Fractional Order Derivative Research Articles

Fractional-Order Mathematical Modeling of Breast Cancer: Comparing Adaptive Immune Responses and Estrogen Dynamics with Experimental Data

Breast cancer is a significant global health concern that requires innovative approaches to understand its behavior and improve treatment strategies. This paper introduces a new fractional-order mathematical model to explain the complex dynamics of breast cancer progression, including adaptive immune responses and estrogen dynamics. Utilizing Caputo fractional derivatives, our model reveals insights into the impact of fractional-order dynamics on cancer cell populations. Simulation results demonstrate a notable increase in cell populations with higher fractional orders, suggesting heightened aggressiveness, while lower orders correspond to subdued progression. Unlike traditional integer-order models, fractional-order derivatives offer a more nuanced depiction of nonlinear dynamics, crucial for capturing the complexities of cancer progression. Importantly, our findings underscore the potential clinical relevance of fractional-order models in informing personalized treatment strategies, particularly through the modulation of estrogen levels. By integrating treatment considerations, such as hormone therapy, our model holds promise for advancing precision medicine approaches tailored to individual patient characteristics.

Fractional-Order Modeling and Stochastic Dynamics Analysis of a Nonlinear Rubbing Overhung Rotor System

To study the nonlinear dynamic behavior and system stability of a rubbing overhung rotor with viscoelastic and memory-effect damping and random uncertain parameters, this paper introduces a fractional-order modeling and stochastic dynamic analysis method for the nonlinear overhung rotor system with frictional impact faults. Firstly, the dynamic equations of the overhung rotor considering friction effect and fractional damping effect are established based on the transfer matrix method and fractional order derivative. Then, the time-domain response of the fractional-order dynamic equations is solved by combining the Runge–Kutta method and the continuous fractional expansion, and the steady-state response characteristics of different fractional damping are analyzed in the deterministic case. Finally, to analyze the response of the system under the effect of stochastic parameters, the sparse grid-based PCE metamodel of the system response is developed. Statistical moments, probability distributions, and sensitivity indices of the response of stochastic systems are revealed. The results of this paper provide a theoretical basis for efficient and accurate prediction of the stochastic response of nonlinear rubbing overhung rotor systems.

Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach

Abstract The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations.

Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.

Stability of impulsive delayed switched systems with conformable fractional-order derivatives

In this paper, exponential stability is addressed for impulsive delayed switched systems with conformable derivatives by Halanay inequalities technique. First, based on the fractional-order monotonicity theorem and the definition of conformable fractional-order derivative, novel conformable fractional-order Halanay inequalities are developed that extend and improve the existing ones. Second, using the obtained Halanay inequalities and the estimate of conformable fractional-order exponential functions, globally exponential stability conditions that depend on the infimum and supremum of impulsive step sizes are proposed for the systems. Third, more relaxing stability conditions are presented that do not depend on the infimum and supremum of impulsive step sizes, which means that the conditions on impulses are weaker than those commonly used in the existing literature. Finally, examples are given to illustrate the validity of the theoretical results.

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0<α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

Faedo-Galerkin method for the time-fractional reaction-diffusion model

This paper investigates a time-fractional reaction-diffusion equation characterized by a Caputo derivative of fractional order , incorporates non-local and memory effects that are essential for accurately modeling complex diffusion processes in such environments. These effects arise from the fractal nature of the media, which leads to anomalous diffusion behavior that cannot be adequately described by classical diffusion models. We start by giving a definition of a weak solution for this problem, ensuring that the solution satisfies the equation in a generalized sense. Subsequently, the Faedo-Galerkin method combined with compactness arguments is employed to establish the existence and uniqueness of the weak solution. This approach involves approximating the solution using a sequence of finite-dimensional functions and then passing to the limit to obtain the desired solution . Moreover, we establish the stability of these solutions, which is fundamental for understanding their behavior under varying conditions and ensuring the reliability of the model . Our findings contribute to a deeper understanding of anomalous diffusion phenomena in fractal media and provide a solid foundation for further research in this area.

Exponential Quasi-Synchronization of Fractional-Order Fuzzy Cellular Neural Networks via Impulsive Control

This paper investigates the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks with parameters mismatch via impulsive control. Firstly, under the framework of the generalized Caputo fractional-order derivative, a new fractional-order impulsive differential inequality is established. Secondly, based on this fractional-order impulsive differential inequality, a general criterion for the quasi-synchronization of fractional-order systems is obtained. Then, specific to the fractional-order fuzzy cellular neural network model in this paper, the criteria and error estimation of the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks can be obtained. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

Estimation of the soluble solid content of citrus based on the fractional-order derivative and optimal band combination algorithm.

The soluble solid content (SSC) is a primary characteristic index for evaluating the internal quality of citrus fruits. The development of rapid and nondestructive SSC detection techniques can help address the current issues of postharvest quality grading in China's citrus industry. In this study, a total of 261 experimental samples, including 70 Murcott, 91 Clementine, and 100 Navel orange, were divided into prediction and validation sets in a 7:3 ratio. After obtaining the reflection spectra and SSCs, SNV-FOD (Standard Normal Variate-Fractional-Order Derivative) was used to process the spectra, and the optimal band combination algorithm was introduced to select SSC-sensitive bands. Then, the obtained optimal dual-band combination was input into eight regression models for comparison, and the best performing models stacked ensemble models was selected. Finally, the H-ELR (HyperOpt-optimized ensemble learning regression) model, optimized using a Bayesian function, was applied for the effective estimation of SSC for three common citrus varieties in Guangxi, Murcott, Clementine, and Navel oranges. The results show that (1) the SNV-FOD preprocessing method proposed in this study improved the correlation coefficient with the SSC from 0.546 to 0.836 compared to that of the original spectrum, (2) the optimal dual-band combination (969 and 1069nm) constructed by integrating the differential index and 1.2-order derivative yielded the most accurate results (RPD=2.13), and (3) the H-ELR model, based on HyperOpt optimization, achieved good estimated performance (RPD=2.46). PRACTICAL APPLICATION: This research contributes to the development of practical SSC prediction instruments with excellent universality and ease of application.

Fractional Einstein–Gauss–Bonnet Scalar Field Cosmology

Our paper introduces a new theoretical framework called the Fractional Einstein–Gauss–Bonnet scalar field cosmology, which has important physical implications. Using fractional calculus to modify the gravitational action integral, we derived a modified Friedmann equation and a modified Klein–Gordon equation. Our research reveals non-trivial solutions associated with exponential potential, exponential couplings to the Gauss–Bonnet term, and a logarithmic scalar field, which are dependent on two cosmological parameters, m and α0=t0H0 and the fractional derivative order μ. By employing linear stability theory, we reveal the phase space structure and analyze the dynamic effects of the Gauss–Bonnet couplings. The scaling behavior at some equilibrium points reveals that the geometric corrections in the coupling to the Gauss–Bonnet scalar can mimic the behavior of the dark sector in modified gravity. Using data from cosmic chronometers, type Ia supernovae, supermassive Black Hole Shadows, and strong gravitational lensing, we estimated the values of m and α0, indicating that the solution is consistent with an accelerated expansion at late times with the values α0=1.38±0.05, m=1.44±0.05, and μ=1.48±0.17 (consistent with Ωm,0=0.311±0.016 and h=0.712±0.007), resulting in an age of the Universe t0=19.0±0.7 [Gyr] at 1σ CL. Ultimately, we obtained late-time accelerating power-law solutions supported by the most recent cosmological data, and we proposed an alternative explanation for the origin of cosmic acceleration other than ΛCDM. Our results generalize and significantly improve previous achievements in the literature, highlighting the practical implications of fractional calculus in cosmology.

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

The Investigation of Nonlinear Time-Fractional Models in Optical Fibers and the Impact Analysis of Fractional-Order Derivatives on Solitary Waves

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering.

Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization.

Optimizing Fractional-Order Convolutional Neural Networks for Groove Classification in Music Using Differential Evolution

This study presents a differential evolution (DE)-based optimization approach for fractional-order convolutional neural networks (FOCNNs) aimed at enhancing the accuracy of groove classification in music. Groove, an essential element in music perception, is typically influenced by rhythmic patterns and acoustic features. While FOCNNs offer a promising method for capturing these subtleties through fractional-order derivatives, they face challenges in efficiently converging to optimal parameters. To address this, DE is applied to optimize the initial weights and biases of FOCNNs, leveraging its robustness and ability to explore a broad solution space. The proposed DE-FOCNN was evaluated on the Janata dataset, which includes pre-rated music tracks. Comparative experiments across various fractional-order values demonstrated that DE-FOCNN achieved superior performance in terms of higher test accuracy and reduced overfitting compared to a standard FOCNN. Specifically, DE-FOCNN showed optimal performance at fractional-order values such as v = 1.4. Further experiments demonstrated that DE-FOCNN achieved higher accuracy and lower variance compared to other popular evolutionary algorithms. This research primarily contributes to the optimization of FOCNNs by introducing a novel DE-based approach for the automated analysis and classification of musical grooves. The DE-FOCNN framework holds promise for addressing other related engineering challenges.

Retraction Note: A study on the reflection of quasi Rayleigh waves in a magneto self reinforced thermoelastic medium with fractional order derivative

Retraction Note: A study on the reflection of quasi Rayleigh waves in a magneto self reinforced thermoelastic medium with fractional order derivative

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

Multidimensional and Nonlinear Fractional Langevin Equations: Advancing the Theory in of Anomalous Diffusion

This paper presents a comprehensive theoretical analysis of fractional Langevin equations (FLEs) and their applications in modeling anomalous diffusion processes. We extend the classical FLE framework in several significant directions to address the complexities observed in various physical, biological, and social systems. First, we derive a multidimensional extension of the FLE, providing a robust tool for studying anomalous diffusion in higher-dimensional spaces. Second, we incorporate non-Gaussian Lévy α-stable noise into the FLE, enabling the description of systems characterized by heavy-tailed distributions and extreme events. Third, we develop a nonlinear variant of the FLE to model more intricate restoring forces and potential landscapes. Finally, we analyze a generalized FLE with different orders of fractional derivatives for the inertial and friction terms, offering a flexible framework capable of capturing a wider range of anomalous diffusion phenomena. For each extension, we prove the existence and uniqueness of solutions, derive the corresponding probability distributions, and analyze the scaling behavior of the mean squared displacement in the long-time limit. Our results demonstrate how these generalized FLEs can describe various regimes of anomalous diffusion, from subdiffusive to superdiffusive behavior. This work provides a unified theoretical foundation for understanding and modeling complex diffusive processes across diverse disciplines, paving the way for more accurate descriptions of systems exhibiting memory effects, non-Gaussian statistics, and multiscale dynamics.

Fractional-order stochastic gradient descent method with momentum and energy for deep neural networks

In this paper, a novel fractional-order stochastic gradient descent with momentum and energy (FOSGDME) approach is proposed. Specifically, to address the challenge of converging to a real extreme point encountered by the existing fractional gradient algorithms, a novel fractional-order stochastic gradient descent (FOSGD) method is presented by modifying the definition of the Caputo fractional-order derivative. A FOSGD with moment (FOSGDM) is established by incorporating momentum information to accelerate the convergence speed and accuracy further. In addition, to improve the robustness and accuracy, a FOSGD with moment and energy is established by further introducing energy formation. The extensive experimental results on the image classification CIFAR-10 dataset obtained with ResNet and DenseNet demonstrate that the proposed FOSGD, FOSGDM and FOSGDME algorithms are superior to the integer order optimization algorithms, and achieve state-of-the-art performance.

On Observer and Controller Design for Nonlinear Hadamard Fractional-Order One-Sided Lipschitz Systems

This paper presents an extensive investigation into the state feedback stabilization, observer design, and observer-based controller design for a specific category of nonlinear Hadamard fractional-order systems. The research extends the conventional integer-order derivative to the Hadamard fractional-order derivative, offering a more universally applicable method for system analysis. Furthermore, the traditional Lipschitz condition is adapted to a one-sided Lipschitz condition, broadening the range of systems amenable to analysis using these techniques. The efficacy of the proposed theoretical findings is illustrated through several numerical examples. For instance, in Example 1, we select an order of derivative r = 0.8; in Example 2, r is set to 0.9; and in Example 3, r = 0.95. This study enhances the comprehension and regulation of nonlinear Hadamard fractional-order systems, setting the stage for future explorations in this domain.

Electric quadrupole transitions of 98–112Ru atomic nuclei via conformable fractional Bohr Hamiltonian

In this paper, we investigate the energy spectra, wave functions and the [Formula: see text] transition rates for [Formula: see text]Ru atomic nuclei, using the conformable fractional Bohr Hamiltonian model. For the [Formula: see text]-part of the potential, the newly proposed Yukawa plus modified exponential potential is considered and the harmonic oscillator potential in [Formula: see text]-part, with [Formula: see text] fixed around [Formula: see text]. By using the conformable fractional Nikiforov–Uvarov method, energy spectra and wave functions are obtained analytically. The sensitivity of the potential parameters and the spectra with respect to the fractional order parameter is investigated. The normalized fractional energies and fractional electric quadrupole transition rates are compared with the available experimental predictions and those from existing theoretical studies. Comparisons are made at different values of the fractional derivative order. The results are presented across a broader range of values of [Formula: see text], providing a systematic analysis of how variations in this parameter influence the energy spectra, wave functions, and [Formula: see text] transition rates in Ru nuclei. Overall, the comparison of our results with experimental data shows the good accuracy of our model, especially when the fractional parameter goes to lower values. It can be concluded that the order of fractional derivative plays a crucial role in refining theoretical predictions of electric quadrupole transition rates, especially for transitions involving higher multipolarities.