Definition Of Fractional Derivative Research Articles

Generalized fractional viscothermoelastic nanobeam under the classical Caputo and the new Caputo–Fabrizio definitions of fractional derivatives

In this paper, we introduce a new unified formula that governs two different definitions of fractional derivative; the classical Caputo definition and Caputo–Fabrizio's new definition. Hence, an analysis has been constructed for fractional viscothermoelastic, isotropic, and homogeneous nanobeams. The governing equations of the viscothermoelastic nanobeam have been constructed in the context of the non-Fourier heat conduction law with one relaxation of Lord and Shulman (L-S). Laplace transform has been applied, and its inversions have been calculated by using the Tzou method of approximation. The numerical results have been validated for a thermoelastic rectangular nanobeam of silicon as a case when it is subjected to ramp-type heating and simply supported. The fractional-order parameter based on the two types of fractional derivatives has significant impacts on all the studied functions except the temperature increment function. The results based on the two types of fractional derivatives are different, although they generate the same behavior of the thermomechanical waves. The ramp-time heat parameter has significant effects on all the studied functions.

Explicit tight frames for simulating a new system of fractional nonlinear partial differential equation model of Alzheimer disease

This paper is devoted to develop a new mathematical model for Alzheimer disease based on a system of fractional-order partial differential equations. The system of Alzheimer disease includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. We consider the Caputo fractional derivative definition to analyze the formulated system by simulating the effect of drugs that either failed or currently in clinical trials. To simulate the model, we use tight frame (framelet) systems generated using the unitaryand oblique extension principle. According to the simulation results, and based on using such new direction of fractional modeling, the progression of Alzheimer disease and its consequences will be slowing down. Which may give clinical insights on intervention measures against the disease and its effective therapies.

A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A <i>ψ</i>-HILFER FRACTIONAL OPERATOR

Boundary value problems driven by fractional operators has drawn the attention of several researchers in the last decades due to its applicability in several areas of Science and Technology. The suitable definition of the fractional derivative and its associated spaces is a natural problem that arise on the study of this kind of problem. A manner to avoid of such problem is to consider a general definition of fractional derivative. The purpose of this manuscript is to contribute, in the mentioned sense, by presenting the $\psi-$fractional spaces $\mathbb{H}_{p}^{\alpha, \beta; \psi}([0, T], \mathbb{R})$. As an application we study a problem, by using the Mountain Pass Theorem, which includes an wide class of equations.

Comparative study of DST interpolation approach of fractional orders

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like Grunwald-Letnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.

A fractional study of generalized Oldroyd-B fluid with ramped conditions via local & non-local kernels

Abstract Convective flow is a self-sustained flow with the effect of the temperature gradient. The density is nonuniform due to the variation of temperature. The effect of the magnetic flux plays a major role in convective flow. The process of heat transfer is accompanied by mass transfer process; for instance condensation, evaporation and chemical process. Due to the applications of the heat and mass transfer combined effects in different field, the main aim of this paper is to do comprehensive analysis of heat and mass transfer of MHD unsteady Oldroyd-B fluid in the presence of ramped conditions. The new governing equations of MHD Oldroyd-B fluid have been fractionalized by means of singular and non-singular differentiable operators. In order to have an accurate physical significance of imposed conditions on the geometry of Oldroyd-B fluid, the ramped temperature, concentration and velocity are considered. The fractional solutions of temperature, concentration and velocity have been investigated by means of integral transform and inversion algorithm. The influence of physical parameters and flow is analyzed graphically via computational software (MATHCAD-15). The velocity profile decreases by increasing the Prandtl number. The existence of a Prandtl number may reflect the control of the thickness and enlargement of the thermal effect. The classical calculus is assumed as the instant rate of change of the output when the input level changes. Therefore it is not able to include the previous state of the system called the memory effect. Due to this reason, we applied the modern definition of fractional derivatives. Obtained generalized results are very important due to their vast applications in the field of engineering and applied sciences.

A reliable numerical method for solving fractional reaction-diffusion equations

The present work aims to solve the fractional reaction–diffusion equation (RDE) using an effective and powerful hybrid analytical scheme, namely q-HASTM. The suggested technique is the combination of Sumudu transform (ST) and HAM technique. The definition of Caputo's fractional derivative has used. The numerical procedure reveals that only few iterations are needed for better approximation of the solution which illustrate the competence and sincerity of the suggested scheme. The impact of the reaction term in the solution of the problem explained through the graph.

Guided wave propagation in functionally graded fractional viscoelastic plates: A quadrature-free Legendre polynomial method

Compared to the traditional integer-order viscoelastic model, a fractional-order derivative viscoelastic model is shown to be advantageous. A quadrature-free Legendre polynomial method combining the Weyl definition of fractional order derivative is for the first time employed to solve guided waves in functionally graded fractional viscoelastic plates. The presented method avoids a lot of numerical integration calculation in traditional polynomial method and greatly improves the computational efficiency. The effects of different graded fields and fractional orders on the dispersion and attenuation curves are illustrated. The displacement distributions, Poynting vectors and power flows are analyzed to reveal guided wave characteristics.

Using asymptotic expansions technique for solving the point heat source problem in the fractional theory of thermoelasticity with the Caputo Fabrizio derivative

In this work, the Caputo Fabrizio definition of fractional derivative is utilized to explore a problem in thermoelasticity with fractional theory. The problem regards an infinite elastic space under the impact of an uninterrupted heat point source. The method of asymptotic expansions is used to find the inverse Laplace transform valid for short times. A summarized discussion about the behavior of wave’s propagation is introduced.

Hermite–Hadamard–Fejér inequalities for generalized conformable fractional integrals

In this paper, we study generalized definitions of left and right conformable fractional derivative and integrals. Some applications of the generalized definition are also given for the nonlinear fractional differential equations. We also establish identities associated with the left‐ and right‐hand side of Hermite–Hadamard–Fejér inequality for preinvex functions. We, then, give some new bounds for the Hermite–Hadamard–Fejér inequalities for preinvex functions using the generalized conformable fractional integrals.

General Response Formula for CFD Pseudo-Fractional 2D Continuous Linear Systems Described by the Roesser Model

In many engineering problems associated with various physical phenomena, there occurs a necessity of analysis of signals that are described by multidimensional functions of more than one variable such as time t or space coordinates x, y, z. Therefore, in such cases, we should consider dynamical models of two or more dimensions. In this paper, a new two-dimensional (2D) model described by the Roesser type of state-space equations will be considered. In the introduced model, partial differential operators described by the Conformable Fractional Derivative (CFD) definition with respect to the first (horizontal) and second (vertical) variables will be applied. For the model under consideration, the general response formula is derived using the inverse fractional Laplace method. Next, the properties of the solution will be considered. Usefulness of the general response formula will be discussed and illustrated by a numerical example.

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>

Thermal shock fracture associated with a unified fractional heat conduction

Recently, a unified fractional heat conduction model is newly proposed, which is a further improvement of the fractional thermoelasticity theory. It would be interesting to know how the newly developed fractional derivative definitions affects the thermal fracture behavior when some defects such as cracks exist in the medium. This work is aimed at analyzing the thermal fracture problem of an elastic half-space and strip with an insulated Griffith crack based on the unified fractional heat conduction model. Fourier and Laplace transforms are employed to solve the mixed boundary problem associated with a time-fractional partial differential equation. Temperature and stress intensity factors are evaluated by solving a system of singular integral equations. Effects of different fractional derivatives and different parameters on the thermoelastic fields are analyzed. A comparison of the temperature and stress intensity factors between the present model and the Fourier model is made. Numerical results show that different fractional definitions have different memory effects on the transient responses, and the effects of fractional definitions on the responses will vary accordingly with the parameter values.

The fractional Kelvin-Voigt model for circumferential guided waves in a viscoelastic FGM hollow cylinder

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be more accurate. A thorough knowledge of the dispersive characteristics of such model is very essential to the application of guided wave testing technique. In this paper, the guided waves in a fractional Kelvin-Voigt viscoelastic FGM hollow cylinder with material changing in the thickness direction are investigated. The Weyl definition of fractional order derivatives and the extended Legendre polynomial approach are employed for the derivations of the governing equations. The presented approach has the advantage that the solution of the complex partial differential wave equations with variable coefficients is reduced to an eigenvalue problem, which overcomes the shortcomings of the existing iterative methods such as the Newton downhill method and improves computation efficiency. The previous methods for dealing with viscoelastic guided wave transform the wave equations into a matrix determinant problem solved by the iterative methods, which has a very slow calculation speed. Comparisons with the related studies are conducted to validate the correctness of the presented approach, and the convergence of the approach is discussed. The full three dimensional spectrum, phase dispersion curves and attenuation curves are illustrated for various fractional order viscoelastic FGM hollow cylinders. The influences of fractional order, grade field and radius-thickness ratio on dispersion and attenuation curves are illustrated. The difference of the dispersion characteristics between the viscoelastic model and the elastic one is discussed. The influences of fractional order on displacement distributions are also studied.

A Multilevel Approach for Stability Conditions in Fractional Time Diffusion Problems

The Caputo definition of fractional derivatives introduces solution to the difficulties appears in the numerical treatment of differential equations due its consistency in differentiating constant functions. In the same time the memory and hereditary behaviors of the time fractional order derivatives (TFODE) still common in all definitions of fractional derivatives. The use of properties of companion matrices appears in reformulating multilevel schemes as generalized two level schemes is employed with the Gerschgorin disc theorems to prove stability condition. Caputo fractional derivatives with finite difference representations is considered. Moreover the effect of using the inverse operator which transmit the memory and hereditary effects to other terms is examined. The theoretical results is applied to a numerical example. The calculated solution has a good agreement with the exact solution.

Solution of One Dimensional Conformable fractional Heat Equation

The aim of this paper is to present the solution of one dimensional conformable fractional heat equation by applying conformable fractional derivative which is considered as more convenient definition of fractional derivative.

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 Caputo Fabrizio fractional order model for control of glucose in insulin therapies for diabetes

Many fresh definitions of fractional derivatives have been suggested and used in recent years to produce mathematical models with memory, background, or non-local effects for a broad range of real-world structures. The primary aim of this article is to create and evaluate a fractional-order derivative for an extensive regulatory scheme for glucose-insulin regulation. The existence and uniqueness are determined by a fixed point theorem and an iterative scheme. We suggest an impulsive differential equation model study plasma glucose control for diabetic patients with impulsive insulin injections. It is regarded as a deterministic mathematical model related to the diabetes mellitus fractional derivatives. For fractional orders, numerical simulations are performed to demonstrate the impacts of varying the fractional-order to achieve the theoretical outcomes and comparison with the Caputo derivative are made. The results of these case studies indicate that this plasma glucose control of the fractional-order model is an appropriate candidate.

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.

A weak Galerkin finite element method for solving time-fractional coupled Burgers' equations in two dimensions

In this paper, we present a continuous and discrete time weak Galerkin finite element method (WG-FEM) for solving two dimensional time fractional coupled Burgers' equations. The stability is proved for discrete time WG-FEM and the optimal order error in L2-norm is obtained based on fractional derivative definition, fractional integral definition and dual argument technique for continues and discrete time WG-FEM. The numerical example is to illustrate the theoretical analysis with polynomial mixture {Pk(K),Pk−1(∂K),[Pk−1(K)]2}.