Concept Of Fractional Derivative Research Articles

Variable order fractional diabetes models: numerical treatment

ABSTRACT In this paper, we present two numerical approximation solutions for the updated nonlinear variable order fractional differential equations of diabetes disease. The variable order fractional differential operator is based on the Caputo fractional derivative concept. The first used method is developed using the nonstandard finite difference process, such that a new quick approximation is applied to tackle the nonlinear terms. By applying some reasonable assumptions to the given data, we can prove that this scheme maintains the positivity and boundedness of the discretized solutions. The second method is based on approximation with shifted Jacobi polynomials. The properties of Jacobi polynomials, together with the shifted Jacobi-Gauss-Lobatto nodes were utilized to reduce the proposed system into a set of algebraic nonlinear equations. Furthermore, some numerical experiments are conducted to compare the performance of the introduced schemes. Our numerical simulations show that the nonstandard finite difference approach requires less computational time compared to the Jacobi-Gauss-Lobatto spectral collocation. While the last one is more accurate when used for solving the variable order fractional diabetes model.

Prabhakar fractional simulations for natural convective hybrid nanofluid mixed with Cu and Al2O3 nanoparticles flowing through a channel

This study seeks to investigate the heat transfer analysis of hybrid nanofluids containing fractionalized water and kerosene oil, which flow through a vertical channel via convection. To represent the problem in terms of fractional partial differential equations, we utilized the Prabhakar time-fractional derivative, a recent advancement in the concept of fractional derivatives. The governing equations were then solved using physical initial and boundary conditions that include momentum, concentration, and energy equations. Stehfest and Tzou's Laplace inversion techniques are used to provide semi-analytical solutions for governed equations, such as temperature, concentration, and momentum profiles, and we applied the fractional Laplace transformation to find solutions in the transform domain. Furthermore, the resultant solutions in tabular form are also derived using Tzou and Stehfest's numerical methods for Laplace inversion to ensure the accuracy of our findings. According to our research, an increase in volumetric fraction causes a reduction in fluid velocity. Due to the physical properties of the investigated nanoparticles, the water-based hybrid nanofluid has a bigger effect on the temperature and momentum profile than the kerosene oil-based hybrid nanofluid.

Fractional Gradient Optimizers for PyTorch: Enhancing GAN and BERT

Machine learning is a branch of artificial intelligence that dates back more than 50 years. It is currently experiencing a boom in research and technological development. With the rise of machine learning, the need to propose improved optimizers has become more acute, leading to the search for new gradient-based optimizers. In this paper, the ancient concept of fractional derivatives has been applied to some optimizers available in PyTorch. A comparative study is presented to show how the fractional versions of gradient optimizers could improve their performance on generative adversarial networks (GAN) and natural language applications with Bidirectional Encoder Representations from Transformers (BERT). The results are encouraging for both state-of-the art algorithms, GAN and BERT, and open up the possibility of exploring further applications of fractional calculus in machine learning.

Modelling Influenza A disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates

The objective of this manuscript is to present a novel approach to modeling influenza A disease dynamics by incorporating the Caputo-Fabrizio (CF) fractional derivative operator into the model. Particularly distinct contact rates between exposed and infected individuals are taken into account in the model under study, and the fractional derivative concept is explored with respect to this component. We demonstrate the existence and uniqueness of the solution and obtain the series solution for all compartments using the Laplace transform method. The reproduction number of the Influenza A model, which was created to show the effectiveness of different contact rates, was obtained and examined in detail in this sense. To validate our approach, we applied the predictor-corrector method in the sense of the Caputo-Fabrizio fractional derivative and demonstrate the effectiveness of the fractional derivative in accurately predicting disease dynamics. Our findings suggest that the use of the Caputo-Fabrizio fractional derivative can provide valuable insights into the mechanisms underlying influenza A disease and enhance the accuracy of disease models.

Nonstationary Stochastic Response of Hysteretic Systems Endowed With Fractional Derivative Elements

AbstractIn this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.

Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo–Katugampola Derivatives

Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area.

A mathematical model for simulating the transmission dynamics of COVID-19 using the Caputo–Fabrizio fractional-order derivative with nonsingular kernel

The emergence of the new coronavirus variant from the coronaviridae family has become a global concern, and all nations, including Bangladesh, are battling to contain the spread of the disease. In this article, we discuss a COVID-19 vaccination model with the Caputo–Fabrizio (CF) fractional order derivative approach to reveal the complicated action in Bangladesh. We determine the existence and uniqueness properties of the outcomes acquired from our proposed model subjected to the Picard–Lindelöf theorem. We perform the stability analysis by using the fixed-point theorem. We utilize Laplace Transform to evaluate the approximate solution of the model. In the numerical simulation, we consider a new approach called the four-step Adams-Bashforth Predictor–Corrector iteration scheme, which simulates that the fractional order provides more precise results. Finally, several numerical results are displayed with the different numbers of the order of the system. We depict the influence of vaccination in the coronavirus model with different parameter values. By analyzing the transmission dynamics of the virus, we bring to light the importance of several doses of vaccination to prevent disease transmission and contagions. We also conducted the sensitivity analysis for the model parameters to evaluate the impact of those parameters on disease outbreaks. The numerical results demonstrate significant information in the Caputo–Fabrizio fractional derivative concept and provide important insights into predicting disease transmission and control policies.

Enhanced fractional adaptive processing paradigm for power signal estimation

Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order).

Generalization of Reset Controllers to Fractional Orders

Reset control is a simple non-linear control technique that can help overcome the structural limitations of linear control. Fractional control uses the concept of fractional derivatives to expand the range of possibilities when modeling a controller, making it more robust. Fractional reset control merges the advantages of both areas and is the object of this paper. Fractional-order versions of different reset controllers were implemented, namely a fractional Clegg integrator, a fractional generalized first-order reset element, a fractional generalized second-order reset element, and fractional “constant in gain lead in phase” controllers with first- and second-order reset elements. These were computed directly from a numerical implementation of the Grünwald–Letnikov definition of fractional derivatives, and their performances were analyzed.

Elastic Thermal Deformation of an Infinite Copper Material Due to Cyclic Heat Supply Using Higher-Order Nonlocal Thermal Modeling

Thermoelastic modeling at nanoscale is becoming more important as devices shrink and heat sources are more widely used in modern industries, such as nanoelectromechanical systems. However, the conventional thermoelastic theories are no longer applicable in high-temperature settings. This study provides an insight into the thermomechanical features of a nonlocal viscous half-space exposed to a cyclic heat source. Using a novel concept of fractional derivatives, introduced by Atangana and Baleanu, it is assumed that the viscoelastic properties follow the fractional Kelvin–Voigt model. The nonlocal differential form of Eringen’s nonlocal theory is employed to consider the impact of small-scale behavior. It is also proposed that the rule of dual-phase thermal conductivity can be generalized to thermoelastic materials to include the higher-order time derivatives. The numerical results for the examined physical variables are presented using the Laplace transform technique. Furthermore, several numerical analyses are performed in-depth, focusing on the effects of nonlocality, structural viscoelastic indicator, fractional order, higher-order and phase-lag parameters on the behavior of the nanoscale half-space. According to the presented findings, it appears that the higher-order terms have a major impact on reactions and may work to mitigate the impact of thermal diffusion. Furthermore, these terms provide a novel approach to categorize the materials based on their thermal conductivities.

Ritz approximate method for solving delay fractional optimal control problems

In this paper, a numerical method based on the Müntz–Legendre polynomials for solving a class of time-delay fractional optimal control problems (TDFOCPs) is presented. The concept of fractional derivatives is considered in the Caputo sense. The key point of the applied method is that the initial and boundary conditions are imposed upon the approximations of state and control functions. First, the unknown state or control and their delayed functions are approximated by the Müntz–Legendre polynomials basis using the Ritz method then the next one is calculated through the given fractional differential equation. By substituting the estimated values in the cost function, the TDFOCP is converted to an unconstrained linear or nonlinear optimization problem. The convergence of the suggested method is extensively argued and several illustrative examples are presented to show the applicability and effectiveness of the method. It is shown that only a few terms of Müntz–Legendre polynomials are needed for obtaining the high accuracy and satisfactory results. The obtained results are compared with the available results in the literature to demonstrate the superiority of the applied approach. Furthermore, the approximated solutions agree with exact solutions and as α approaches an integer value, the method provides solution for the integer-order optimal control problem.

New Result for the Analysis of Katugampola Fractional-Order Systems—Application to Identification Problems

In the last few years, a new class of fractional-order (FO) systems, known as Katugampola FO systems, has been introduced. This class is noteworthy to investigate, as it presents a generalization of the well-known Caputo fractional-order systems. In this paper, a novel lemma for the analysis of a function with a bounded Katugampola fractional integral is presented and proven. The Caputo–Katugampola fractional derivative concept, which involves two parameters 0 < α < 1 and ρ > 0, was used. Then, using the demonstrated barbalat-like lemma, two identification problems, namely, the “Fractional Error Model 1” and the “Fractional Error Model 1 with parameter constraints”, were studied and solved. Numerical simulations were carried out to validate our theoretical results.

Thermal transport model for Brinkman type nanofluid containing carbon nanotubes with sinusoidal oscillations conditions: a fractional derivative concept

With efficient thermal activities, nanofluids have a novel role in different scientific engineering fields due to their unique and numerous applications. For example, these fluids can be used in magnetic resonance imaging (MRI), magnetic refrigeration (MR), cancer treatment (hyperthermia), and drug delivery. By inspiriting these applications and the significance of electrically conducting nanofluids, in this article, we have studied blood-based nanofluid containing carbon nanotubes (CNTs). The Brinkman type nanofluid modal is established in terms of an efficient mathematical fractional technique namely Prabhakar fractional derivative with ramped temperature and sinusoidal oscillations conditions and for the generalized solutions of temperature and velocity profile, Laplace transformation scheme is utilized. For the heat transfer of nanofluids, the Prabhakar fractional derivative which is based on generalized Fourier’s law of thermal flux is determined. The physical behavior of different parameters is examined by graphical illustrations. As a result, we have concluded that the velocity profile is a bit higher for multi-walled carbon nanotubes (MWCNTs) as compared to single-walled carbon nanotubes (SWCNTs). Furthermore, velocity and temperature fields represent decaying behavior by varying the values of fractional parameters.

How Many Fractional Derivatives Are There?

In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also some myths of Fractional Calculus and false fractional derivatives. The results are expected to contribute to limit the appearance of derivatives that differ from existing ones just because they are defined on distinct domains, and to prevent the ambiguous use of the concept of fractional derivative.

Right Fractional Sobolev Space via Riemann–Liouville Derivatives on Time Scales and an Application to Fractional Boundary Value Problem on Time Scales

Using the concept of fractional derivatives of Riemann–Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some embeddings. Finally, as an application, we propose a recent method to study the existence of weak solutions of fractional boundary value problems on time scales by using variational methods and critical point theory, and, by constructing an appropriate variational setting, we obtain two existence results of the problem.

Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

By the concept of fractional derivative of Riemann-Liouville on time scales, we first introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Then, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using fibering mapping and Nehari manifolds, the existence of weak solutions for a class of fractional boundary value problems on time scales is studied, and a result of the existence of weak solutions for this problem is obtained.

Numerical simulation of time partial fractional diffusion model by Laplace transform

<abstract><p>In the present work, the authors developed the scheme for time Fractional Partial Diffusion Differential Equation (FPDDE). The considered class of FPDDE describes the flow of fluid from the higher density region to the region of lower density, macroscopically it is associated with the gradient of concentration. FPDDE is used in different branches of science for the modeling and better description of those processes that involve flow of substances. The authors introduced the novel concept of fractional derivatives in term of both time and space independent variables in the proposed FPDDE. We provided the approximate solution for the underlying generalized non-linear time PFDDE in the sense of Caputo differential operator via Laplace transform combined with Adomian decomposition method known as Laplace Adomian Decomposition Method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by aforementioned techniques. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate technique to handle nonlinear partial differential equations as compared to the other available numerical techniques. At the end of this paper, the obtained numerical solution is visualized graphically by Matlab to describe the dynamics of desired solution.</p></abstract>

Cattaneo–Friedrich and Crank–Nicolson analysis of upper-convected Maxwell fluid along a vertical plate

The concept of fractional derivative is used to solve a variety of viscoelastic fluid problems. However, researchers mostly overlooked the consequences of nonlinear convection in the fractional viscoelastic fluid models and were concerned only with situations where the governing equations are linear. Most importantly, the nonlinear fluid models, whether classical or fractional, are solved for steady-state conditions. To overcome these limitations, this research presents unsteady flow and heat transfer of nonlinear fractional upper-convected Maxwell (UCM) viscoelastic fluid along a vertical plate. The governing equations of the fractional Maxwell fluid are developed by introducing Friedrich shear stress and Cattaneo heat flux models to the classical UCM fluid model. An additional feature to the invention of the constructed fractional model is the consequence of an external magnetic field. Moreover, the considered model comprises nonlinear, coupled, fractional partial differential equations. Therefore, a numerical scheme is developed with the aid of the L1-approximation of Caputo derivative and the Crank–Nicolson method. The effects of different regulating parameters on fluid features have been thoroughly investigated. The obtained results are exhibited graphically and discussed in detail. It is observed that the skin friction increases for the velocity relaxation time parameter, but an opposite behavior is observed against the velocity fractional derivative parameter. Moreover, a significant enhancement is noticed in the Nusselt number for increasing estimates of the Prandtl number.

Analysis and simulation of herd behaviour dynamics based on derivative with nonlocal and nonsingular kernel

A two-component system modelling the interaction between a prey and predator which cohabit together in nonlinear fashion in a given habitat is considered in this paper using the concept of fractional derivative. In the herb dynamical system, the classical time-derivative is modeled with the Atangana-Baleanu fractional-order operator which combines both nonlocal and nonsingular kernels in its formulation. A recent numerical scheme based on the Adams-Bashforth method is applied to approximate the fractional derivatives. To ensure the correct choice of parameters that are biologically meaningful in such dynamics, we examine the model for stability analysis. Numerical results are given for different parameter values of γ∈(0<γ⩽1] to justify our theoretical findings.

Fractional mathematical modeling for epidemic prediction of COVID-19 in Egypt

In this work the concept of fractional derivative is used to improve a mathematical model for the spread of the novel strain of corona virus disease COVID-19 in Egypt. We establish the dynamics model to predict the transmission of COVID-19. The results predicted by the model show a good agreement with the actual reported data. The effect of precautionary measures on the behavior of the model was studied and it was confirmed that the quarantine period should be long enough to achieve the desired result.