Fractional Derivative Parameter Research Articles

Numerical and analytical investigation to fractional third grade viscoelastic fluids in rotating frame with slip boundary conditions

In order to obtain the mixture or separation of the biological or chemical fluids with viscoelastic characteristics, the rotating electro-osmotic flow of fractional third grade fluids in a microchannel is investigated. The momentum equation is solved by Laplace and Fourier Transforms after linearization by small parameter perturbation method. Furthermore, the numerical solutions also are obtained by finite difference method. The velocity distribution influenced by different physical parameters is discussed. Two interesting phenomena can be observed. One of them is that reverse flow appears at the center of the channel for some physical parameters. Another one is that the velocity in the microchannel will arrive at a steady state from oscillation at last. However, with the increase of fractional derivative parameters, the amplitude of the oscillation becomes bigger at the center and the time arriving at the steady state is delayed.

Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects

Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature.

Cattaneo–Friedrich and Crank–Nicolson analysis of upper-convected Maxwell fluid along a vertical plate

The concept of fractional derivative is used to solve a variety of viscoelastic fluid problems. However, researchers mostly overlooked the consequences of nonlinear convection in the fractional viscoelastic fluid models and were concerned only with situations where the governing equations are linear. Most importantly, the nonlinear fluid models, whether classical or fractional, are solved for steady-state conditions. To overcome these limitations, this research presents unsteady flow and heat transfer of nonlinear fractional upper-convected Maxwell (UCM) viscoelastic fluid along a vertical plate. The governing equations of the fractional Maxwell fluid are developed by introducing Friedrich shear stress and Cattaneo heat flux models to the classical UCM fluid model. An additional feature to the invention of the constructed fractional model is the consequence of an external magnetic field. Moreover, the considered model comprises nonlinear, coupled, fractional partial differential equations. Therefore, a numerical scheme is developed with the aid of the L1-approximation of Caputo derivative and the Crank–Nicolson method. The effects of different regulating parameters on fluid features have been thoroughly investigated. The obtained results are exhibited graphically and discussed in detail. It is observed that the skin friction increases for the velocity relaxation time parameter, but an opposite behavior is observed against the velocity fractional derivative parameter. Moreover, a significant enhancement is noticed in the Nusselt number for increasing estimates of the Prandtl number.

Nonlinear dynamics and chaos in fractional differential equations with a new generalized Caputo fractional derivative

In this paper, novel systems of fractional differential equations involving a new generalized Caputo fractional derivative were proposed. The complex dynamic behavior of these systems was studied by numerical simulation. Nonlinear dynamics and chaos in hybrid fractional order systems were investigated using a predictor–corrector algorithm. In particular, the effect of the new generalized fractional derivative parameters on the dynamics of the proposed systems was discussed. The rich variation obtained from the characteristics of the studied systems recommends the implementation of the new generalized derivative in fractional calculus applications.

Fractional blood flow in rotating nanofluid with different shapes nanoparticles in the influence of activation energy and thermal radiation.

A fractional blood flow model, in the presence of magnetic nanoparticles, is considered in this work. The effects of activation energy and thermal radiation on the blood flowing in the oscillating elastic tube are studied. The nanofluid inside the tube is activated by the rotating effect of the charged particles, a constant external magnetic field, and the activation energy. The blood is assumed to be at a temperature and a concentration that vary with the speed of the particles. The study takes advantage of a model, which includes a fractional-order derivative of Caputo's type. The shape of nanoparticles and the speed of blood and the distributions of temperature and concentration are assimilated to Brownian motion and thermophoresis. They are calculated numerically using the L1-algorithm method. The results show that the applied magnetic field and the effects of the fractional-order parameter reduce the velocity of the nanofluid and nanoparticles, which considerably affects the temperature and concentration of the fluid. It is also found that the particle shape and fractional derivative parameters significantly influence velocities and heat transfer.

Influence of fractional time order on W-shaped and Modulation Instability gain in fractional Nonlinear Schrödinger Equation

In this paper, we study the influence of fractional time derivative on W-shaped profile and Modulation Instability gain in fractional Nonlinear Schrodinger Equation (NLSE) which could be used to describe the propagation of pulses in random media, optical metamaterials and others nonlinear systems. We first imply the auxiliary equation method to set up bright, dark and W-shape optical solitons solutions under certain conditions. Thereafter, we graphically depict the obtained results which deeply show the potency of the fractional parameter order on the width and shape of the solitons. We then study the Modulation Instability (MI) gain spectra and we end up remarking that the MI gain sidebands and the MI gain shapes are mainly influenced by the fractional time derivative parameter compared to the previous works reported in nonlinear optic fibers (Wyller et al., 2002; Zhanga et al., 2017).

Clout of fractional time order and magnetic coupling coefficients on the soliton and modulation instability gain in the Heisenberg ferromagnetic spin chain

In this paper we utilize the auxiliary equation method to show the leverage of the fractional derivative parameter and the Magnetic Coupling Coefficients (MCC) on the procured soliton solutions and the Modulation Instability (MI) gain in Heisenberg ferromagnetic spin. The pedestal of the work is set on the (2+1)-fractional time order dimensional Heisenberg ferromagnetic spin chain equation which demonstrate the propagation of nonlinear waves in Heisenberg ferromagnetic spin chain system. The results collected have enclosed bright-dark soliton, dark soliton and rational solutions. Compared the acquired results with Zhao et al. (2016) [26], Akbar (2021) [27], Hosseini et al. (2021b) [28], further soliton-like solutions have been drop in and the marvel of the fractional parameter has been exposed across the spatiotemporal and plot evolution of the bright and dark soliton solutions. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the MCC jointed with the fractional time derivative order. The collected results are new in literature in our knowledge and will give the pipe to the solitary waves theory in Heisenberg ferromagnetic spin including inhomogeneity parameters.

Study on the generalized k-Hilfer–Prabhakar fractional viscoelastic–plastic model

The general fractional operator shows its great predominance in the construction of constitutive model owing to its agility in choosing the embedded parameters. A generalized fractional viscoelastic–plastic constitutive model with the sense of the k-Hilfer–Prabhakar (k-H-P) fractional operator, which has the character recovering the known classical models from the proposed model, is established in this article. In order to describe the damage in the creep process, a time-varying elastic element is used in the proposed model with better representation of accelerated creep stage. According to the theory of the kinematics of deformation and the Laplace transform, the creep constitutive equation and the strain of the modified model are established and obtained. The validity and rationality of the proposed model are identified by fitting with the experimental data. Finally, the influences of the fractional derivative order and parameter k on the creep process are investigated through the sensitivity analyses with two- and three-dimensional plots.

Solitary and Rogue Wave Solutions to the Conformable Time Fractional Modified Kawahara Equation in Mathematical Physics

Utilizing of illustrative scheming programming, the study inspects the careful voyaging wave engagements from the nonlinear time fractional modified Kawahara equation (mKE) by employing the advanced exp − φ ξ -expansion policy in terms of trigonometric, hyperbolic, and rational function through some treasured fractional order derivative and free parameters. The undercurrents of nonlinear wave answer are scrutinized and confirmed by MATLAB in 3D and 2D plots, and density plot by specific values of the convoluted parameters is designed. Our preferred advanced exp − φ ξ -expansion technique which is parallel to ( G ′ / G ) expansion technique is trustworthy dealing for searching significant nonlinear waves that progress a modification of dynamic depictions that ascend in mathematical physics and engineering grounds.

Bio-heat transfer analysis based on fractional derivative and memory-dependent derivative heat conduction models

The fractional derivative (FD) and memory-dependent derivative (MDD) are efficient methods to predict the transient thermal responses. In the present paper, for the first time, the FD and MDD are introduced into classical Pennes bio-heat conduction equation with single phase-lag and the corresponding bio-heat transfer models have been formulated with the aids of thermal energy conservation equation. The Laplace transform and numerical inverse transform method are adopted to get access to thermal responses due to temperature shock. Reductions of the present models to classical Pennes and single phase-lag bio-heat transfer models are presented and comparison between different FD and MDD in transient bio-heat transfer is carried out firstly. An approximate formulation of propagation speed of thermal signal in FD models is given according to results firstly. Influences of fractional derivative parameter and tempered parameter in FD, delay time in MDD, phase-lag time and rate of blood perfusion on temperature variation and distribution are carried out numerically. Both fractional derivative and memory-dependent derivative have the ability to predict thermal response between diffusion and wave behavior and CF fractional derivative and MDD with K3 may be suitable methods.

The Space-Time Coupled Fractional Cattaneo-Friedrich Maxwell Model with Caputo Derivatives.

In the current article, we have thoroughly investigated the collective impact of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid (Cattaneo–Friedrich Maxwell-CFM model) over a permeable surface embedded in a porous medium. Here we have utilized the Caputo time-fractional derivatives and mechanical laws (generalized shear stress constitutive equation and generalized Fourier’s and Fick’s laws) are being used to fractionalize the presented model. The effects of radiative heat flux, Ohmic dissipation, and internal absorption are presented through generalized Fourier’s law while Fick’s law or mass transfer equation offers the effects of first order chemically reactive species. The finite element method and finite difference method are being utilized to numerically solve the nonlinear coupled differential equations. It is established, through compression of numerical and analytical solutions, that the presented model is convergent. Further, error analysis of the subject model is also carried out. Moreover, for better illustration of results, we have also offered a graphical and tabular presentation of impacts of the parameters of interest on velocity, temperature, concentration profile, local skin friction coefficient, and heat and mass transfer. It is evident from the obtained results that velocity near and away from the surface increases with the enhancement of fractional derivative parameter whereas an opposite trend is observed in the case of temperature. Furthermore, it is noticed that temperature shows a decreasing behavior for the value {Lambda }_{theta }<2 and {Lambda }_{phi }<2, on the other hand entirely opposite trend is witnessed for {Lambda }_{theta }ge 3 and {Lambda }_{phi }ge 3. From an engineering perspective, we have acquired comprehensive outcomes such that the heat transfer offers an increasing trend in the case of TR and thermal fractional parameter beta_{1} . Additionally, the chemical reaction parameter and Sc significantly contribute towards the mass transfer rate. Since, in literature, one cannot refer to such results with non-integer Caputo fractional derivatives thus the results obtained through the current assessment hold significance for future research avenues. Moreover, the numerical inferences of the subject study may contribute to an advanced thermal processing method in the food industry to swiftly increase the temperature for cooking or sterilization drives.

The influence of awareness campaigns on the spread of an infectious disease: a qualitative analysis of a fractional epidemic model.

Mass-media coverage is one of the most widely used government strategies on influencing public opinion in times of crisis. Awareness campaigns are highly influential tools to expand healthy behavior practices among individuals during epidemics and pandemics. Mathematical modeling has become an important tool in analyzing the effects of media awareness on the spread of infectious diseases. In this paper, a fractional-order epidemic model incorporating media coverage is presented and analyzed. The problem is formulated using susceptible, infectious and recovered compartmental model. A long-term memory effect modeled by a Caputo fractional derivative is included in each compartment to describe the evolution related to the individuals’ experiences. The well-posedness of the model is investigated in terms of global existence, positivity, and boundedness of solutions. Moreover, the disease-free equilibrium and the endemic equilibrium points are given alongside their local stabilities. By constructing suitable Lyapunov functions, the global stability of the disease-free and endemic equilibria is proven according to the basic reproduction number R_0. Finally, numerical simulations are performed to support our analytical findings. It was found out that the long-term memory has no effect on the stability of the equilibrium points. However, for increased values of the fractional derivative order parameter, each solution reaches its equilibrium state more rapidly. Furthermore, it was observed that an increase of the media awareness parameter, decreases the magnitude of infected individuals, and consequently, the height of the epidemic peak.

Mathematical model of the polymer concrete by fractional calculus with respect to a spatial variable

The article presents a solution of the vibration string equation containing the fractional derivative concerning the spatial variable, in which the fractional derivative is defined by Caputo. This model is utilized for characterizing the oscillation mechanical process in a viscoelastic medium. The theoretical solution is presented, and the fractional derivative parameter is determined. We compared the theoretical solution and the experimental data for polymer concrete samples. A sample of the problem of structural mechanics has been considered. This sample allows the demonstration of some advantages of the application of the suggested approach to solve the fractional vibration equation.

Simulating the joint impact of temporal and spatial memory indices via a novel analytical scheme

The prime concern of this study is to simulate the joint effect for the presence of two fractional derivative parameters (memory indices) by providing a novel analytical solution scheme for the fractional initial value problems. Our goal has been fulfilled by extending the residual power series method into the two-dimensional time and space, with time and space endowed with fractional derivative orders $$\alpha $$ and $$\gamma $$ , respectively (simply denoted by fractional $$(\alpha ,\gamma )$$ -space), by virtue of a new $$(\alpha ,\gamma )$$ -fractional power series representation ( $$(\alpha ,\gamma )$$ -FPS). The necessary theoretical framework for the convergence and the error bound is also provided to enrich our analytical study. Among other main findings, it is deserved to mention that the fractional derivative parameters act like the homotopy parameters, in a topological sense, to generate a rapidly convergent series solution for the classical integer version of the problem under consideration, which promotes the idea that these parameters describe a remnant memory. The efficiency of the proposed approach is assessed by projecting the obtained solutions of several well-known (non)linear problems into lower-dimensional fractal space and/or into integer space and then comparing them with the corresponding results of the literature. Overall, the method shows a wide versatility and adequacy in dealing with such hybrid problems.

Finite volume method for mixed convection boundary layer flow of viscoelastic fluid with spatial fractional derivatives over a flat plate

A new fractional finite volume method is developed for the mixed convection boundary layer flow and heat transfer of viscoelastic fluid over a flat plate. The spatial fractional derivative of the Riemann–Liouville type is employed in the constitutive relation and modified Fourier’s law respectively. Nonlinear and coupled boundary layer governing equations are formulated with non-uniform boundary conditions. The discretized scheme combined with the shifted Grunwald–Letnikov formula is proved to be conditionally stable, further the convergence and accuracy of the numerical solutions are presented. Results demonstrate that space fractional derivative parameters have strong effects on the velocity and temperature distributions. Moreover, the viscoelastic fluid with spatial fractional derivative performs stress relaxation with distance from the intersections of velocity profiles.

Traveling wave with beta derivative spatial-temporal evolution for describing the nonlinear directional couplers with metamaterials via two distinct methods

This work is reported the analytical solutions for describing the nonlinear directional couplers with metamaterials by including spatial–temporal fractional beta derivative evolution. The auxiliary ordinary differential equation method and the generalized Riccati method with the properties of beta derivative are implemented to secure such solutions. The solutions are obtained in the new forms by involving of some useful mathematical functions. The constraint conditions among the traveling wave parameters are evaluated. Some of the obtained solutions are presented graphically to illustrate the effectiveness of beta derivative parameter and mathematical techniques. It is investigated that the amplitudes of soliton are increased with the increase of fractional beta derivative parameter. The obtained results would be very useful to examine and understand the physical issues in nonlinear optics, especially in twin-core couplers with optical metamaterials.

Numerical modeling of fractional viscoelastic non-Newtonian fluids over a backward facing step – Buoyancy driven flow and heat transfer

The unsteady flow and natural convection heat transfer of fractional Maxwell viscoelastic fluid over sudden expansion geometry such as vertical backward facing step is studied. The time–space fractional derivatives are calculated based on Caputo fractional derivative. The nonlinear governing equations are solved using finite difference method mixed with a L-1 algorithm. The step, upstream and downstream walls were set at a constant temperature. The range of step length is 0≤S≤0.5 and dimensionless temperature is 0.25≤θw≤1. The effects of velocity and temperature fractional derivatives (αand β) and various physical parameters, such as step length, dimensionless temperature, on Nusselt number, friction coefficient, profile of velocity and temperature, are investigated numerically. The results reveal that the fractional derivative parameters, the variation of wall temperature and step length have significant effects on the rate of heat transfer, the friction coefficient, the velocity and temperature of backward facing step. The average of Nusselt number variation increases with increasing of β and decreases of α parameters. The results show that the velocity and thermal boundary layer of the non-Newtonian fluid (ordinary Maxwell) is higher than Newtonian fluid. Also, the Nusselt number for the ordinary Maxwell fluid flow is greater than the Newtonian fluid flow. With the increase of the step length of backward facing step, the average Nusselt number, friction coefficient, velocity boundary layer, and temperature boundary layer increases. The obtained results can improve significantly the performance of the systems design that operate with the non-Newtonian fluids.

Unsteady stagnation-point flow and heat transfer of fractional Maxwell fluid towards a time dependent stretching plate with generalized Fourier’s law

PurposeThe purpose of this study is to investigate the unsteady stagnation-point flow and heat transfer of fractional Maxwell fluid towards a time power-law-dependent stretching plate. Based on the characteristics of pressure in the boundary layer, the momentum equation with the fractional Maxwell model is firstly formulated to analyze unsteady stagnation-point flow. Furthermore, generalized Fourier’s law is considered in the energy equation and boundary condition of convective heat transfer.Design/methodology/approachThe nonlinear fractional differential equations are solved by the newly developed finite difference scheme combined with L1-algorithm, whose convergence is verified by constructing a numerical example.FindingsSome interesting results can be revealed. The larger fractional derivative parameter of velocity promotes the flow, while the smaller fractional derivative parameter of temperature accelerates the heat transfer. The temperature boundary layer is thicker than the velocity boundary layer, and the velocity enlarges as the stagnation parameter raises. This is because when Prandtl number < 1, the capacity of heat diffusion is greater than that of momentum diffusion. It is to be observed that all the temperature profiles first enhance a little and then reduce rapidly, which indicates the thermal retardation of Maxwell fluid.Originality/valueThe unsteady stagnation-point flow model of Maxwell fluid is extended from integral derivative to fractional derivative, which has more flexibility to describe viscoelastic fluid’s complex dynamic process and provide a theoretical basis for industrial processing.

Axisymmetric convection flow of fractional Maxwell fluid past a vertical cylinder with velocity slip and temperature jump

A novel finite volume method is developed to investigate the axisymmetric convection flow and heat transfer of fractional viscoelastic fluid past a vertical cylinder. Fractional cylindrical governing equations are formulated by fractional Maxwell model and generalized Fourier's law. The velocity slip and temperature jump boundary conditions are considered across the fluid-solid interface. Numerical results are validated by exact solutions of special case with source terms. The effects of fractional derivative parameter and boundary condition parameters on flow and heat transfer characteristics are discussed. The viscoelastic fluid performs evident shear thickening property in the fractional Maxwell constitutive relation. Moreover, the boundary condition parameters have remarkable influence on velocity and temperature distributions.

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ̅-Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ̅-Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ̅-Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics.