Sequential Fractional Derivatives Research Articles

Helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium

This paper presents an analysis for helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, where the motion is due to the longitudinal time-dependent shear stress and the oscillating velocity in boundary. The exact solutions are established by using the sequential fractional derivatives Laplace transform coupled with finite Hankel transforms in terms of generalized G function. Moreover, the effects of various parameters (relaxation time, fractional parameter, permeability and porosity) on the flow and heat transfer are analyzed in detail by graphical illustrations.

Coupled systems of fractional equations related to sound propagation: Analysis and discussion

In this paper, we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive, and also memoryless processes such as classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework, we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations.

Couette Flow of a Generalized Oldroyd-B Fluid due to Constantly Accelerated Shear Stress Coupled with the Pressure Gradient

This paper presented an analysis for the couette flow of a generalized Oldroyd-B fluid within an infinite cylinder subject to a time-dependent shear stress with the influence of the internal constantly decelerated pressure gradient. The exact solutions are established by means of the combine of the sequential fractional derivatives Laplace transform and finite Hankel transform and presented by integral and series form in terms of the Mittag-Leffler function. Moreover, the effects of various parameters are analyzed in detail by graphical illustrations.

Impulsive periodic boundary value problems for fractional differential equation involving Riemann–Liouville sequential fractional derivative

In this paper, we investigate the existence of solutions of the periodic boundary value problem for nonlinear impulsive fractional differential equation involving Riemann–Liouville sequential fractional derivative by using monotone iterative method. An example is presented to illustrate our main result.

Exact solutions for the unsteady rotating flows of a generalized Maxwell fluid with oscillating pressure gradient between coaxial cylinders

This paper deals with the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative model between two infinite straight circular cylinders, where the flow is due to an infinite straight circular cylinder rotating and oscillating pressure gradient. The velocity field and the adequate shear stress are determined by means of the combine of the sequential fractional derivatives Laplace transform and finite Hankel transform. The exact solutions are presented by integral and series form in terms of the generalized G and Mittag-Leffler functions. The similar solutions can be easily obtained for ordinary Maxwell and Newtonian fluids as limiting cases. Finally, the influence of the relaxation time and the fractional parameter on the fluid dynamic characteristics, as well as a comparison between models, is shown by graphical illustrations.

Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative

This paper presents an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate. Where the no-slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. Closed form solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete Laplace transform of the sequential fractional derivatives. The solutions for no-slip condition and no magnetic field can be derived as the special cases. Furthermore, the effects of various parameters on the corresponding flow and shear stress characteristics are analyzed and discussed in detail.

Slip Effects on Fractional Viscoelastic Fluids

Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no‐slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions pΨq, by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no‐slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is θ → 0. Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no‐slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.

Periodic boundary value problems for Riemann-Liouville sequential fractional differential equations

In this paper, we shall discuss the properties of the well-known Mittag–Leffler function, and consider the existence of solution of the periodic boundary value problem for a fractional differential equation involving a Riemann–Liouville sequential fractional derivative by means of the method of upper and lower solutions and Schauder fixed point theorem.

Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag–Leffler function, and consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving Riemann–Liouville sequential fractional derivative by using monotone iterative method.

Unsteady Flow of Non-Newtonian Visco-Elastic Fluid in Dual-Porosity Media with The Fractional Derivative

The fractional order derivative was introduced to the seepage flow research to establish the relaxation models of non-Newtonian viscoelastic fluids in dual porous media. The flow characteristics of non-Newtonian viscoelastic fluids through a dual porous medium were studied by using the Hankel transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions were obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation were also resulted. The pressure transient behavior of non-Newtonian viscoelastic fluids flow through an infinite dual porous media was studied by using Stehfest's inversion method of the numerical Laplace transform. It shows that the characteristics of the fluid flow are appreciably affected by the order of the fractional derivative.

Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels

Transient electro-osmotic flow of viscoelastic fluids in rectangular micro-channels is investigated. The general twofold series solution for the velocity distribution of electro-osmotic flow of viscoelastic fluids with generalized fractional Oldroyd-B constitutive model is obtained by using finite Fourier and Laplace transforms. Under three limiting cases, the generalized Oldroyd-B model simplifies to Newtonian model, fractional Maxwell model and generalized second grade model, where all the explicit exact solutions for velocity distribution are found through the discrete Laplace transform of the sequential fractional derivatives. These exact solutions may be able to predict the flow behavior of viscoelastic biological fluids in BioMEMS and Lab-on-a-chip devices and thus could benefit the design of these devices.

Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model

This paper is concerned with the exact analytic solutions for the velocity field and the associated tangential stress corresponding to a potential vortex for a fractional Maxwell fluid. The fractional calculus approach is taken into account in the constitutive relationship of a non-Newtonian fluid model. Exact analytic solutions are obtained by using the Hankel transform and the discrete Laplace transform of sequential fractional derivatives. The solutions for a Maxwell fluid appear as the limiting cases of our general solutions by setting α = 1 . The influence of fractional coefficient on the decay of vortex velocity is also analyzed by graphical illustrations.

Linear fractional differential equations with variable coefficients

This work is devoted to the study of solutions around an α -singular point x 0 ∈ [ a , b ] for linear fractional differential equations of the form [ L n α ( y ) ] ( x ) = g ( x , α ) , where [ L n α ( y ) ] ( x ) = y ( n α ) ( x ) + ∑ k = 0 n − 1 a k ( x ) y ( k α ) ( x ) with α ∈ ( 0 , 1 ] . Here n ∈ N , the real functions g ( x ) and a k ( x ) ( k = 0 , 1 , … , n − 1 ) are defined on the interval [ a , b ] , and y ( n α ) ( x ) represents sequential fractional derivatives of order k α of the function y ( x ) . This study is, in some sense, a generalization of the classical Frobenius method and it has applications, for example, in obtaining generalized special functions. These new special functions permit us to obtain the explicit solution of some fractional modeling of the dynamics of many anomalous phenomena, which until now could only be solved by the application of numerical methods. 1 1 This research was funded, in part, by MEC (MTM2004-00327 and MTM2007-60246) and by ULL.

Flow of a generalized second-grade fluid between two side walls perpendicular to a plate with a fractional derivative model

In this paper, the exact analytic solutions are obtained for flow of a generalized second-grade fluid between two side walls perpendicular to a plate. The fractional calculus approach in the constitutive relationship model of a non-Newtonian fluid is used. The exact analytic solutions are constructed by using Fourier sine transforms and the discrete Laplace transform of the sequential fractional derivatives. The solutions for a second-grade fluid appear as the limiting cases of the presented solutions on taking β = 1 . Moreover, in the absence of side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate.

Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model

The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.

α-Analytic solutions of some linear fractional differential equations with variable coefficients

This paper investigates the solutions, around an ordinary point x 0 ∈ [ a, b] for fractional linear differential equations of the form: [ L n α ( y ) ] ( x ) = g ( x , α ) , where [ L n α ( y ) ] ( x ) = y ( n α ) ( x ) + ∑ k = 0 n - 1 a k ( x ) y ( k α ) ( x ) with α ∈ (0, 1]. Here n ∈ N, the real functions g( x) and a k ( x) ( k = 0, 1, … , n − 1) are defined on the interval [ a, b], and y ( nα) ( x) represents sequential fractional derivatives of order kα of the function y( x). This study is an extension of the corresponding works by Al-Bassam.

The flow analysis of fiuids in fractal reservoir with the fractional derivative

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the flow models of fluids in fractal reservoirs with the fractional derivative. The flow characteristics of fluids through a fractal reservoir with the fractional order derivative are studied by using the finite integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of fluids flow through an infinite fractal reservoir is studied by using the Stehfest's inversion method of the numerical Laplace transform. It shows that the order of the fractional derivative affect the whole pressure behavior, particularly, the effect of pressure behavior of the early-time stage is larger The new type flow model of fluid in fractal reservoir with fractional derivative is provided a new mathematical model for studying the seepage mechanics of fluid in fractal porous media.

Decay of vortex velocity and diffusion of temperature for fractional viscoelastic fluid through a porous medium

In this paper, we investigate a circular motion of generalized second grade fluid through a porous medium. The velocity and temperature fields of the flow through a porous medium of a generalized second grade fluid with fractional derivative model are described by fractional partial differential equations. Exact analytical solutions of these differential equations are obtained in terms of the Fox's H-function by using the discrete Laplace transform of the sequential fractional derivatives and Hankel transform.

Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid was introduced. The velocity and temperature fields of the vortex flow of a generalized second fluid with fractional derivative model were described by fractional partial differential equations. Exact analytical solutions of these differential equations were obtained by using the discrete Laplace transform of the sequential fractional derivatives and generalized Mittag-Leffler function. The influence of fractional coefficient on the decay of vortex velocity and diffusion of temperature was also analyzed.

Analysis of the flow of non-Newtonian viscoelastic fluids in fractal reservoir with the fractional derivative

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the relaxation models of non-Newtonian viscoelastic fluids with the fractional derivative in fractal reservoirs. A new type integral transform is introduced, and the flow characteristics of non-Newtonian viscoelastic fluids with the fractional order derivative through a fractal reservoir are studied by using the integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of non-Newtonian viscoelastic fluids flow through an infinite fractal reservoir is studied by using the Stehfest’s inversion method of the numerical Laplace transform. It is shown that the clearer the viscoelastic characteristics of the fluid, the more the fluid is sensitive to the order of the fractional derivative. The new type integral transform provides a new analytical tool for studying the seepage mechanics of fluid in fractal porous media.