Fractional Derivative Approach Research Articles

Modeling creep/recovery behavior of cold-set gels using different approaches

This study aimed to characterize cold-set unfilled and emulsion-filled gels (EFG) of commercial soy protein isolate (SPI), with and without xanthan gum (XG) and locust bean gum (LBG), using creep/recovery tests. For this purpose, a comparative study of the data was developed using two different approaches:(1) the classical Burger's model and (2) a fractional derivative approach. The polysaccharides incorporation decreased the gels' creep compliance and increased the recovery rates. Systems with XG presented lower creep compliance and higher recovery capacity. EFG were stronger and exhibited higher elasticity than unfilled gels. Burger's model parameters allowed some comparisons among the systems, however, due to their lack of clear physical meanings they did not allow an in-depth discussion on the gels' properties. Besides, this model did not describe well recovery data as the fractional derivative approach, whose parameters with clearer physical meanings, allowed a more intuitive discussion about the characteristics of the gels.

Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the error bound does not blowup when α (the order of fractional derivative) approaches 1−. It is shown both in theoretically and numerically that the temporal convergence order of the ADI scheme is O(τ2α+τtnα−1) at time t=tn; hence the scheme is globally O(τα) accurate in temporal direction, but it is O(τmin⁡{2α,1}) when t is away from 0.

The cubic B-spline interpolation method for numerical point solutions of conformable boundary value problems

In this analysis, the cubic B-spline method is employed for constructing the approximate solutions of a class of fractional two-point boundary value problems. These fractional problems are expressed in terms of the conformable fractional derivative approach. More deeply, a class of conformable Lane-Emden model is considering and modifying with singularities. Many numerical applications are presented and discussed to exhibit the feasibility and efficiency of the procedure involved in both linear and non-linear cases. The numerical findings are quite similar to those of the exact solutions as well as require relatively less computational work. Latterly, remarks and future research work are provided.

Thermal activity of conventional Casson nanoparticles with ramped temperature due to an infinite vertical plate via fractional derivative approach

With inspired thermal characteristics and dynamic applications of nanoparticles, the researchers are motivated to suggest multidisciplinary significances of nano-materials in various thermal engineering and industrial processes. In this analysis, a mixed free convection nanofluid flowing on a vertical plate is discussed with heat transfer in the drilling of nanofluid. The Casson liquid is assumed as a base fluid with uniform suspension of clay nanoparticles. The leading partial differential governing equations with physical properties of nanoparticles are demonstrated with corporal flow phenomenon and imposed thermal conditions. In the governing partial differential equations, the partial derivative with respect to time is replaced by the most recent definition of fractional derivatives i.e. Atangana-Baleanu time-fractional derivative, and then the solution of temperature and velocity field is found by utilizing the Laplace transformation. The effects of different parameters with different values are deliberated and plotted graphically and numerically for temperature and momentum equations. The main results and conclusions are accomplished at the end of this exertion and compared with existing literature. It is concluded that the velocity profile boost up with Grashof number. The results of temperature and momentum equations show a more decaying tendency as compared to Caputo-Fabrizio fractional derivative.

A mathematical model for thermography on viscous fluid based on damped thermal flux

Abstract Thermography is a fully noninvasive technique that discerns the thermal profiles of highly viable rheological parameters in heat and mass transference. In this paper, the free convection flow of viscous fluid among two vertical and parallel plates in the existence of a transverse magnetic field is investigated. The Caputo time-fractional derivative is manipulated for introducing a thermal transport equation along with a weak memory. The analytical and closed-form fractional solution for the temperature and velocity profiles are obtained through Laplace paired in conjunction with the finite Sine-Fourier transforms technique. The solution to the classical model is concluded as a special case for the solutions to the fractional modeled problem when the memory factor (the order of fractional derivative) approaches 1. Also, the solutions are stated in connection with the Mittag–Leffler function. The influences of variations of fractional and material parameters are depicted through MathCad15.

Stability analysis and optimization of a hybrid rotating machinery support combining journal bearings with viscoelastic supports

Hybrid oil-bearing-viscoelastic-support systems are promising mechanical devices aiming to provide stability and vibration control in rotating systems. This paper proposes a mathematical model that combines analytical calculation of oil-film coefficients, fractional derivative approach for the viscoelastic materials dynamics and generalized equivalent parameters for dynamically modelling multi-DOF supports as stiffness coefficients. This general model accounts double frequency dependence of the hybrid system, i.e., spin speed and external excitation frequency, and temperature dependence of the viscoelastic material. The generalized equivalent parameter approach is fundamental for determining natural frequencies and ordaining the eigenvalues and eigenvectors of a rotating system that combines viscous and viscoelastic behaviors. The simulations show that viscoelastic supports improve the rotating system stability when compared to the system using oil bearings. Furthermore, the optimization can be applied to the hybrid system, improving the vibration suppression characteristics of the device.

Fractional Fourier Transform based Riesz fractional derivative approach for edge detection and its application in image enhancement

Edge detection plays a key role in detecting the boundaries of an object in the image to improve the quality of image edges. The edge detection techniques based on integer order derivatives are more prevalent in literature. But these techniques have the limitation of providing thicker edges and being sensitive to noise. Therefore, in order to deal with this issue, an edge detection technique based on Riesz fractional derivative (RFD) in Fractional Fourier Transform (FrFT) domain is proposed in this paper. The RFD mask used for edge detection is obtained by using various interpolation methods. The selection of mask size is done on the basis of Figure of Merit (FOM) and Edge Preservation Index (EPI). The edges obtained with proposed approach in FrFT domain are further used for image enhancement. The effectiveness of proposed approach is confirmed using test images from different standard datasets by considering performance metrics such as FOM, EPI, Information Entropy, Average Gradient, Edge Intensity, etc. Further validation of the proposed approach is done by considering various uncontrolled features such as Poisson noise, different illumination conditions, JPEG compression artifacts, etc. Experimental results show that proposed technique outperforms the existing techniques by providing improvement in various performance parameters.

A generalized model for quantitative analysis of sediments loss: A Caputo time fractional model

Sediment loss is an indispensable geological phenomenon that determines the reduction of mass in time interval with the differential influence of controlling factors. In this study, the competence of rocks is considered as a controlling factor while weathering and erosion are termed as the decaying parameter. Attributed to the reduction of the total mass in unit time. A mathematical model is [proposed for the above-mentioned phenomenon and generalized using the Caputo fractional derivatives approach to better understand and predict the sediment loss. The model is solved for exact solutions using the Laplace transform technique. The results obtained for β and -β are in strong agreement with the field images of selected outcrop sections. The memory effect is also shown with different values of αThe results revealed that the mathematical model better explains the geological processes.

Diffusion through skin in the light of a fractional derivative approach: progress and challenges.

This review is focussed on modelling the transport processes of different drugs across the intact human skin by introducing a memory formalism based on the fractional derivative approach. The fundamental assumption of the classic transport equation in the light of the Fick's law is that the skin barrier behaves as a pseudo-homogeneous membrane and that its properties, summarized by the diffusion coefficient D, do not vary with time and position. This assumption does not hold in the case of a highly heterogeneous system as the skin is, whose outermost layer (the stratum corneum) is comprised of a multi-layered structure of keratinocytes embedded in a lamellar matrix of hydrophobic lipids, followed by the dermis that contains a network of capillaries that connect to the systemic circulation. A possible way to overcome these difficulties resides in the introduction of mathematical models which involve fractional derivatives to describe complex systems with interactions in space and time, following the model originally developed by Caputo in order to consider the memory effects in materials. Although the introduction of fractional derivatives to model memory effects is completely phenomenological, i.e., characterized by a single parameter, i.e., the fractional derivative order [Formula: see text] a number of authors have found that this approach can provide a better comparison to experimental data and that this technique may be alternative to integer-order derivative models. In this review, we aim to summarize some our recent results, concerning the transport of different diffusing compounds of different structural complexity across the intact skin.

Revisiting Lévy flights on bounded domains: a Fock space approach

The statistical description of a one-dimensional superdiffusive Lévy flier restricted to a finite domain is well known to be technically involving. For example, in this type of process the probability distribution P(x, t) and survival probability S(t) cannot be obtained from the method of images. Other methods, such as the fractional derivative approach, also find technical difficulties due to the long jumps combined with the presence of absorbing boundaries. Here we revisit this problem through a different point of view. We map the corresponding master equation to a Schrödinger-like equation and then describe the Lévy flier evolution in a Fock space. The system states are assigned to the available positions in the discrete space. The Hamiltonian-like matrix is calculated for any Lévy index α ∈ (0, 2]. For the system sizes studied here the computation of its eigenvalues and eigenvectors are performed using a symbolic computing software. This method allows to build the time evolution operator, the distribution P(x, t) and the survival probability S(t). We compare our results for P(x, t) with direct Monte Carlo simulations and find a good agreement for all α ∈ (0, 2]. Similarly, our results for S(t) nicely agree with the numerical simulations for any time t, including the short-term behavior. In the long-term asymptotic limit we identify the crossover between the power-law and exponential behaviors, which emerge respectively when only one or both boundaries are reached by the Lévy flier. Comparisons with some exact expressions for Lévy flights in the continuous space limit also display good agreement. We conclude our analysis by discussing the possibility of extending the present framework to general bounded random walks and flights.

On modelling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method

We present a fractional-order epidemic model for childhood diseases with the new fractional derivative approach proposed by Caputo and Fabrizio. By applying the Laplace Adomian decomposition method (LADM), we solve the problem and the solutions are presented as infinite series converging to the solution. We prove the existence, uniqueness, and stability of the solution by using the fixed point theory. Also, we provide some numerical results to illustrate the effectiveness of the new derivative.

Concrete Based Jeffrey Nanofluid Containing Zinc Oxide Nanostructures: Application in Cement Industry

Concrete is a non-Newtonian fluid which is a counterexample of Jeffrey fluid. The flow of Jeffrey fluid is considered containing nanostructures of zinc oxide in this study. The flow of the nanofluid is modeled in terms of partial fractional differential equations via Atangana–Baleanu (AB) fractional derivative approach and then solved using the integral transformation. Specifically, the applications are discussed in the field of concrete and cement industry. The variations in heat transfer rate and skin friction have been observed for different values of volume fractions of nanoparticles. The results show that by adding 4% Z n O nanoparticles increase skin friction up to 15%, ultimately enhancing the adhesion capacity of concrete. Moreover, Z n O increase the density of concrete, minimizing the pores in the concrete and consequently increasing the strength of concrete. The solutions are simplified to the corresponding solutions of the integer ordered model of Jeffrey-nanofluid. Applications of this work can be found in construction engineering and management such as buildings, roads, tunnels, bridges, airports, railroads, dams, and utilities.

On the fractional-order dynamics of a double pendulum with a forcing constraint using the nonsingular fractional derivative approach

On the fractional-order dynamics of a double pendulum with a forcing constraint using the nonsingular fractional derivative approach

Using fractional derivatives for improved viscoelastic modeling of textile composites. Part II: Fabric under different temperatures

An improved modelling of the viscoelastic behavior of fabric composites at different temperatures is presented by a new means of fractional calculus. To this end, a four-parameter fractional derivative Zener model is first used to describe the dynamic behavior of the fabric, where the constitutive relation’s response to a unit step of strain (for determination of the time dependent stress relaxation modulus) can be readily found in the literature. This response function is then fitted to the experimental data of samples of a typical prepreg composite made of E-glass fibers comingled with polypropylene fibers, both at the dry and consolidated conditions. It is shown that the Zener model can better capture the experimental data when compared to the conventional approaches such as integer order modelling and Prony series. The results, particularly, showed that the viscoelastic behavior of the material at lower temperature regimes is far more accurate using the fractional derivatives approach. Finally, the variation of storage and loss moduli of the dry fabric with external cyclic loading were scrutinized, to determine the rubbery and glassy regions of the frequency response.

Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate

Nanofluids are a novel class of heat transfer fluid that plays a vital role in industries. In mathematical investigations, these fluids are modeled in terms of traditional integer-order partial differential equations (PDEs). It is recognized that traditional PDEs cannot decode the complex behavior of physical flow parameters and memory effects. Therefore, this article intends to study the mixed convection heat transfer in nanofluid over an inclined vertical plate via fractional derivatives approach. The problem in hand is modeled in connection with Atangana–Baleanu fractional derivatives without singular and local kernel with a strong memory. Human blood is considered as base fluid and carbon nanotube (CNTs) (single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs)) are dispersed into it to form blood-CNTs nanofluid. The nanofluid is considered to flow in a saturated porous medium under the influence of an applied magnetic field. The exact analytical expressions for velocity and temperature profiles are acquired using the Laplace transform technique and plotted in various graphs. The empirical results indicate that the memory effect decreases with increasing fractional parameters in the case of both temperature and velocity profiles. Moreover, the temperature profile is higher for blood SWCNTs because of higher thermal conductivity whereas this trend is found opposite in the case of velocity profile due to densities difference.

A hysteretic model for fiber-reinforced composites at finite strains: fractional derivatives, computational aspects and analysis

Deep drawing complex composite parts including strong geometrical non-linearities are currently of major interest. Such forming process induces non-monotonous variations in bending and in-plane shear deformation modes. Moreover, it is experimentally shown that these specific variations lead to an hysteretic behavior of the material. Hence, non dissipative models are no longer appropriate to accurately describe the behaviour since these variations of load cannot longer be neglected. The objective of this paper is therefore to propose an anisotropic model under large strains capable of describing the hysteretic trajectories of the material’s behavior. For this purpose, a fractional derivative approach was applied and identified. Through this hysteretic approach, it is now possible to predict the shape, shear and bending strain more accurately when the fabric undergoes heavy transformation. This model also gives the possibility to predict the residual stress and plastic strain. These predictions lead to quantify the spring back of the material when the punches are removed, which is very important since new deep drawing strategies are emerging. Discretization procedures for fractional derivatives models are described and Matlab source codes are also provided.

Modeling and analysis of the polluted lakes system with various fractional approaches

Abstract The purpose of this paper is to investigate the dynamics of the polluted lakes system through fractional derivative approach in the sense of fractal-fractional operator. Initially, we derive the model based on the assumptions and then apply the fractal-fractional operator. The model with fractal-fractional approach in the sense of Atangana–Baleanu derivative is considered. When, choosing the fractal order one we obtain fractional order, and when choosing the fractional order one a fractal system is obtained. Considering both the operators together we present a model with fractal-fractional. We present a novel numerical approach to solve the fractal-fractional system and then present the existence and uniqueness results for the system. We present various graphical results for the fractal and fractional operators and also for the fractal-fractional orders. The graphical results reveals the significance of the new operator to a practical problem in a better way.

Heat and mass transfer of fractional second grade fluid with slippage and ramped wall temperature using Caputo-Fabrizio fractional derivative approach

Unsteady free convection slip flow of second grade fluid over an infinite heated inclined plate is discussed. The effects of mass diffusions in the flow are also eligible. Caputo-Fabrizio fractional derivative is used in the constitutive equations of heat and mass transfer respectively. Laplace transform is utilized to operate the set of fractional governing equations for both ramped and stepped wall temperature. Expression for Sherwood number and Nusselt number with non-integer order are found by differentiating the analytical solutions of fluid concentration and temperature. Numerical results of Sherwood number, Nusselt number and skin friction are demonstrated in tables. Solutions are plotted graphically to analyze the impact of distinct parameters i.e. Caputo-Fabrizio fractional parameter, second grade parameter, slip parameter, Prandtl number, Schmidt number, thermal Grashof number and mass Grashof number to observe the physical behavior of the flow.

An applied model for predicting memory effects of flexible polyurethane foams

Purpose The efficiency of fractional derivative and hereditary combined approach in modeling viscoelastic behavior of soft foams was successfully addressed in Elfarhani et al. (2016a). Since predictions obtained on flexible polyurethane foam (FPF) type A (density 28 kg m−3) were found very promoting, the purpose of this paper is to apply the approach basing on two other types of foams. Both soft polyurethane foams type B of density 42 kg m−3 and type C of density 50 kg m−3 were subjected to multi-cycles compressive tests. Design/methodology/approach The total foam response is assumed to be the sum of a non-linear elastic component and viscoelastic component. The elastic force is modeled by a seven-order polynomial function of displacement. The hereditary approach was applied during the loading half-cycles to simulate the short memory effects while the fractional derivative approach was applied during unloading cycles to simulate the long memory effects. An identification methodology based on the separation of the measurements of each component force was developed to avoid parameter admixture problems. Findings The proposed model reveals good reliability in predicting the responses of the two considered flexible foams. Predictions as measurements establish that residual responses were negligible compared to elastic and viscoelastic damping responses. Originality/value The development of a new combined model reveals good reliability in predicting the responses of the two polyurethane foams type A and B.

On the fuzzy fractional differential equation with interval Atangana–Baleanu fractional derivative approach

Abstract The fuzzy systems with interval approach use an infinite valued parameter in the range of [0,1] as a confidence degree of belief. This parameter makes more complicity but plays the main role in creating the fuzzy solution of the fuzzy systems. In solving process of the model, the Atangana–Baleanu derivative in the fractional case of differential equations has a memory to use all the previous information. Therefore this is as a key point and advantage of using this derivative to reduce the complicity of numerical results in comparison with other known derivatives. In this paper, first, the ABC fractional derivative on fuzzy set-valued functions in parametric interval form is defined. Then it is applied for proving the existence and uniqueness of the solution of fuzzy fractional differential equation with ABC fractional derivative. In general, it is shown that the last interval model is as a coupled system of nonlinear equations. To solve the final system an efficient numerical method called ABC-PI is used. For more illustration, several examples are solved numerically and analyzed by the figures.