New Definition Of Fractional Derivative Research Articles

A New conformable fractional derivative and applications

The motivation for this paper comes from other papers treating the fractional derivatives. We introducea new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, mean value theorem and Taylor series. Usage of thedefined derivative is given in the example section which shows how our derivative can be used in solving differential equations. Comparison of our derivative with the derivative defined by Abdejjawad and overall conclusions are given in the conclusion section.

A Novel Formulation of the Fractional Derivative with the Order α≥0 and without the Singular Kernel

A new definition of fractional derivative (NFD) with order α≥0, is developed in this paper. The new derivative has a smooth kernel that takes on two different representations for the temporal and spatial variables. The advantage of the proposed approach over traditional local theories and fractional models with a singular kernel lies in the possibility that there is a class of problems capable of describing scale-dependent fluctuations and material heterogeneities. Moreover, it has been shown that the NFD converges to the classical derivative faster than some other fractional derivatives.

Thermoelastic diffusion responses of sandwich structures associated with new definitions of fractional derivative

The last five years have witnessed great developments of tempered fractional diffusion equation, both theoretically and numerically. In this work, new insights on fractional order thermoelastic diffusion are provided in two aspects: first, a novel form of fractional order model is proposed directly from Fourier’s law and Fick’s law; second, tempered fractional derivative is adopted, which shares the property that: it may transit from power-law behavior to exponential behavior in large time scale. The new model is applied to study the transient responses of a sandwich structure with ideal interfacial conditions subjected to symmetrical thermal and chemical shock loadings. In the numerical part, Laplace transform method is adopted, and the effects of different fractional order definitions, material characteristic parameters and fractional derivative parameters on the thermoelastic diffusion fields are analyzed and presented graphically. The results show that different fractional order definitions have different memory effects on the transient responses, and the effects of fractional order definitions on the responses will vary accordingly with the parameter values. This may provide insight to optimize the thermodiffusion management by selecting different materials and fractional derivative forms.

A New Conformable Fractional Derivative and Applications

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative

<p style='text-indent:20px;'>In recent years, a new definition of fractional derivative which has a nonlocal and non-singular kernel has been proposed by Atangana and Baleanu. This new definition is called the Atangana-Baleanu derivative. In this paper, we present a new technique to obtain the numerical solution of advection-diffusion equation containing Atangana-Baleanu derivative. For this purpose, we use the operational matrix of fractional integral based on Genocchi polynomials. An error bound is given for the approximation of a bivariate function using Genocchi polynomials. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method.</p>

Fractional order theory of Cattaneo-type thermoelasticity using new fractional derivatives

Transient thermoelastic interactions between materials and the moving heat sources, i.e. Laser additive manufacturing, Laser-assisted thermotherapy, high speed sliding and rolling contacts, are becoming increasingly important. In this work, a unified fractional thermoelastic theory is developed, and applied to study transient responses caused by a moving heat source. Theoretically, new insights on fractional thermoelasticity are provided by introducing new definitions of fractional derivative, i.e. Caputo-Fabrizio, Atangana–Baleanu and Tempered-Caputo type. Numerically, a semi-infinite medium subjected to a source of heat moving with constant velocity is considered within the present model under two different sets of boundary conditions: stress free and temperature given for the first, displacement fixed and thermally adiabatic for the second. Analytical solutions to all responses are firstly formulated in Laplace domain, and then transformed into time domain through numerical method. The numerical results show that Caputo-Fabrizio and Atangana–Baleanu type models predict smaller transient responses than Caputo type theory, while Tempered-Caputo model may give larger results by increasing the tempered parameter. Meanwhile, the effect of fractional order, tempered parameter of Tempered-Caputo model, and the velocity of heat source on all responses is discussed in detail. The time history of responses shows that: for long-term process, the exponential function of TC definition will make sense, and the temperature from TC model is greatly different from that of C model. This work may provide comprehensive understanding for thermoelastic interactions due to moving heat source, and open up possibly wide applications of such new fractional derivatives.

Fractional thermoelasticity revisited with new definitions of fractional derivative

Fractional thermoelastic models have been formulated from classical thermoelasticity or extended thermoelasticity, i.e. Lord-Shulman or Green-Naghdi theory. It seems different authors have different physical pictures on such topic, for instance, there exist several approaches from Lord-Shulman model to fractional order ones. This work is aimed to simplify the theoretical frameworks by clarifying connections between existed models. To this end, a new fractional thermoelasticity is established by directly extending classical thermoelasticity with the aids of new forms of fractional derivatives, i.e. Caputo-Fabrizio, Atangana–Baleanu and Tempered-Caputo definitions. All definitions are introduced into Fourier's law by a unified way with relaxation time incorporated. Theoretically, the present model may be simplified into existed fractional theory. Numerically, it has the capability of describing thermoelastic behaviors from fractional Lord-Shulman models. For numerical studies, Laplace transform method is adopted, and a two-layered structure subjected to thermal heating is considered, from which the effects of different fractional derivatives and relaxation time are firstly uncovered. And then, the influences of material constants of two layers, especially with interfacial conditions, are discussed in detail. Finally, some concluding remarks are made.

A New Generalized Definition of Fractional Derivative with Non-Singular Kernel

This paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann–Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann–Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented.

Time and Space Fractional Schrödinger Equation with Fractional Factor**Supported by National Natural Science Foundation of China under Grant Nos. 11472247 and 11872335

In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrödinger equation (FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrödinger equation to the fractional one. When applying this form of Schrödinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well, the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrödinger equation.

An approximate method for solving fractional partial differential equation by using an embedding process

In this article, solving of fractional partial differential equation (FPDE) with optimal time control approach is developed. Conformable derivative as a new definition of fractional derivative is considered. At first the (FPDE) with stationary regime is converted to optimal time control problem, then by using an embedding process, the obtained system is converted to finite-dimensional linear programming (LP) problem and finally optimal time corresponded by (FPDE) system is approximated. For example, this method is used on the conformable fractional heat equation with the initial and boundary conditions.

A new definition of fractional derivative

In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grünwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition.

The approximate solution of non-linear time-delay fractional optimal control problems by embedding process

Abstract In this paper, a class of time-delay fractional optimal control problems (TDFOCPs) is studied. Delays may appear in state or control (or both) functions. By an embedding process and using conformable fractional derivative as a new definition of fractional derivative and integral, the class of admissible pair (state, control) is replaced by a class of positive Radon measures. The optimization problem found in measure space is then approximated by a linear programming problem (LPP). The optimal measure which is representing optimal pair is approximated by the solution of a LPP. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to non-linear TDFOCPs. The usefulness of the used idea in this paper is that the method is not iterative, quite straightforward and can be applied to non-linear dynamical systems.

Hybrid Method for Solution of Fractional Order Linear Differential Equation with Variable Coefficients

Abstract In this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.

Fractional viscoelastic models with non-singular kernels

New definitions of fractional derivative with non-singular kernels are proposed recently. In the present paper, we apply the classical and new definitions of fractional derivative to four fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin–Voigt model, fractional Zener model and fractional Poynting–Thomson model. For each fractional viscoelastic model, the stress relaxation modulus, creep compliance and dynamic modulus are derived analytically under the classical and new fractional derivative definitions. The performance of these models under different fractional derivative definitions is further compared. The results show that the fractional Zener model and fractional Poynting–Thomson model are equivalent in all conditions. Compared with the classical fractional derivative definition with a power function kernel, the fractional derivative definition with a logarithmic function kernel can be used to describe the ultraslow creep and relaxation behaviors. However, the performance of fractional Maxwell model with the exponential function kernel is close to that of integer-order Maxwell model. Fractional Maxwell model and fractional Zener model with the Mittag–Leffler function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. Thus, specific modification is needed when applying the new definitions of fractional derivative to viscoelasticity.

Addendum to: A comment on some new definitions of fractional derivative

This is a brief Addendum to Giusti (Nonlinear Dyn 1–7, 2018. https://doi.org/10.1007/s11071-018-4289-8 ), whose aim is to properly acknowledge a result that had been obtained in a previous paper, of which the author was unfortunately not aware of until very recently.

A comment on some new definitions of fractional derivative

After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann-Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann-Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott-Blair model.

A new Definition of Fractional Derivative and Fractional Integral

A new Definition of Fractional Derivative and Fractional Integral

Technical Note on a New Definition of Fractional Derivative

Technical Note on a New Definition of Fractional Derivative

Fractional derivatives in the transport of drugs across biological materials and human skin

Abstract The diffusion of drugs across a composite structure such as a biological membrane is a rather complex phenomenon, because of its inhomogeneous nature, yielding a diffusion rate and a drug solubility strongly dependent on the local position across the membrane itself. These problems are particularly strengthened in composite structures of a considerable thickness like, for example, the human skin, where the high heterogeneity provokes the transport through different simultaneous pathways. In this note, we propose a generalization of the diffusion model based on Fick’s 2nd equation by substituting a diffusion constant by means of the memory formalism approach (diffusion with memory). In particular, we employ two different definitions of the fractional derivative, i.e., the usual Caputo fractional derivative and a new definition recently proposed by Caputo and Fabrizio. The model predictions have been compared to experimental results concerning the permeation of two different compounds through human skin in vivo , such as piroxicam, an anti-inflammatory drug, and 4-cyanophenol, a test chemical model compound. Moreover, we have also considered water penetration across human stratum corneum and the diffusion of an antiviral agent employed as model drugs across the skin of male hairless rats. In all cases, a satisfactory good agreement based on the diffusion with memory has been found. However, the model based on the new definition of fractional derivative gives a better description of the experimental data, on the basis of the residuals analysis. The use of the new definition widens the applicability of the fractional derivative to diffusion processes in highly heterogeneous systems.

3D Memory Constitutive Equations for Plastic Media

AbstractIn this paper, the authors begin by observing that using the new definition of fractional derivative with an exponential kernel makes it is possible to obtain again some generalized constitutive equation of continuum mechanics as a Kelvin-Voight Maxwell standard linear solid. However, this equivalence holds only in a space of smooth solutions. In the second part, two different fractional derivatives to describe the behavior of some types of plastic materials are considered. Before a kernel with a negative power of the variable and then a kernel with a negative exponential are assumed. The problems related with 1D (one dimensional) stress and a 1D strain respectively are applied to the 3D (three dimensional) body as part of a discussion of the Poisson phenomenon.