earthquake-induced vibrations on uncertain building structures. Firstly, a new variable fractional-order sliding mode controller (VFOSMC) is established by embedding the concepts of variable fractional-order (VFO) calculus into a sliding mode controller. Then, the FVFOSMC is designed by using VFO fuzzy logic with online adaptive approximation to avoid chattering or inaccurate switching. The global stability of the closed-loop system is proved based on VFO Lyapunov stability theory. Three examples illustrate the performance of the proposed FVFOSMC. The results confirm the robustness and superior performance of the new method in the presence of uncertain structure parameters and external unknown earthquake excitation.

The boundedness, global Mittag-Leffler sability and S-asymptotic ω-periodic of fractional-order fuzzy inertial neural networks with delays

The boundedness, global Mittag-Leffler stability (GMLS), and S-asymptotic ω-periodic of fuzzy fractional-order inertial neural networks (FINN) with delays are discussed. Using the properties of Riemann-Liouville fractional-order calculus, variable substitutions and the property of fuzzy functions are adopted to get the boundedness, the GMLS, and the S-asymptotic ω-periodic of the system. Furthermore, a numerical example is given to demonstrate the theorems.

Fractional‐order fault‐tolerant containment control of multiple fixed‐wing UAVs via disturbance observer and interval type‐2 fuzzy neural network

AbstractThis article investigates the fault‐tolerant containment control (FTCC) problem for a group of fixed‐wing unmanned aerial vehicles with simultaneous considerations of faults and collision avoidance. A fractional‐order (FO) FTCC scheme is established to steer all follower UAVs into the convex hull formed by the leader UAVs with the involvements of FO calculus, disturbance observers (DOs), and interval type‐2 fuzzy neural networks (IT2FNNs). In the proposed control protocol, FO sliding‐mode surfaces with artificial potential functions are first designed to revamp the filtered containment errors. Then, the DOs with FO calculus are constructed to estimate the FO lumped disturbances due to faults and external disturbances. Moreover, to compensate for the DO estimation errors, IT2FNN learning mechanisms are introduced to improve the FTCC capability. It is shown by Lyapunov stability analysis that all follower UAVs can successfully converge into the convex hull spanned by the leader UAVs without collisions even when a portion of UAVs is encountered by faults. Simulation results are presented to show the effectiveness of the developed control scheme.

Filtering Structures for α-Stable Systems

Recent years have brought to attention filtering and state estimation paradigms in systems that exhibit rapidly changing states. Dynamics of such systems falls beyond what can be accurately modelled using the ubiquitous Gaussian statistics, and thus, classical state estimation and filtering techniques have started to lose their effectiveness. In order to formulate a more general filtering approach, in this letter, the paradigm of adaptive filtering is revised from the perspective of the characteristic function. The so formulated approach is based on fractional-order calculus and is designed to deal with the more general setting of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -stable statistics in an efficient manner. Formulating the filtering structure form perspective of the characteristic function accommodates for general adaptive filtering structures parametrised by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> factor, while fractional-order calculus allows for mathematically tractable solutions. Thus, accommodating demands of modern filtering applications. Finally, effectiveness of the derived framework is demonstrated over a number of numerical simulations.

Computational modeling of financial crime population dynamics under different fractional operators

&lt;abstract&gt;&lt;p&gt;This paper presents an analysis and numerical simulation of financial crime population dynamics using fractional order calculus and Newton's polynomial. The dynamics of financial crimes are modeled as a fractional-order system, which is then solved using numerical methods based on Newton's polynomial. The results of the simulation provide insights into the behavior of financial crime populations over time, including the stability and convergence of the systems. The study provides a new approach to understanding financial crime populations and has potential applications in developing effective strategies for combating financial crimes. Fractional derivatives are commonly applied in many interdisciplinary fields of science because of its effectiveness in understanding and analyzing complicated phenomena. In this work, a mathematical model for the population dynamics of financial crime with fractional derivatives is reformulated and analyzed. A fractional-order financial crime model using the new Atangana-Baleanu-Caputo (ABC) derivative is introduced. The reproduction number for financial crime is calculated. In addition, the relative significance of model parameters is also determined by sensitivity analysis. The existence and uniqueness of the solution in consideration of the ABC derivative are discussed. A number of conditions are established for the existence and Ulam-Hyers stability of financial crime equilibria. A numerical scheme is presented for the proposed model, starting with the Caputo-Fabrizio fractional derivative, followed by the Caputo and Atangana-Baleanu fractional derivatives. Finally, we solve the models with fractal-fractional derivatives.&lt;/p&gt;&lt;/abstract&gt;

FRACTIONAL CALCULUS OPERATORS–BLOCH–TORREY PARTIAL DIFFERENTIAL EQUATION–ARTIFICIAL NEURAL NETWORKS–COMPUTATIONAL COMPLEXITY MODELING OF THE MICRO–MACROSTRUCTURAL BRAIN TISSUES WITH DIFFUSION MRI SIGNAL PROCESSING AND NEURONAL MULTI-COMPONENTS

Fractional calculus and fractional-order calculus are arranged in lineage as regards the mathematical models with complexity-theoretical tenets capable of capturing the subtle molecular dynamics by the integration of power-law convolution kernels into time- and space-related derivatives emerging in equations concerning the Magnetic Resonance Imaging (MRI) phenomena to which the fractional models of diffusion and relaxation are applied. Endowed with an intricate level of complexity and a unique physical and structural scaffolding at molecular and cellular levels with numerous synapses forming elaborate neural networks which entail in-depth probing and computing of patterns and signatures in individual cells and neurons, human brain as a heterogeneous medium is constituted of tissues with cells of different sizes and shapes, distributed across an extra-cellular space. Characterization of the unique brain cells is sought after to unravel the connections between different cells and tissues for accurate, reliable, robust and optimal models and computing. Accordingly, Diffusion Magnetic Resonance Imaging (DMRI), as a noninvasive and experimental imaging technique with clinical and research applications, provides a measure related to the diffusion characteristics of water in biological tissues, particularly in the brain tissues. Compatible with these aspects and beyond the diffusion coefficients’ measurement, DMRI technique aims to exceed the spatial resolution of the MRI images and draw inferences from the microstructural properties of the related medium. Thus, novel tools become essential for the description of the biological (organelles, membranes, macromolecules and so on) and neurological (axons, dendrites, neurons and so forth) tissues’ complexity. Mathematical model-based computational analyses with multifaceted methods to extract information from the DMRI with SpinDoctor into neuronal dynamics can provide quantitative parametric instruments in order to reflect the tissue properties focusing on the precise link between the tissue microstructure and signals acquired by employing advanced medical imaging technologies. Coalesced with accurate neuron geometry models as well as numerical DMRI simulations, a novel extended and multifaceted predictive mathematical model based on SpinDoctor and Bloch–Torrey partial differential equation (BTPDE) with the Caputo fractional-order derivative (FOD) with three-parameter [Formula: see text] Mittag-Leffler function (MLF) has been proposed and developed in our study by extending for the application on Brain Neuron Spin Unit dataset with the relevant multi-stage application-related steps. The feedforward neural networks (FFNNs) with BFGS Quasi-Newton equation, as one of the artificial neural network (ANN) algorithms, are applied on BTPDE with Caputo fractional-order derivative for the neurons and their algorithmic complexity is computed by building a BTPDE with Caputo FOD Neuron model based on different fractional orders. The fractional-order degree of the proposed and developed model is applied in relation to their corresponding complexity degrees. Consequently, experimentation and observations from the simulation-driven FFNN (with BFGS Quasi-Newton equation) learning scheme applied to the Bloch–Torrey PDE–Caputo FOD with MLF Neuron model (named as FFNN–BTPDE–CFODMLF Neuron model) proposed in this study, are made. Thus, by investigating whether the mathematical models based on the accurate neuron geometry models obtained can be optimized by comparing the errors in order to define the order parameter and identify to what optimal extent the errors are in relation to the prediction results with a particular focus on the neuron model, we have been able to estimate and predict brain microstructure through DMRI, accentuating mathematical and medical contributions based on the exploitation and corroboration of powerful modeling as well as computational capabilities.

Image Enhancement Using Adaptive Fractional Order Filter

Image enhancement is an important preprocessing task as the contrast is low in most of the medical images, Therefore, enhancement becomes the mandatory process before actual image processing should start. This research article proposes an enhancement of the model-based differential operator for the images in general and Echocardiographic images, the proposed operators are based on Grunwald-Letnikov (G-L), Riemann-Liouville (R-L) and Caputo (Li & Xie), which are the definitions of fractional order calculus. In this fractional-order, differentiation is well focused on the enhancement of echocardiographic images. This provoked for developing a non-linear filter mask for image enhancement. The designed filter is simple and effective in terms of improving the contrast of the input low contrast images and preserving the textural features, particularly in smooth areas. The novelty of the proposed method involves a procedure of partitioning the image into homogenous regions, details, and edges. Thereafter, a fractional differential mask is appropriately chosen adaptively for enhancing the partitioned pixels present in the image. It is also incorporated into the Hessian matrix with is a second-order derivative for every pixel and the parameters such as average gradient and entropy are used for qualitative analysis. The wide range of existing state-of-the-art techniques such as fixed order fractional differential filter for enhancement, histogram equalization, integer-order differential methods have been used. The proposed algorithm resulted in the enhancement of the input images with an increased value of average gradient as well as entropy in comparison to the previous methods. The values obtained are very close (almost equal to 99.9%) to the original values of the average gradient and entropy of the images. The results of the simulation validate the effectiveness of the proposed algorithm.

EDITORIAL SPECIAL ISSUE: PART IV-III-II-I SERIES

EDITORIAL SPECIAL ISSUE: PART IV-III-II-I SERIES

Application of Fractional Derivatives in Characterization of ECG graphs

In this paper, we have discussed the prediction of Left Ventricular Hypertrophy (LVH) and Right Ventricular Hypertrophy (RVH) of the heart from an Electrocardiogram (ECG) Graph using fractional order calculus. An ECG is a rough or unreachable curve that is continuous everywhere but non-differentiable at some points or all points where classical calculus fails. The purpose of this paper is to find left and right fractional derivatives at those non-differentiable points and then predict LVH and RVH by calculating the phase transition values (absolute difference of left and right fractional derivatives). Fractal dimension and Hurst exponents of V1, V2, V5, and V6 leads of the ECGs have been calculated for both problematic and normal ECGs. All such measures will help doctors to diagnose LVH and RVH from ECG in a more accurate manner as compared to the techniques which are under the classical order method.

Control Tuning for the Quadcopter Unmanned Aerial Vehicle Based on Genetic Evolutionary Algorithm

The quadcopter is an unmanned aerial vehicle that has been used in civil and military applications because of its relatively low cost, simple design, and ease of maintenance in addition to its unique flying characteristics and vast potential. Six Proportional-Integral-Derivative (PID) controllers are usually built for the position and attitude control of the quadcopter. However, tuning the controllers’ parameters poses a real challenge, as the system is highly coupled and contains two cascaded control loops while being underactuated (only four control inputs for the six degrees of freedom). In this study, a quadcopter model has been implemented by using MATLAB / SIMULINK where the model has been constructed to be integrated with the evolutionary genetic algorithm in addition to a third-party toolbox that is used to simulate fractional order calculus. The gains of the traditional and the fractional-order proportional-integral-derivative controllers have been tuned by using the genetic algorithm against different path shapes. The performance of the manually and the genetic algorithm tuned proportional-integral-derivative controllers have been compared with the genetic algorithm tuned fractional-order proportional-integral-derivative controllers. The genetic algorithm tuned fractional-order proportional-integral-derivative controllers have been superior and more robust in all cases with a lower fitness function value, much smother responses, and higher trajectory tracking ability.

Differentiation of benign and malignant breast lesions using diffusion-weighted imaging with a fractional-order calculus model

PurposeTo assess the feasibility of using three diffusion parameters (D, β, and μ) derived from fractional-order calculus (FROC) diffusion model for improving the differentiation between benign and malignant breast lesions. MethodIn this prospective study, 103 patients with breast lesions were enrolled. All subjects underwent diffusion-weighted imaging (DWI) with 12b values. Inter-observer agreement with respect to quantification of parameters by two radiologists was assessed using intraclass coefficient. Conventional apparent diffusion coefficient (ADC) and three FROC model parameters D, β, and μ were compared between the benign lesion and malignant lesion groups using the Mann-Whitney U test. Then, a comprehensive prediction model was created by using binary logistic regression. Receiver operating characteristic (ROC) curve analysis was performed to evaluate the diagnostic performance of the parameters using histopathological diagnosis as the reference standard. ResultsThe FROC parameters and ADC all exhibited significant differences between benign lesions and malignant lesions (P＜0.001). Among the individual parameters, the sensitivity of μ was higher than ADC (95.92% for μ vs 91.84% for ADC), and the specificity of β was higher than ADC (72.22% for β vs 70.37% for ADC). The combination of ADC and FROC parameters (D and β) generated the largest area under the ROC curve (0.841) when compared with individual parameters, indicating an improved performance for differentiating benign lesions from malignant lesions. ConclusionsThis study demonstrated the feasibility of using the FROC diffusion model to improve the accuracy of identifying malignant breast lesions.

A hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system for the travelling salesman problem

AbstractThe ant colony optimization (ACO) is one efficient approach for solving the travelling salesman problem (TSP). Here, we propose a hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system (SSMFAS) to address the TSP. The state-adaptive slime mold (SM) model with two targeted auxiliary strategies emphasizes some critical connections and balances the exploration and exploitation ability of SSMFAS. The consideration of fractional-order calculus in the ant system (AS) takes full advantage of the neighboring information. The pheromone update rule of AS is modified to dynamically integrate the flux information of SM. To understand the search behavior of the proposed algorithm, some mathematical proofs of convergence analysis are given. The experimental results validate the efficiency of the hybridization and demonstrate that the proposed algorithm has the competitive ability of finding the better solutions on TSP instances compared with some state-of-the-art algorithms.