Definition Of Fractional Derivative Research Articles

New class of conformable derivatives and applications to differential impulsive systems

In this paper, first we introduce new definition of conformable fractional derivatives and study their algebraic properties. As application, we consider certain classes of conformable differential linear systems subject to impulsive effects and establish qualitative behavior of the nontrivial solutions such as stability, disconjugacy, nonexistence, upper bound estimation for maximum number of zeros of the nontrivial solutions and some oscillatory properties. The main tool of this establishment concerns with Lyapunov inequalities of under study fractional order linear systems.

Steady-state response of systems with fractional dampers

In this paper, steady state vibrations of systems with built-up viscoelastic dampers are considered. The dampers are modeled using the fractional-derivative rheological models. The Caputo type fractional derivative definition is used. In particular, the steady state vibrations of systems are analysed. The solution to the steady state vibrations is written using real quantities. The effects induced by changes of environmental temperature are also considered and, in this context, the time-temperature superposition principle is adopted. The results of several parametric studies are also described and discussed in detail.

Efficient algorithms for analyzing the singularly perturbed boundary value problems of fractional order

In this paper, we were concerned with the description of the singularly perturbed boundary value problems in the scope of fractional calculus. We should mention that, one of the main methods used to solve these problems in classical calculus is the so-called matched asymptotic expansion method. However we shall note that, this was not achievable via the existing classical definitions of fractional derivative, because they do not obey the chain rule which one of the key elements of the matched asymptotic expansion method. In order to accommodate this method to fractional derivative, we employ a relatively new derivative so-called the local fractional derivative. Using the properties of local fractional derivative, we extend the matched asymptotic expansion method to the scope of fractional calculus and introduce a reliable new algorithm to develop approximate solutions of the singularly perturbed boundary value problems of fractional order. In the new method, the original problem is partitioned into inner and outer solution equations. The reduced equation is solved with suitable boundary conditions which provide the terminal boundary conditions for the boundary layer correction. The inner solution problem is next solved as a solvable boundary value problem. The width of the boundary layer is approximated using appropriate resemblance function. Some theoretical results are established and proved. Some illustrating examples are solved and the results are compared with those of matched asymptotic expansion method and homotopy analysis method to demonstrate the accuracy and efficiency of the method. It can be observed that, the proposed method approximates the exact solution very well not only in the boundary layer, but also away from the layer.

Numerical approximation of Riemann‐Liouville definition of fractional derivative: From Riemann‐Liouville to Atangana‐Baleanu

In the last decade, theoretical and applied studies were done in order to provide a suitable definition of fractional derivative, which meets all the requirement of a derivative in its primary sense. It was concluded by some eminent researchers that the Riemann‐Liouville version was the most suitable. However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann‐Liouville definition, from power‐law kernel to generalized Mittag‐Leffler‐law via exponential‐decay‐law.

On the New Solutions of the Conformable Time Fractional Generalized Hirota-Satsuma Coupled KdV System

Abstract In this paper, generalized Hirota Satsuma coupled KdV system is solved with tanh method and q-Homotopy analysis method. New fractional derivative definition called “conformable fractional derivative” used in the solution procedure. Tanh method with conformable derivative firstly introduced in the literature. By the graphics of analytical and approximate solutions, it is shown that, both methods provide an effective and powerful mathematical tool for solving nonlinear PDEs containing conformable fractional derivative.

On the families of fractional dynamical models

In this paper, we reveal the uncertainty and its fractional generalized Hamiltonian representation for the fractional and nonlinear problem and find general methods of constructing a family of fractional dynamical models. By using the definition of combined fractional derivative, we construct a unified fractional generalized Hamiltonian equation, a fractional generalized Hamiltonian equation with combined Riemann–Liouville derivative, and a fractional generalized Hamiltonian equation with combined Caputo derivative; also, as special cases, under the different definitions of fractional derivatives, we, respectively, obtain a series of different kinds of fractional generalized Hamiltonian equations. In particular, we present a new concept of the family of fractional dynamical models, and it is found that, using the fractional generalized Hamiltonian method, we can construct a series of families of fractional dynamical models. And then, as the new method’s application, we construct three new kinds of families of fractional dynamical models, which include a family of fractional Lorentz–Dirac models, a family of fractional Lotka–Volterra models and a family of fractional Henon–Heiles models.

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

PurposeRecently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.Design/methodology/approachIn this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.FindingsThe theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.Originality/valueA new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

Technical Note on a New Definition of Fractional Derivative

Technical Note on a New Definition of Fractional Derivative

FDTD Method for Wave Propagation in Havriliak–Negami Media Based on Fractional Derivative Approximation

A finite-difference time-domain (FDTD) method for simulating wave propagation in Havriliak-Negami (H-N) dispersive media is presented. In an H-N medium, the time-domain polarization relation involves a fractional differential operator, whose analytical evaluation is troublesome. Moreover, if the fractional differential operator is approximated by a sum of fractional time derivatives, the numerical treatment of the polarization relation results in significant memory storage demands, because fractional derivatives are not local. However, by an appropriate approximation of the Grunwald-Letnikov definition of fractional derivatives, an FDTD scheme with reasonable memory requirements is feasible. In particular, the coefficients that appear in the Grunwald-Letnikov definition are approximated by sums of decaying exponentials. As a result, the FDTD scheme is based on recursive equations, while a limited number of auxiliary vectors need to be stored. The proposed scheme has been applied successfully to the simulation of the excitation of an H-N medium by a wideband Gaussian electromagnetic pulse.

Solutions with Wright Function for Time Fractional Free Convection Flow of Casson Fluid

A fractional model of Casson fluid coupled with energy equation is developed. Casson fluid in the presence of heat transfer is considered over an oscillating vertical plate with constant wall temperature. Definition of fractional Caputo derivative is used in the mathematical formulation of the problem. Exact solutions via Laplace transform are obtained and presented in terms of Wright function. Parametric studies were undertaken, and the obtained solutions are illustrated through plots for various physical parameters.

Analytical Solutions of Riccati Fractional Equation with Coefficients Satisfying Differential Conditions with Arbitrary Function

New exact solutions of the Fractional Riccati Differential equation are presented. Exact solutions are obtained in cases that the coefficient functions and an arbitrary function satisfy some differential conditions. In view of the conditions imposed on Riccati equation's coefficients we define and formulate the coefficients of the Riccati equation to obtain the differential cases, for each case the general solution is presented. The results obtained are useful and confirm the simplicity of the conformable definition of fractional derivative.

Dynamic preserving method with changeable memory length of fractional-order chaotic system

In this paper, an asymptotically stability condition α+β≥3γ of the fractional-order Lü system is proposed by using the theory of stability. Under this asymptotically stability condition and the Riemann-Liouville fractional derivative definition, the numerical efficiency is obtained by combining the nonstandard finite difference method with the Grünwald-Letnikov method. In addition, the reported dynamic preserving properties of the nonstandard finite difference method are verified by comparing with the predictor-corrector algorithm. Moreover, in order to reduce the computation time of fractional derivatives, a model with changeable memory length of short memory principle is introduced and solved by the nonstandard finite difference method. In the numerical examples, about 30% of computation time can be reduced by applying the changeable memory length model.

A New Fractional Model for the Falling Body Problem

Recently, a conformable fractional derivative has been proposed to calculate the derivative of non-integer order of time functions. It has been shown that this new fractional derivative definition obeys many advantages over the preceding definitions. For mathematical models in applied sciences and to preserve the dimensionality of the physical quantities, an auxiliary parameter (σ) which has the dimension of seconds should be implemented in the fractional derivative definition. We obtain analytic solutions for the resulting conformable fractional differential equations describing the vertical velocity and the height of the falling body. It is shown that the dimensions of velocity and height are always correct without any restrictions on the auxiliary parameter σ which contradicts with the results in the literature when applying the Caputo definition to the same problem. This may open the door for many future works either to describe the role of such an auxiliary parameter or to derive a more suitable definition for the fractional derivative.

Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives

This article is focused on natural convection of unsteady flow of generalized Maxwell fluid over an oscillating vertical flat plate with constant temperature at the boundary. The Maxwell fluid with classical derivatives, describing one dimensional flow has been generalized to non-integer order derivatives known as fractional derivative with term of buoyancy. A modern definition of fractional derivative, recently introduced by Caputo and Fabrizio has been used to formulate the considered problem. Semi analytical solutions of the dimensionless problem have been obtained by using the Laplace transform. The solutions for temperature, velocity and shear stress are obtained with numerical inversion techniques of Laplace transform namely, Stehfest’s and Tzou’s algorithms. At the end, graphical illustrations for temperature, velocity, Nusselt number and shear stress are plotted. We have studied especially the influence of fractional parameter on temperature, velocity and shear stress respectively. We have observed that temperature can be enhanced for increasing the fractional parameter α while velocity and shear stress can be increased by decreasing the value of fractional parameter α.

Stability Analysis of Distributed Order Hilfer-Prabhakar Differential Equations

In the current study we presented a distributed order form of Hilfer-Prabhakar (DHP) derivative, which in special cases reduces to the existent definitions of fractional or distributed order derivatives. Moreover, we analyzed the stability of DHP differential equations, which are the generalized form of all previous distributed or fractional differential equations. The obtained results showed that sufficient conditions on asymptotic stability of these systems have been obtained through the generalized properties of Mittag-Leer functions and the Laplace transform. Moreover, a number of conditions on stability analysis of such systems have been introduced by using a new definition of inertia of a matrix.

The Numerical Solution of Fractional Bratu-Type Differential Equations

This paper introduce the differential transform method (DTM) to solve the fractional Bratu-type differential equation modeling a combustion in numerical slab. For the definition of fractional derivative, the Caputo sense is used. The result corresponds to the exact solution when the obtained solution is constructed as power series for some values of fractional order. Finally, some examples are presented to indicate the efficiency of applied method. Comparison of the results obtained by DTM with those obtained by other methods is given.

Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs

By using a counterexample, we proved the fractional chain rule appeared in many references does not hold under Riemann–Liouville definition and Caputo definition of fractional derivative. It shows that this chain rule is invalid in investigating exact solutions of nonlinear fractional partial differential equations (PDEs). In this paper, based on the homogenous balanced principle, the function-expansion method of separation variable type are introduced. By using this method, a series of nonlinear time fractional PDEs such as time fractional KdV equation and Burgers equation, time fractional diffusion–convection equations are studied from mathematical viewpoint. The dynamical properties of these exact solutions are discussed and the profiles of several representative exact solutions are illustrated.

Solving Time-Fractional Order Telegraph Equation Via Sinc–Legendre Collocation Method

In this paper, we introduce a numerical method for solving time-fractional order telegraph equation. The method depends basically on an expansion of approximated solution in a series of Sinc function and shifted Legendre polynomials. The fractional derivative is expressed in the Caputo definition of fractional derivatives. The expansion coefficients are then determined by reducing the time-fractional order telegraph equation with its boundary and initial conditions to a system of algebraic equations for these coefficients. This system can be solved numerically using the Newton’s iteration method. Several numerical examples are introduced to demonstrate the reliability and effectiveness of the introduced method.

Random walk's models for fractional diffusion equation

Fractional diffusion equations are used for mass spreading in inhomogeneous media. They are applied to model anomalous diffusion, where a cloud of particles spreads in a different manner than the classical diffusion equation predicts. Thus, they involve fractional derivatives. Here we present acontinuous variant of Grünwald-Letnikov's formula, which is useful to compute the flux of particles performing random walks, allowing for heavy-tailed jump distributions. In fact, we set a definition of fractional derivatives yielding the operators which enable us to retrieve the space fractional variant of Fick's law, for enhanced diffusion in disordered media, without passing through any partial differential equation for the space and time evolution of the concentration.

An analytical approach to reconstruct the initial distribution of a sub-diffusion model with time-fractional derivative

The inverse problem has been widely used in optical design, image processing and many other fields. This study is to employ an analytical approach to reconstruct the initial distribution of the fractional sub-diffusion model with Caputo’s definition of fractional derivative in time. The basic strategy is to solve the corresponding direct problem via separation of variables and Laplace transform and then to convert the initial inverse problem into an integral equation of the first kind. The key point is that we employ the Picard’s theorem to design an analytical solution of initial diffusion distribution. The proposed scheme is tested to some benchmark problems. Numerical results show that the analytical approach performs efficiently.