Fractional Method Research Articles

A New Fractional Approach for the Higher-Order q-Taylor Method

The main goal of this work is to propose a new fractional approach of the higher order q-Taylor method with Initial Value Problems (IVPs) for fractional q-difference equations which is called the Fractional Higher-Order q-Taylor Method (FHOqTM). By applying the generalised q-Taylor theorem, this would be achieved. As a consequence, we calculate the FHOqTM’s local truncation error. Finally, we present numerical applications to validate our results by comparing the exact solution and the approximatesolution resulting from (FHOqTM).

Cerebral white matter myelination is associated with longitudinal changes in processing speed across the adult lifespan

Abstract Myelin's role in processing speed is pivotal, as it facilitates efficient neural conduction. Its decline could significantly affect cognitive efficiency during aging. In this work, myelin content was quantified using our advanced MRI method of myelin water fraction (MWF) mapping. We examined the relationship between MWF at the time of MRI and retrospective longitudinal change in processing speed among 121 cognitively unimpaired participants, aged 22 to 94 years, from the Baltimore Longitudinal Study of Aging and the Genetic and Epigenetic Signatures of Translational Aging Laboratory Testing (mean follow-up duration of 4.3 ± 6.3 years) using linear mixed-effects models, adjusting for demographics. We found that higher MWF values correlated with longitudinally better-maintained processing speed, with particularly significant associations in several white matter regions. Detailed voxel-wise analysis provided further insight into the specific white matter tracts involved. This research underscores the essential role of myelin in preserving processing speed and highlights its potential as a sensitive biomarker for interventions targeting age-related cognitive decline, thereby offering a foundation for preventative strategies in neurological health.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

Numerical Simulation of Droplet Coalescence Using Meshless Radial Basis Function and Domain Decomposition Method

The present investigation of the dynamic two-binary droplet interactions has gained attention since its use to expand and improve several numerical methods. Generally, its interactions are classified into coalescence, bouncing, reflective, and stretching separation. This study simulated droplet coalescence using the meshless radial basis function (RBF) method. These methods are used to solve the Navier-Stokes equations combined with the Cahn-Hilliard equations to track the interface between two fluids. This work uses the fractional step method to calculate the pressure-velocity coupling in the Navier-Stokes equations. The numerical results were compared with the available data in the literature to validate the proposed method. Based on the validation, the proposed method conforms well with the literature. To identify further coalescence characteristics, the model considered different values in viscosity (2, 4, and 8 cP), collision velocity (1.5 m/s and 3 m/s), and surface tension (0.014, 0.028, and 0.056 N/m) parameters. The increasing viscosity was linearly proportional to the collision time, whereas increased surface tension and collision velocity shortened the collision time.

Fractional‐order fast terminal sliding mode trajectory tracking control of quadrotor UAV based on finite time disturbance observer

AbstractIn this paper, a novel trajectory tracking control problem of the quadrotor UAV is addressed, which considers the system uncertainties and unknown external disturbances meanwhile. To ensure a rapid convergence rate of the states, fast terminal sliding mode control (FTSMC) algorithm is improved combining with a fractional order method. In addition, finite‐time observer is designed to handle disturbances and enhance the robustness of the system. The closed‐loop finite‐time stability of the quadrotor UAV system is established using Lyapunov theory in accordance with the novel control law. To evaluate the effectiveness of our proposed strategy, various simulations under different scenarios are conducted respectively. These simulation results clearly demonstrate the superiority of our control scheme, which we refer to as fractional‐order fast terminal sliding mode control (FOFTSMC).

Fractional differentiation method: Some applications to the theory of subdiffusion-controlled reactions

Fractional differentiation method: Some applications to the theory of subdiffusion-controlled reactions

Anchored Chitosan-Functionalized Magnetite Nanoparticles for Crystal Violet Decolorization from Aqueous Samples.

In this research, 3-(triethoxysilyl)propyl isocyanate (TESPIC) functionalized chitosan was successfully synthesized to fabricate silica-coated magnetite nanoparticles (Fe3O4@SiO2-CS MNPs). The synthesized MNPs were characterized using XRD, FT-IR, SEM, and TEM instruments and were utilized for the decolorization of Crystal Violet cationic dye (CV). The affecting variables controlling CV removal efficiency were investigated using the Taguchi fractional factorial design method (L16 array). Under the optimized removal conditions (adsorbent amount = 0.12 g (4.8 g L-1), pH = 4, ionic strength = 0.05 mol L-1 NaCl, and 30 min stirring), 98.2% of the CV dye was eliminated. The kinetic and equilibrium adsorption isotherms were explained by the pseudo-second-order kinetic (R2 = 0.999) and Freundlich isotherm models, respectively. MATLAB's fmincon function as an efficient solution was applied in order to compare the Redlich-Peterson three-parametric isotherm model with two-parametric models. Moreover, the Fe3O4@SiO2-CS-TESPIC MNPs showed recyclability and reusability for subsequent runs. The findings confirmed that these functional MNPs can be considered as proper adsorbents for the removal of CV dye from the aqueous solutions.

An optimized fractionation method reveals insulin-induced membrane surface localization of GLUT1 to increase glycolysis in LβT2 cells

Insulin is an important regulator of whole-body glucose homeostasis. In insulin sensitive tissues such as muscle and adipose, insulin induces the translocation of glucose transporter 4 (GLUT4) to the cell membrane, thereby increasing glucose uptake. However, insulin also signals in tissues that are not generally associated with glucose homeostasis. In the human reproductive endocrine axis, hyperinsulinemia suppresses the secretion of gonadotropins from gonadotrope cells of the anterior pituitary, thereby linking insulin dysregulation to suboptimal reproductive health. In the mouse, gonadotropes express the insulin receptor which has the canonical signaling response of IRS, AKT, and mTOR activation. However, the functional outcomes of insulin action on gonadotropes are unclear. Here, we demonstrate through use of an optimized cell fractionation protocol that insulin stimulation of the LβT2 gonadotropic cell line results in the unexpected translocation of GLUT1 to the plasma membrane. Using our high purity fractionation protocol, we further demonstrate that though Akt signaling in response to insulin is intact, insulin-induced translocation of GLUT1 occurs independently of Akt activation in LβT2 cells.

A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease.

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (Sα), exposed (Eα), infected (Uα), quarantined (Qα), vaccinated (Vα), and recovered (Rα) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R0), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R0 sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R0 have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy's significant possibilities in the struggle against the COVID-19 pandemic.

A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids

A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids

Contradiction Hidden in Values of Urban Raster Data

Abstract. This paper discusses that raster data identified by the independent largest fraction method may include both manifest and hidden contradictions, and describes the underlying mechanisms of contradictions. Using the examples of zones of land use and floor-area ratio, we demonstrate that the cells including more than three different zone category combinations might be identified with a non-existing combination of the zone categories. For the zones in which the ratio of contradictory cells is comparatively large, this problem can be quite significant. We model, therefore, the probability of contradictory identification using the adjacency relationship of different categories, and demonstrate the good fitness of the model using actual urban raster data. The results reveal that there can be a significant proportion of hidden contradictions.

A Fractional Tikhonov Regularization Method for Identifying a Time-Independent Source in the Fractional Rayleigh–Stokes Equation

The aim of this paper is to identify a time-independent source term in the Rayleigh–Stokes equation with a fractional derivative where additional data are considered at a fixed time point. This inverse problem is proved to be ill-posed in the sense of Hadamard. By using a fractional Tikhonov regularization method, we construct a regularized solution. Then, according to a priori and a posteriori regularization parameter selection rules, we prove the convergence estimates of the regularization method. Finally, we provide some numerical examples to prove the effectiveness of the proposed method.

A q-theorem of Pólya using Hurwitz partial fraction method

AbstractWe establish a q-counterpart of the method of partial fraction developed by Hurwitz-Pólya to investigate the zeros of q-cosine and q-sine transforms, where $$q \in (0,1)$$ q ∈ ( 0 , 1 ) is a fixed number. We prove reality and simplicity of the zeros and give a precise description of their distribution. The conditions imposed on both q and the integrand are less restrictive than previously assumed in the literature. A direct infinite partial fraction expansion is obtained via q-sampling theory.

A Procedure for Solid-Phase Extractions Using Metal-Oxide-Coated Silica Column in Lipidomics.

Lipid enrichment is indispensable for enhancing the coverage of targeted molecules in mass spectrometry (MS)-based lipidomics studies. In this study, we developed a simple stepwise fractionation method using a titanium- and zirconium-dioxide-coated solid-phase extraction (SPE) silica column that separates neutral lipids, phospholipids, and other lipids, including fatty acids (FAs) and glycolipids. Chloroform was used to dissolve the lipids, and neutral lipids, including steryl esters, diacylglycerols, and triacylglycerols, were collected in the loading fraction. Second, methanol with formic acid (99:1, v/v) was used to retrieve FAs, ceramides, and glycolipids, including glycosylated ceramides and glycosylated diacylglycerols, by competing for affinity with the Lewis acid sites on the metal oxide surface. Finally, phospholipids strongly retained via chemoaffinity interactions were eluted using a solution containing 5% ammonia and high water content (45:50 v/v, 2-propanol:water), which canceled the electrostatic and chelating interactions with the SPE column. High average reproducibility of <10% and coverage of ∼100% compared to those of the non-SPE samples were demonstrated by untargeted lipidomics of human plasma and mouse brain, testis, and feces. The advantage of our procedure was showcased by characterizing minor lipid subclasses, including dihexosylceramides containing very long-chain polyunsaturated FA in the testis, monogalactosyl and digalactosyl monoacylglycerols in feces, and acetylated and glycolylated derivatives of gangliosides in the brain that were not detected using conventional solvent extraction methods. Likewise, the value of our method in biology is maximized during glycolipidome profiling in the absence of neutral lipids and phospholipids that cover more than 80% of the chromatographic peaks.

Enhanced CO2 hydrogenation reaction by Tuning interfacial Cu/ZnOx through synergistic interactions in the precursors

The Cu/ZnO/Al2O3 is a typical industrial catalyst for water–gas-shift reaction and methanol synthesis, and is also gaining momentum in CO2 hydrogenation reaction. The dynamic evolution of the phases and microstructures of the precursor of this catalyst leads to a notable synergistic effect that defines its overall catalytic function and performance. To gain insights into the role and interaction between the relevant precursors, we compared Cu/ZnO/Al2O3 catalysts using a conventional co-precipitation and a fractional precipitation method, where the latter one shows an enhanced Cu/ZnOx interface due to a thorough and strong interaction between two components in the precursor. The ZnOx decoration on Cu with unsaturated Znδ+ species boosted the methanol formation to a rate of 508 gCH3OH‧kgcat−1‧h−1 with 58 % selectivity at 513 K and 3 MPa. This work provides mechanistic insights into the synergistic interplay between the involved phases in the Cu/ZnO/Al2O3 catalyst.

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

A Small-Sample Classification Strategy for Extracting Fractional Cover of Native Grass Species and Noxious Weeds in the Alpine Grasslands.

The fractional cover of native grass species (NGS) and noxious weeds (NW) provides a more comprehensive understanding of grassland health in the alpine grasslands. However, coverage extraction of NGS and NW from satellite hyperspectral imagery can be challenging due to the small spectral and spatial feature difference, insufficient training samples, and the lack of effective fractional cover extraction methods. In this research, firstly, a feature optimization method is proposed to optimize the difference feature between NGS and NW. Secondly, a spectral-spatial constrained re-clustering training sample extension method (SSCTSE) is proposed to increase the number of training samples. Thirdly, a composite three-kernel SVM method (CTK-SVM) is developed to produce fractional cover maps of NGS and NW. The experimental results show that (1) the feature optimization method is effective in preserving the spectral and spatial difference features while eliminating invalid features; (2) the SSCTSE algorithm is capable of significantly increasing the number of training samples; (3) the fractional cover maps of NGS and NW are produced with the CTK-SVM method with overall accuracies of approximately 65%, and the RMSEs of NGS and NW are approximately 16% and 11%, respectively. The results provide a foundation for the fractional cover extraction of different grass species in alpine grasslands based on satellite hyperspectral imagery.

Fractional-order spike-timing-dependent gradient descent for multi-layer spiking neural networks

Accumulated detailed knowledge about the neuronal activities in human brains has brought more attention to bio-inspired spiking neural networks (SNNs). In contrast to non-spiking deep neural networks (DNNs), SNNs can encode and transmit spatiotemporal information more efficiently by exploiting biologically realistic and low-power event-driven neuromorphic architectures. However, the supervised learning of SNNs still remains a challenge because the spike-timing-dependent plasticity (STDP) of connected spiking neurons is difficult to implement and interpret in existing backpropagation learning schemes. This paper proposes a fractional-order spike-timing-dependent gradient descent (FO-STDGD) learning model by considering a derived nonlinear activation function that describes the relationship between the quasi-instantaneous firing rate and the temporal membrane potentials of nonleaky integrate-and-fire neurons. The training strategy can be generalized to any fractional orders between 0 and 2 since the FO-STDGD incorporates the fractional gradient descent method into the calculation of spike-timing-dependent loss gradients. The proposed FO-STDGD model is tested on the MNIST and DVS128 Gesture datasets and its accuracy under different network structure and fractional orders is analyzed. It can be found that the classification accuracy increases as the fractional order increases, and specifically, the case of fractional order 1.9 improves by 155 % relative to the case of fractional order 1 (traditional gradient descent). In addition, our scheme demonstrates the state-of-the-art computational efficacy for the same SNN structure and training epochs.