Application Of Fractional Calculus Research Articles

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

Applying Laplace Transformation on Epidemiological Models as Caputo Derivatives

This paper delves into the application of fractional calculus, with a focus on Caputo derivatives, in epidemiological models using ordinary differential equations. It highlights the critical role Caputo derivatives play in modeling intricate systems with memory effects and assesses various epidemiological models, including SIR variants, demonstrating how Caputo derivatives capture fractional-order dynamics and memory phenomena found in real epidemics. The study showcases the utility of Laplace transformations for analyzing systems described by ordinary differential equations with Caputo derivatives. This approach facilitates both analytical and numerical methods for system analysis and parameter estimation. Additionally, the paper introduces a tabular representation for epidemiological models, enabling a visual and analytical exploration of variable relationships and dynamics. This matrix-based framework permits the application of linear algebra techniques to assess stability and equilibrium points, yielding valuable insights into long-term behavior and control strategies. In summary, this research underscores the significance of Caputo derivatives, Laplace transformations, and matrix representation in epidemiological modeling. We assume that by using this type of methodology we can get analytic solutions by hand when considering a function as constant in certain cases and it will not be necessary to search for numerical methods.

Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative

Abstract This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

Caputo Fractional Derivative Inequalities for Refined Modified ( h , m ) -Convex Functions

This paper introduces a novel type of convex function known as the refined modified (h,m)-convex function, which is a generalization of the traditional (h,m)-convex function. We establish Hadamard-type inequalities for this new definition by utilizing the Caputo k-fractional derivative. Specifically, we derive two integral identities that involve the nth order derivatives of given functions and use them to prove the estimation of Hadamard-type inequalities for the Caputo k-fractional derivatives of refined modified (h,m)-convex functions. The results obtained in this research demonstrate the versatility of the refined modified (h,m)-convex function and the usefulness of Caputo k-fractional derivatives in establishing important inequalities. Our work contributes to the existing body of knowledge on convex functions and offers insights into the applications of fractional calculus in mathematical analysis. The research findings have the potential to pave the way for future studies in the area of convex functions and fractional calculus, as well as in other areas of mathematical research.

An efficient computational scheme for solving coupled time-fractional Schrödinger equation via cubic B-spline functions.

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions.

Certain geometric properties of the fractional integral of the Bessel function of the first kind

<abstract><p>This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order $ \lambda $ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular.</p></abstract>

An enhanced respiratory mechanics model based on double-exponential and fractional calculus.

We address mathematical modelling of respiratory mechanics and put forward a model based on double-exponential and fractional calculus for parameter estimation, model simulation, and evaluation based on actual data. Our model has been implemented on a publicly available executable code with adjustable parameters, making it suitable for different applications. Our analysis represents the first application of fractional calculus and double-exponential modelling to respiratory mechanics, and allows us to propose a hybrid model fitting experimental data in different ventilation modes. Furthermore, our model can be used to study the mechanical features of the respiratory system, improve the safety of ventilation techniques, reduce ventilation damages, and provide strong support for fast and adaptive determination of ventilation parameters.

FRACTIONAL CALCULUS AND ITS APPLICATIONS TO DIFFERENTIAL EQUATIONS

FRACTIONAL CALCULUS AND ITS APPLICATIONS TO DIFFERENTIAL EQUATIONS

The application of a novel variable-order fractional calculus on rheological model for viscoelastic materials

Viscoelastic materials have become popular for novel multifunctional materials in emerging modern technologies, which have complex elastic-viscous-plastic behaviors. The variable-order fractional derivative has been gradually applied to characterize such variable memory effects of complex physical systems, while existing definitions are somewhat controversial. The new Scarpi’s variable-order fractional calculus has rigorous mathematical logic compared with the existed definitions. However, its practicality and feasibility in engineering are ambiguous due to the fewer applications up to now. In this paper, we aimed at introducing the Scarpi’s variable-order fractional derivative into the rheological applications of viscoelastic materials. Based on the Scarpi’s variable-order fractional derivative, the fractional Zener model and fractional viscoelastic-visoplastic models are established, and the parameters sensitivity on stress-strain relation is discussed. Then, the proposed models are then applied to depict different rheological behaviors of polymers at various temperatures and strain rates, and the results show that they have excellent agreement for experimental data compared to the reference model. It is observed that the order increases linearly with strain of the viscoelastic model and decreases exponentially with strain of the viscoplastic model. Furthermore, the physical significance of the corresponding fractional order parameters is investigated. It is found that they have the variation relation with temperatures and strain rates. It reveals that the variable fractional order can be regarded as an index to characterize the evolution of mechanical behaviors of polymers.

Existence and Uniqueness Results for a Pantograph Boundary Value Problem Involving a Variable-Order Hadamard Fractional Derivative

This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii’s fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach.

Applications of fractional calculus to thermodynamics analysis of hydromagnetic convection in a channel

The present analysis signifies the impacts of fraction calculus on the MHD analysis of an incompressible fluid flow with entropy generation, viscous dissipation, and joule heating carried out between two endless vertical plates. The left vicinity of the MHD fluid flow channel has a convectively hot MHD fluid with a specific temperature that produces the heat transfer coefficient inside the flow channel. The MHD fluid flow model has been formulated with the fractional order derivative, which is an exciting area of research. The transient form of the system of dimensionless PDEs with imposed boundary conditions is transformed to a fractional order derivative model, which was formerly solved via the Finite Difference Scheme (FDS). The graphical illustrations signify the significance of the transverse magnetic field, viscous dissipation, and Ohamic heating alongside Biot and Bejan numbers. The velocity and temperature illustrations have been shown graphically to observe the influences within the flow channel. With predetermined values of the corresponding dimensionless parameters, the coefficient of skin friction and Nusselt number accompanied by a range of convection constraints are displayed graphically within the flow channel.

Viscoelastic phenomena in methylcellulose aqueous systems: Application of fractional calculus

Fractional calculus models can potentially describe the viscoelastic phenomena in soft solids. Nevertheless, their successful application is limited. This paper explored the potential of using fractional calculus models to describe the viscoelastic properties of soft solids, focusing on methylcellulose aqueous systems. Methylcellulose is an important food additive, and it is known for its complex rheological behaviors, including thermogelation, which still puzzle rheologists. Through dynamic mechanical analysis and fractional rheology, we demonstrated that fractional calculus described the frequency- and temperature-dependent rheology of methylcellulose. This paper also showcased how including one springpot could potentially replace numerous spring-dashpot arrangements. Our findings using fractional calculus suggested that the thermogelation of methylcellulose involves the cooperative mobility of polymer chains and can be described as a process analogous to the glass transition in polymers. This study highlighted the power of combining fractional calculus and rheology to understand complex viscoelastic phenomena in soft solids.

Some New Estimates of Hermite–Hadamard Inequalities for Harmonical cr-h-Convex Functions via Generalized Fractional Integral Operator on Set-Valued Mappings

The application of fractional calculus to interval analysis is vital for the precise derivation of integral inequalities on set-valued mappings. The objective of this article is to reformulated the well-known Hermite–Hadamard inequality into various new variants via fractional integral operator (Riemann–Liouville) and generalize the various previously published results on set-valued mappings via center and radius order relations using harmonical h-convex functions. First, using these notions, we developed the Hermite–Hadamard (H–H) inequality, and then constructed some product form of these inequalities for harmonically convex functions. Moreover, to demonstrate the correctness of these results, we constructed some interesting non-trivial examples.

Finite Time Stability Results for Neural Networks Described by Variable-Order Fractional Difference Equations

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied.

On the formulation of a predictor–corrector method to model IVPs with variable‐order Liouville–Caputo‐type derivatives

Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described.

A Proposed Application of Fractional Calculus on Time Dilation in Special Theory of Relativity

Time dilation (TD) is a principal concept in the special theory of relativity (STR). The Einstein TD formula is the relation between the proper time t0 measured in a moving frame of reference with velocity v and the dilated time t measured by a stationary observer. In this paper, an integral approach is firstly presented to rededuce the Einstein TD formula. Then, the concept of TD is introduced and examined in view of the fractional calculus (FC) by means of the Caputo fractional derivative definition (CFD). In contrast to the explicit standard TD formula, it is found that the fractional TD (FTD) is governed by a transcendental equation in terms of the hyperbolic function and the fractional-order α. For small v compared with the speed of light c (i.e., v≪c), our results tend to Newtonian mechanics, i.e., t→t0. For v comparable to c such as v=0.9994c, our numerical results are compared with the experimental ones for the TD of the muon particles μ+. Moreover, the influence of the arbitrary-order α on the FTD is analyzed. It is also declared that at a specific α, there is an agreement between the present theoretical results and the corresponding experimental ones for the muon particles μ+.

Dynamic of Some Relapse in a Giving Up Smoking Model Described by Fractional Derivative

Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model that captures the dynamics of relapse in a smoking cessation context and presents the dynamic behavior of the proposed model utilizing Caputo fractional derivatives. The model incorporates four compartments representing potential, persistent (heavy), temporally recovered, and permanently recovered smokers. The basic reproduction number R0 is computed, and the local and global dynamic behaviors of the free equilibrium smoking point (Y0) and the smoking-present equilibrium point (Y*) are analyzed. It is demonstrated that the free equilibrium smoking point (Y0) exhibits global asymptotic stability when R0≤1, while the smoking-present equilibrium point (Y*) is globally asymptotically stable when R0>1. Additionally, analytical results are validated through a numerical simulation using the predictor–corrector PECE method for fractional differential equations in Matlab software.

Fractional-order model and experimental verification of granules-beam coupled vibration

Granular media generally exhibit fluid phase, solid phase, or two coexisting phases, and the mechanism of the phase transition is not clear enough. Therefore, unlike the airflow-structure or fluid–structure coupled vibrations, the granules-structure coupled vibrations have not yet been theoretically modeled, and the interest in such coupled vibrations, which has just started in recent years, is mainly focused on experiments and simulations. In order to develop a theoretical model of the coupled vibrations of granular media and beam structures, in this paper, a novel concept called the inertia-viscous effect is introduced in terms of terminology and mathematical expression, and surprisingly, its constitutive relation can be summarized in a unified framework based on fractional calculus, together with Hooke spring, Newtonian dashpot, and Abel sticky pot. Furthermore, the existence of this effect in the granules-beam coupled vibrations is experimentally confirmed, and more importantly, the effect is more suitable for modeling by fractional-order derivatives rather than the conventional integer-order ones due to the frequency-dependent characteristics of the effect. Based on this, the theoretical model of the granules-beam coupled vibrations and the associated parameters identification procedure are successfully developed, in which a fractional-order model is proposed to describe the additional dynamic load (ADL) of the granular media on the beam. The numerical and experimental results suggest that the system response can be accurately predicted by the developed theoretical model without the computational cost as in the discrete element methods, which is the main contribution of this work. Finally, the effectiveness and robustness of the developed theoretical model are further verified by the results of numerous parametric experiments, and importantly, by combining these results with the analytical results, the mechanism of ADL is revealed, thus providing insights for modeling coupled vibrations between other engineering structures and granular media. From another perspective, this work can also be seen as a novel application of fractional calculus in the dynamics of mechanical systems.

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.

Numerical Approach for Solving a Fractional-Order Norovirus Epidemic Model with Vaccination and Asymptomatic Carriers

This paper explored the impact of population symmetry on the spread and control of a norovirus epidemic. The study proposed a mathematical model for the norovirus epidemic that takes into account asymptomatic infected individuals and vaccination effects using a non-singular fractional operator of Atanganaa–Baleanu Caputo (ABC). Fixed point theory, specifically Schauder and Banach’s fixed point theory, was used to investigate the existence and uniqueness of solutions for the proposed model. The study employed MATLAB software to generate simulation results and demonstrate the effectiveness of the fractional order q. A general numerical algorithm based on Adams–Bashforth and Newton’s Polynomial method was developed to approximate the solution. Furthermore, the stability of the proposed model was analyzed using Ulam–Hyers stability techniques. The basic reproductive number was calculated with the help of next-generation matrix techniques. The sensitivity analysis of the model parameters was performed to test which parameter is the most sensitive for the epidemic. The values of the parameters were estimated with the help of least square curve fitting tools. The results of the study provide valuable insights into the behavior of the proposed model and demonstrate the potential applications of fractional calculus in solving complex problems related to disease transmission.