Definition Of Fractional Derivative Research Articles

Noether’s currents for conformable fractional scalar field theories

The construction of fractional derivatives with the right properties for use in field theory is reputed to be a difficult task, essentially because of the absence of a unique definition and uniform properties. The conformable fractional derivative introduced in 2014 by Khalil et al. in their seminal paper is a novel and well-behaved definition of fractional derivative for a function that is derivable in the usual sense. In this paper, we investigate the consistency of the Euler–Lagrange formalism for a field theory defined on such a fractional space–time. We especially focus on the relation between symmetries and conservation laws (Noether’s currents), about the symmetry group introduced to construct the Lagrangian of the field. In particular, we show that the use of the conformable derivative induces additional terms in the calculation of the action variation. We also investigate the conservation of the Noether current and show that this property only takes place on condition that the equations of motion are verified with a new definition of the conserved law.

On the distributional fractional derivative: From unidimensional to multidimensional

In this paper, we use the generalized notions of Riemann–Liouville (fractional calculus with respect to a regular function) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, and the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.

Heat transfer model analysis of fractional Jeffery-type hybrid nanofluid dripping through a poured microchannel

This study investigates the impact of coupled heat and mass transfer on the peristaltic migration of a magnetohydrodynamic (MHD) stress-strain Jeffery-type hybrid nanofluid flowing through an inclined asymmetric micro-channel with a porous medium. The fundamental two-dimensional momentum and energy transport equations are simplified under the assumptions of long wavelength and low Reynolds number. To solve the momentum and heat transfer problems, two advanced fractional derivative approaches are employed: the fractal integral and the Prabhakar fractional derivative. The pressure difference is determined using numerical integration techniques, such as the Stehfest and Tzou's algorithms. The results are presented through graphs and tables, which illustrate the effects of various parameters on the velocity, heat transfer, and trapping phenomena. As a result, we concluded that, pressure gradients grow with higher Reynolds numbers and channel-inclined angles. The heat transfer rate is observed to decrease as the Darcy number and the orientation of the electromagnetic field increase. When comparing the fractional derivative approaches, the fractal operator exhibits a more significant impact on the momentum profiles compared to the Prabhakar fractional operator. This difference is attributed to the distinct characteristics of the integral kernels associated with each fractional derivative definition. Furthermore, when comparing hybrid nanofluids, water-based (H2O + Ag + TiO2) hybrid fluids have a somewhat more significant effect than (C6H9NaO7 + Ag + TiO2) hybrid nanofluids.

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

Fractional-order rate-dependent thermoelastic diffusion theory based on new definitions of fractional derivatives with non-singular kernels and the associated structural transient dynamic responses analysis of sandwich-like composite laminates

Nowadays, the ultrafast heating technologies (e.g., laser-aided material processing, etc.) have been extensively applied in the manufacturing and micro-machining of microelectronic and semiconductor devices (e.g., field-effect transistor, etc.). In recent years, there have been a large number of literatures contributed on constitutive modeling and transient dynamic responses analysis of the coupled thermoelastic diffusion to provide a comprehensive understanding on multi-field coupling behavior on this topic and avoid the unwanted vibration response in intelligent active vibration control. Although the rate-dependent thermoelastic diffusion theories have been historically proposed, the memory-dependence characteristics of strain relaxation as well as heat conduction and mass diffusion are still not considered in ultrafast heating condition. To compensate for such deficiency, a fractional-order rate-dependent thermoelastic diffusion theory is established in this work by introducing the new fractional derivatives with non-singular kernels (i.e., the Atangana-Baleanu and the Tempered-Caputo definitions) into the strain rate dependent constitutive equation as well as classical Fourier heat conduction and Fick mass diffusion laws. With the aids of the extended thermodynamic principles, the new constitutive and governing equations are obtained. The proposed theory is applied to investigate the structural dynamic responses of sandwich-like composite laminates subjected to axisymmetric transient thermal and chemical shock loadings by using Laplace transformation approach. The influences of the different fractional derivatives and material constants ratios on wave propagations and structural dynamic responses are evaluated and discussed.

Basic fractional nonlinear-wave models and solitons

This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin’s fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective Lévy indices. Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.

Dynamical analysis of a novel fractional order SIDARTHE epidemic model of COVID-19 with the Caputo–Fabrizio(CF) derivative

Fabrizio and Caputo suggested an extraordinary definition of fractional derivative, which has been used in many fields. The SIDARTHE infectious disease model with regard to COVID-19 is studied by the new notion in this paper. Making use of the Banach fixed point theorem, the existence and uniqueness of the model’s solution are demonstrated. Then, an efficient method is utilized to deduce the iterative scheme. Finally, some numerical simulations of the model under various fractional orders and parameters are shown. From the computed result, we can see that it not only supports the theoretical demonstration, but also has an intensive insight into the characteristics of the model.

A New Mixed Fractional Derivative with Applications in Computational Biology

This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application in computational biology is presented.

Spectroscopic analysis of residual radicals of gamma sterilized UHMWPE with fractional order differential operators

The purpose of this research was to evaluate the Grunwald-Letnikov definition of fractional order derivatives for determining the fractional derivative of ESR signals from UHMWPE free radicals using the fitted Gaussian distribution method. Specifically, the study focused on analyzing two long-lasting oxygen-induced residual radicals di- or tri-enyls with a carbon center radical (R1) and the oxygen-containing radical (R2). The impact of the derivative order on ESR spectral parameters, such as the Landé g-factor and peak-to-peak separation, was analyzed, and new spectral parameters were established for both radicals. The samples were measured in an ESR tube using ESR power saturation techniques at room temperature. Following the measurements, the fitted Gaussian distribution of the ESR signals was used to determine the fractional derivative using the Grunwald-Letnikov definition. Two estimators (I and II) were developed for both radicals, and their values were found to be 9.19 and 4.27 for Radical R1 and 11.51 and 2060.62 for Radical R2. Our results showed that the fractional derivative approach provided accurate and reliable estimations for the radicals. This method can be easily implemented for various ESR signals and can be useful in many applications, including materials science and biomedical research. The accuracy and reliability of our estimators were confirmed by comparing them with the results obtained from the conventional method. Our findings have important implications for materials science and biomedical research, where ESR signals are widely used. This method can help researchers to obtain more accurate and reliable estimations of radicals, which can ultimately lead to more accurate and reliable conclusions in their studies. Additional research is required to assess the effectiveness of this approach on various ESR signals and to investigate its potential use in other fields of study.

A study on the existence results of boundary value problems of fractional relaxation integro-differential equations with impulsive and delay conditions in Banach spaces

<abstract><p>The aim of this paper was to provide systematic approaches to study the existence of results for the system fractional relaxation integro-differential equations. Applied problems require definitions of fractional derivatives, allowing the utilization of physically interpretable boundary conditions. Impulsive conditions serve as basic conditions to study the dynamic processes that are subject to sudden changes in their state. In the process, we converted the given fractional differential equations into an equivalent integral equation. We constructed appropriate mappings and employed the Schaefer's fixed-point theorem and the Banach fixed-point theorem to show the existence of a unique solution. We presented an example to show the applicability of our results.</p></abstract>

Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method

The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. The LRPS approach offers both computational efficiency and solution accuracy, making it an effective technique for solving nonlinear fractional partial differential equations (NFPDEs). The results are presented in the form of graphs for various values of the order of the fractional derivative and time, and the essential objective is to reduce computation effort.

Numerical Schemes and Stability Analysis for Fractional Models of Transient Uni- and Bidimensional flow in Petroleum Reservoirs

Objective: To propose a modeling for the flow in oil reservoirs using Caputo’s definition of fractional derivative applied to the spatial coordinate of the problem. Furthermore, delimit a region of numerical stability for the explicit method of solving the model equations. Methodology: Starting from the material balance in a differential control volume and the generalized Darcy's law, and coupling it with an equation of state, models describing flow in porous media were derived. A numerical solution scheme using an explicit discretization was proposed. For this purpose, the L1 method for approximating spatial fractional derivatives and finite difference approximation for temporal derivatives were employed.. Results and Discussion: Through the Von Neumann stability analysis, it was observed that the explicit solution schemes for the models are conditionally stable and dependent on the order of the fractional derivative. Additionally, numerical stability regions were constructed for one and two-dimensional transient schemes, revealing that the one-dimensional scheme is stable for , and the two-dimensional scheme is stable for for all derivative orders between 0 and 1. Implications of the research: Historically, flow models in petroleum reservoirs have been based on the application of Darcy's law; however, its use is limited due to some constraints. Recently, fractional calculus has played a significant role in generalizing and obtaining more accurate models in various application areas. In this context, Darcy's law has been generalized, allowing for the derivation of fractional models that are more suitable for describing flow in porous media. These models, in general, are solved using a numerical method, and it is crucial to understand the stability region of the method for their resolution.

Comparative Analysis and Definitions of Fractional Derivatives

Fractional Calculus (FC) has emerged as a valuable tool in various fields. This study explores the historical development of (FC) and examines prominent definitions regarding Fractional Derivatives (FD), such as the Riemann-Liouville, Grunwald-Letnikov, Caputo Fractional Derivative, Katugampula derivatives, Caputo Fractional Derivative, Caputo-Fabrizio Fractional Derivative and as well as Atangana-Baleanu Fractional Derivative. It critically evaluates their strengths, weaknesses and implications on (FD) equations. The findings contribute to establishing a clearer understanding of Fractional Derivatives (FD) and guiding their appropriate use in practical scenarios. It investigates the impact of different definitions on the properties and behaviors of (FD), providing valuable insights for researchers.

Computational results of convective heat transfer for fractionalized Brinkman type tri-hybrid nanofluid with ramped temperature and non-local kernel

Engineers have recently become attracted to electrically conducting nanofluids (NFs) due to various applications in several applied science and engineering disciplines. They have been employed in magnetic refrigeration, cancer treatment (hyperthermia), medicine, and magnetic resonance imaging, among other things. Considering the importance of electrically conducting NFs, in this article, we have proposed a fractionalized MHD (Magnetohydrodynamics) and thermal transmission of a Brinkman-type tri-hybrid nanofluid over an infinite plate saturated through a porous medium with generalized velocity and ramped conditions. For the solution of the governed fractional model, we have utilized a recent definition of fractional derivatives, known as Atangana-Baleanu (AB) fractional derivative and Laplace transformation. The computational results are exhibited for tri-hybrid (TiO2-Al2O3-CuO/H2O) NF and described through graphical diagrams and tables to discover the physics of numerous relevant flow parameters on temperature and velocity profiles. It is detected that both thermal transmission and momentum profile for tri-hybrid NF is a better technique as compared to hybrid NF and NF for both graphical and numerical comparisons.

New definitions of fractional derivatives and integrals for complex analytic functions

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus.

Asymptotic Boundedness of Conformable Fractional Delay Differential Systems

This article focuses on the problem of asymptotic boundedness for non-autonomous conformable fractional delay differential systems. Firstly, by utilizing the definition of conformable fractional derivative and fractional comparison principle, we propose non-autonomous conformable fractional Halanay inequalities. And then, based on the Halanay inequalities, sufficient conditions for the asymptotic boundedness are constructed for the systems. Finally, a numerical example shows the applicability of the proposed technique.

New Results by Using Advanced Definition of Fractional Derivative Without Singular Kernel

New Results by Using Advanced Definition of Fractional Derivative Without Singular Kernel

Q-HOMOTOPY SHEHU ANALYSIS TRANSFORM METHOD OF TIME-FRACTIONAL COUPLED BURGERS EQUATIONS

In this study, numerical solutions to time-fractional coupled Burgers equations are obtained utilizing the q-homotopy Shehu analysis transform method. The definition of fractional derivatives in the sense of Caputo. q-homotopy Shehu analysis transform method is also used to find the numerical solutions of the time-fractional coupled Burgers equations. In addition, the MAPLE software is utilized to plot the graphs of the solutions. These results demonstrate that the presented method is accurate and simple to implement.

Piecewise power law approximation of the interlayer relaxation curve for the long term viscoelastic fractional modeling of laminated glass

Laminated glass is composed of glass layers coupled by thin polymeric films which, being highly viscoelastic, influence the bending response over time. Here, an approximation of the polymer relaxation curve is proposed by means of continuously-connected power law branches. In the bi-logarithmic stress-time plane, this corresponds to a piecewise trilinear path, which well reproduces the experimental tests on most commercial polymers. A power law is naturally suited to be modeled via fractional calculus, as the Boltzmann convolution integral coincides with the definition of fractional derivative. However, the piecewise characterization through multiple power laws requires to properly extend the Grünwald–Letnikov integration scheme for fractional differential equations. For the paradigmatic case of a simply supported three-layer laminated beam, numerical results show an excellent correspondence with experiments under long duration loads. The comparison with the classic Prony series approach highlights, on the one hand, how calibration of material parameters is simplified; on the other hand, how the fractional approach is computationally more efficient and stable, even when the integration time step is wide enough to follow long term phenomena.

Memory effects on the thermal fracture behavior of cracked plates via fractional heat conduction models

Studies have shown that fractional derivatives have important memory effect on the thermal shock fracture of structures. In the present work, four generalized fractional models are applied to investigate the thermal fracture problem of cracked plates. Laplace transform and finite sine transform are employed to obtain transient temperature and thermal stresses. Stress intensity factors around the crack tips are evaluated through the weight function method for edge and center cracks. Numerical results show that memory effects on the thermoelastic fields are different with different fractional derivative definition and fractional orders, and the effects will vary accordingly with crack geometry parameters.