Application Of Fractional Calculus Research Articles

Fractional Order Calculus and Derivative Implementation

The concept of fractional calculus dates back to the early days of calculus and may be traced back to Arbogast's publications in the 1800s. Multiple derivatives are defined in this situation by powers of D that are logically discovered to be integral in nature. However, this gave birth to the idea of evaluating this operator's fractional power and attempting to determine its equivalent form or operating it on some function in a meaningful way.This part of calculus is classified as special calculus, and it did not find much application in engineering until the advent of electronic computers and control systems, where it began to show its capability, such as increasing the number of control parameters in a PID control system, which essentially increases its ability to be optimised, albeit at a higher complexity. This project begins with introducing the fundamentals of fractional calculus, followed by fractional derivatives of standard functions and their interpretation, and then provides fractional calculus applications in the context of electronics.

Concept and application of interval-valued fractional conformable calculus

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results.We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world.

Inclined surface slip flow of nanoparticles with subject to mixed convection phenomenon: Fractional calculus applications

The interest of researchers towards the nanofluids is noticed in recent years due to leading applications in thermal systems and industrial framework. Referring to such motivations, current study explores the role of velocity slip effects for the mixed convection flow of nanofluid endorsed due to inclined surface. The Casson base fluid model for which the thermal impact needs to be improved. The analysis is observed when the role of velocity slip is important. The modeling of unsteady free convective flow problem yields partial differential system. The Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional operators are implemented in order to simulates the computation of problem. The graphical presentations are prepared in order to check the physical dynamic of parameters.

Best proximity point results with their consequences and applications

In the commenced work, we establish some best proximity point results for multivalued generalized contractions on partially ordered complete metric spaces along with the tactic of altering distance function. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications of fractional calculus and also invoke our findings to the equation of motion modeling to differential equations.

A fractional-order model for calendar aging with dynamic storage conditions

Due to the increasing importance of lithium-ion batteries in electric vehicle and renewable energy applications, battery aging is a subject of intense research. Although many laboratory experiments are performed under well-controlled static conditions, batteries are stored and operated under varying conditions of temperature and state of charge in their real-life performance, so that a suitable model for predicting the effects of calendar aging in lithium-ion batteries with dynamic conditions is highly desirable. In this paper, we review previous models to calculate capacity loss due to calendar aging under variable temperature and state-of-charge conditions according to experimentally observed power-law behavior, and propose a novel model based on fractional calculus. To validate the new model, we compare its predictions with experimental results showing that it can reproduce the non-monotonic behavior that is observed when the state of charge or the temperature change significantly. This is an interesting application of fractional calculus since this characteristic is not obtained with non-fractional models.

Applications of Fractional Calculus on a Certain Class of Univalent Functions Associated with Wanas Operator

The purpose of this work is to use fractional integral and Wanas operator to define a certain class of analytic and univalent functions defined in the open unit disk U. Also, we obtain some results for this class such as integral representation, inclusion relationship and argument estimate.

Applications of fractional calculus in computer vision: A survey

Fractional calculus is an abstract idea exploring interpretations of differentiation having non-integer order. For a very long time, it was considered as a topic of mere theoretical interest. However, the introduction of several useful definitions of fractional derivatives has extended its domain to applications. Supported by computational power and algorithmic representations, fractional calculus has emerged as a multifarious domain. It has been found that the fractional derivatives are capable of incorporating memory into the system and thus suitable to improve the performance of locality-aware tasks such as image processing and computer vision in general. This article presents an extensive survey of fractional-order derivative-based techniques that are used in computer vision. It briefly introduces the basics and presents applications of the fractional calculus in six different domains viz. edge detection, optical flow, image segmentation, image de-noising, image recognition, and object detection. The fractional derivatives ensure noise resilience and can preserve both high and low-frequency components of an image. The relative similarity of neighboring pixels can get affected by an error, noise, or non–homogeneous illumination in an image. In that case, the fractional differentiation can model special similarities and help compensate for the issue suitably. The fractional derivatives can be evaluated for discontinuous functions, which help estimate discontinuous optical flow. The order of the differentiation also provides an additional degree of freedom in the optimization process. This study shows the successful implementations of fractional calculus in computer vision and contributes to bringing out challenges and future scopes.

New Methods to Taking Fractional Derivatives and Its Applications

Recently, interest in fractional calculus has been encouraged by its applications in the different fields ofscience. The applications of fractional calculus include stochastic processes, anomalous diffusion andrheology. In this paper we present new formulas for taking fractional derivatives and some considerationsof its applications for the harmonic oscillator

Generalized Mellin transform and its applications in fractional calculus

Generalized Mellin transform and its applications in fractional calculus

The asymptotic analysis of novel coronavirus disease via fractional-order epidemiological model

We develop a model and investigate the temporal dynamics of the transmission of the novel coronavirus. The main sources of the coronavirus disease were bats and unknown hosts, which left the infection in the seafood market and became the major cause of the spread among the population. Evidence shows that the infection spiked due to the interaction between humans. Hence, the formulation of the model proposed in this study is based on human-to-human and reservoir-to-human interaction. We formulate the model by keeping in view the esthetic of the novel disease. We then fractionalize it with the application of fractional calculus. Particularly, we will use the Caputo–Fabrizio operator for fractionalization. We analyze the existence and uniqueness of the well-known fixed point theory. Moreover, it will be proven that the considered model is biologically and mathematically feasible. We also calculate the threshold quantity (reproductive number) to discuss steady states and to show that the particular epidemic model is stable asymptotically under some restrictions. We also discuss the sensitivity analysis of the threshold quantity to find the relative impact of every epidemic parameter on the transmission of the coronavirus disease. Both the global and local properties of the proposed model will be analyzed for the developed model using the mean value theorem, Barbalat’s lemma, and linearization. We also performed some numerical simulations to verify the theoretical work via some graphical representations.

Fractional time-varying grey traffic flow model based on viscoelastic fluid and its application

Existing traffic flow models are insufficient to excavated the characteristics of fluids and are usually a constant coefficient differential model. They cannot reflect the characteristics of a traffic flow system changing with time, resulting in their poor adaptability. Firstly, the viscoelastic traffic flow model is simplified. The fractional viscoelastic traffic flow model is established in combination with modelling principle of the Bass model and successful application of fractional calculus in viscoelastic fluid. Conformable fractional derivative and the fractional grey model are then introduced to establish a fractional grey viscoelastic traffic flow model that can reflect time-varying characteristics. Finally, the new model is compared with traditional statistical models in terms of model efficiency and stability and is applied to the modelling of traffic flow and traffic congestion level in multiple scenarios. The modelling results are compared with six other models. Results show that the new model has better stability and modelling effect.

Asymptotic cycles in fractional maps of arbitrary positive orders

Many natural and social systems possess power-law memory, and their mathematical modeling requires the application of discrete and continuous fractional calculus. Most of these systems are nonlinear and demonstrate regular and chaotic behavior, different from the behavior of memoryless systems. Finding periodic solutions is essential for understanding the regular and chaotic behavior of nonlinear systems. Fractional systems do not have periodic solutions except fixed points. Instead, they have asymptotically periodic solutions which, in the case of stable regular behavior, converge to the periodic sinks (similar to regular dissipative systems) and, in the case of unstable/chaotic behavior, act as repellers. In one of his recent papers, the first author derived equations that allow calculations of asymptotically periodic points for a wide class of discrete maps with memory. All fractional and fractional difference maps of the orders 0 < \alpha < 1 belong to this class. In this paper, we derive the equations that allow calculations of the coordinates of the asymptotically periodic points for a wider class of maps which include fractional and fractional difference maps of the arbitrary positive orders \alpha > 0. The maps are defined as convolutions of a generating function -G_K(x), which may be the same as in a corresponding regular map x_{n+1} = -G_K(x_n)+x_n, with a kernel U_{\alpha}(k), which defines the type of a map. In the case of fractional maps, it is U{\alpha}(k) = k^{\alpha}, and it is U{\alpha}(k) = k^{(\alpha)}, the falling factorial function, in the case of fractional difference maps. In this paper, we define the space of kernel functions that allow calculations of the periodic points of the corresponding maps with memory. We also prove that in fractional maps of the orders 1 < \alpha < 2 the total of all physical momenta of period-l points is zero.

Transient Propagation of Spherical Waves in Porous Material: Application of Fractional Calculus

A fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates. An equivalent fluid model is used in which the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case of Biot’s theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The basic solution of the wave equation is obtained in the time domain by analytically calculating Green’s function of the porous medium and using the properties of the Laplace transforms. Fractional derivatives are used to describe, in the time domain, the fluid–structure interactions, which are of the inertial, viscous, and thermal kind. The solution to the fractional-order wave equation represents the radiation field in the porous medium emitted by a point source. An important result obtained in this study is that the solution of the fractional equation is expressed by recurrence relations that are the consequence of the modified Bessel function of the third kind, which represents a physical solution of the wave equation. This theoretical work with analytical results opens up prospects for the resolution of forward and inverse problems allowing the characterization of a porous medium using spherical waves.

FRACTIONAL ORDER LOG BARRIER INTERIOR POINT ALGORITHM FOR POLYNOMIAL REGRESSION IN THE ℓ p -NORM

Fractional calculus is the branch of mathematics that studies the several possibilities of generalizing the derivative and integral of a function to noninteger order. Recent studies found in literature have confirmed the importance of fractional calculus for minimization problems. However, the study of fractional calculus in interior point methods for solving optimization problems is still new. In this study, inspired in applications of fractional calculus in many fields, was developed the so-called fractional order log barrier interior point algorithm by replacing some integer derivatives for the corresponding fractional ones on the first order optimality conditions of Karush-Kuhn-Tucker to solve polynomial regression models in the ℓ p −norm for 1 < p < 2. Finally, numerical experiments are performed to illustrate the proposed algorithm.

Modeling Drug Concentration Level in Blood Using Fractional Differential Equation Based on Psi‐Caputo Derivative

This article studies a pharmacokinetics problem, which is the mathematical modeling of a drug concentration variation in human blood, starting from the injection time. Theories and applications of fractional calculus are the main tools through which we establish main results. The psi‐Caputo fractional derivative plays a substantial role in the study. We prove the existence and uniqueness of the solution to the problem using the psi‐Caputo fractional derivative. The application of the theoretical results on two data sets shows the following results. For the first data set, a psi‐Caputo with the kernel ψ = x + 1 is the best approach as it yields a mean square error (MSE) of 0.04065. The second best is the simple fractional method whose MSE is 0.05814; finally, the classical approach is in the third position with an MSE of 0.07299. For the second data set, a psi‐Caputo with the kernel ψ = x + 1 is the best approach as it yields an MSE of 0.03482. The second best is the simple fractional method whose MSE is 0.04116 and, finally, the classical approach with an MSE of 0.048640.

The mathematical description of the bulk fluid flow and that of the content impurity dispersion, obtained by replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective

In the field of fractional calculus applications, there is a tendency to admit that “integer-order derivatives cannot simply be replaced by fractional-order derivatives to develop fractional-order theories”. There are different arguments for that: initialization problem, inconsistency, use of nonsingular or singular kernels, loss of objectivity. In this paper it is shown that the mathematical description of the bulk fluid flow and that of the content impurity spread replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. More precisely, it is proven that, the mathematical description of the bulk fluid 2D flow and that of the content impurity spread, in a horizontal unconfined aquifer, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. It is also proven that, the mathematical description of a Newtonian, incompressible, viscous bulk fluid 3D flow and that of the contained impurity dispersion, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. The obtained results show the compatibility of the general temporal Caputo and general temporal Riemann-Liouville fractional order derivatives with the understanding of the “measured time” evolution. In the same time these results reveal that, the objectivity violation, when integer order temporal derivatives are replaced by classic temporal Caputo or classic temporal Riemann-Liouville fractional order derivatives, is originated in the incompatibility of the classic fractional order derivatives, with the understanding of the “measured time” evolution.

Fractional integral representation in statistical thermodynamics of confined systems

Confined systems are usually treated as integer dimensional systems, like two dimensional (2D), 1D, and 0D, by considering extreme confinement conditions in one or more directions. This approach costs piecewise representations, some limitations in confinement interval, and the deviations from the true behaviors, especially when the confinement is neither strong nor weak. In this study, fractional integral representation (FIR) is proposed as a methodology to calculate the infinite summations in statistical thermodynamics for any dimension and confinement values. FIR directly incorporates the dimension as a control variable into calculation procedures and allows us to get solutions valid for the whole confinement and dimension scales, including the fractional ones. We define the dimension of a summation and used it in the proposed FIR to calculate the partition function. The first and the higher-order FIR are introduced and high accuracy results are achieved. FIR is then extended for a generalized function to calculate thermodynamic properties directly from their fundamental expressions based on infinite sums. By using the proposed FIR approach, the thermodynamic properties of a noninteracting Maxwell-Boltzmann gas confined in an elongated rectangular domain are determined. The excess quantities induced by confinement are examined for different confinement scenarios. FIR successfully predicts the true behavior of thermodynamic properties for the whole range of confinement and dimension scales. Defining and controlling the dimension allows designing new types of thermodynamic cycles. Besides the infinite-well potential for the confinement of particles with quadratic and linear dispersion relations, quadratic and quartic confining potentials are also considered to show the success of FIR. The proposed method not only incorporates the dimension into the calculation procedures but also constitutes an application of fractional calculus in statistical thermodynamics. FIR has many potential applications especially for Bose-Einstein condensation phenomenon which inherently contains dimensional transitions.

Nonlinear dynamics and chaos in fractional differential equations with a new generalized Caputo fractional derivative

In this paper, novel systems of fractional differential equations involving a new generalized Caputo fractional derivative were proposed. The complex dynamic behavior of these systems was studied by numerical simulation. Nonlinear dynamics and chaos in hybrid fractional order systems were investigated using a predictor–corrector algorithm. In particular, the effect of the new generalized fractional derivative parameters on the dynamics of the proposed systems was discussed. The rich variation obtained from the characteristics of the studied systems recommends the implementation of the new generalized derivative in fractional calculus applications.

Some applications of fractional calculus for analytic functions

For analytic functions $f\left( z\right) $ in the class $A_{n},$ fractional calculus (fractional integrals and fractional derivatives) $D_{z}^{\lambda }f\left( z\right) $ of order $\lambda $ are introduced. Applying $% D_{z}^{\lambda }f\left( z\right) $ for $f\left( z\right) \in A_{n},$ we introduce the interesting subclass $A_{n}\left( \alpha _{m},\beta ,\rho ,\lambda \right) $ of $A_{n}.$ The object of this paper is to discuss some properties of $f\left( z\right) $ concerning $D_{z}^{\lambda }f\left( z\right) .$

Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches

The paper addresses diffusion approximations of magnetic field penetration of ferromagnetic materials with emphasis on fractional calculus applications and relevant approximate solutions. Examples with applications of time-fractional semi-derivatives and singular kernel models (Caputo time fractional operator) in cases of field independent and field-dependent magnetic diffusivities have been developed: Dirichlet problems and time-dependent boundary condition (power-law ramp). Approximate solutions in all theses case have been developed by applications of the integral-balance method and assumed parabolic profile with unspecified exponents. Tow version of the integral method have been successfully implemented: SDIM (single integration applicable to time-fractional semi-derivative model) and DIM (double-integration model to fractionalized singular memory models). The fading memory approach in the sense of the causality concept and memory kernel effect on the model constructions have been discussed.