Couette flow of viscoelastic fluid with constitutive relation involving general Caputo-type fractional derivative

In this paper, the general Caputo-type fractional differential operator introduced by Pr. Anatoly N. Kochubei is applied to the linear theory of viscoelasticity. Firstly, using the general Caputo-type derivative, a generalized linear viscoelastic constitutive equation is proposed for the first time. Secondly, the momentum equation for the plane Couette flow of viscoelastic fluid with the constitutive relation is given as an integrodifferential equation and the analytical solution of the equation is established by employing the separation of variables method. Lastly, for special cases of the general constitutive relation, the analytical solutions are obtained in terms of the Mittag-Leffler functions.     

Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives

The link between the treatments of constrained systems with fractional derivatives by using both Hamiltonian and Lagrangian formulations is studied. It is shown that both treatments for systems with linear velocities are equivalent.     

Mathematical modeling of fractional reaction-diffusion systems with different order time derivatives

The linear stability analysis is studied for a two-component fractional reaction-diffusion system with different derivative indices. Two different cases are considered: when the activator index is larger than the inhibitor one and when the inhibitor variable index is larger than the activator one. The general analysis is confirmed by computer simulation of the system with cubic nonlinearity. It is shown that systems with a higher activator variable index lead to a much more complicated space-time dynamics.     

Asymptotic behaviour of a closed‐loop thermosyphon with linear friction and viscoelastic binary fluid

Viscoelastic fluids represent a major challenge both from an engineering and from a mathematical point of view. Recently, we have shown that viscoelasticity induces chaos in closed‐loop thermosyphons even when we consider binary fluids, this is, when we consider a solute in the fluid, as water and antifreezes, for example. In this work, we consider a linear friction law, and we show that in this case with the addition of a solute to the fluid we can prove, under some conditions, chaotic asymptotic behavior for suitable geometry of the circuit and heat flux or ambient temperature functions.     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

Solution operators of three time variables for fractional linear problems

It is well known that the time fractional equation where is the fractional time derivative in the sense of Caputo of u does not generate a dynamical system in the standard sense. In this paper, we study the algebraic properties of the solution operator T(t,s,τ) for that equation with u(s) = v. We apply this theory to linear time fractional PDEs with constant coefficients. These equations are solved by the Fourier multiplier techniques. It appears that their solution exhibits some singularity, which leads us to introduce a new kind of solution for abstract time fractional problems. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

利用Adomain分解法求时间分数阶薛定谔方程的近似解

非线性薛定谔方程是现代科学中非常普遍的非线性模型之一。通过 Adomain分解,得到了（2＋1）维和（3＋1）维非零势阱时间分数阶薛定谔方程的近似解。利用Adomain 分解不用像相关文献中那样将解函数的实部和虚部分别去求解,从而简化了求解过程。     

Calculus of variations with fractional derivatives and fractional integrals

We prove the Euler–Lagrange fractional equations and the sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann–Liouville.     

3-D time-dependent analysis of multilayered cross-anisotropic saturated soils based on the fractional viscoelastic model

The fractional Merchant viscoelastic model is introduced to simulate the viscoelasticity of soil skeleton in this study. According to the elastic-viscoelastic correspondence principle, elastic parameters including shear modulus G_v, horizontal elastic modulus E_h and vertical elastic modulus E_v are replaced by the reciprocal of the flexibility coefficient of viscoelastic media in the Laplace transformed domain. Then, based on the precise integration solutions of multilayered cross-anisotropic elastic saturated soils, 3-D solutions of viscoelastic saturated soils are derived. The final solutions in the physical domain are obtained by the Laplace numerical inversion. The correctness of theories and programs is verified by comparing the numerical results with existing references. Sensitivity analyses are conducted to investigate the effects of viscoelastic parameters, cross-anisotropic parameters and stratification of soils on time-dependent displacement and excess pore water pressure.     

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$\alpha =1$. The influence of fractional coefficient on the decay of vortex velocity is also analyzed by graphical illustrations.     

基于分数阶混沌时间序列的图像加密算法

基于分数阶logistic映射提出了洗牌加密方法.通过离散分数阶微积分得到分数阶序列并把它作为密钥.利用位异或算子,提出了一种新的图像加密算法.对该算法的密钥空间、密钥敏感性和统计特性进行相应的仿真分析.结果表明,该算法可以达到较好的加解密效果,具有很高的安全性,可以满足图像加密安全性的要求.     

A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives

In this study a new framework for solving three-dimensional (3D) time fractional diffusion equation with variable-order derivatives is presented. Firstly, a θ-weighted finite difference scheme with second-order accuracy is introduced to perform temporal discretization. Then a meshless generalized finite difference (GFD) scheme is employed for the solutions of remaining problems in the space domain. The proposed scheme is truly meshless and can be used to solve problems defined on an arbitrary domain in three dimensions. Preliminary numerical examples illustrate that the new method proposed here is accurate and efficient for time fractional diffusion equation in three dimensions, particularly when high accuracy is desired.     

A viscoelastic constitutive modeling of rubber-like materials with the Payne effect

This paper aims to present a viscoelastic constitutive model of rubber-like materials, which can capture the Payne effect under dynamic cyclic loadings. The Payne effect is induced by a damage process of bond rupture inside the rubber-like materials, which leads to the storage and loss moduli changing with the dynamic strain amplitude. A viscoelastic constitutive relation is established based on the nonequilibrium thermodynamics for the rubber-like materials by constructing the Helmholtz free energy as the superposition of a hyperelastic model and a convolution viscous model. The neo-Hookean hyperelastic model and the convolution viscous model in terms of the Prony series are then employed in a modification that the material parameters concerned are treated as internal variables and can be identified through a simple but effective approach. At last, the Payne effect is effectively predicted in a good agreement with experimental results.     

Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel

The paper presents the transportation of viscoelastic fluid with fractional Maxwell model by peristalsis through a channel under long wavelength and low Reynolds number approximations. The propagation of wall of channel is taken as sinusoidal wave propagation (contraction and relaxation). Homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are used to obtain the analytical approximate solutions of the problem. The expressions of axial velocity, volume flow rate and pressure gradient are obtained. The effects of fractional parameters (α), relaxation time (λ₁) and amplitude (?) on the pressure difference and friction force across one wavelength are calculated numerically for different particular cases and depicted through graphs.     

Exponential stability of a linear viscoelastic bar with thermal memory

In this paper we study a one-dimensional evolution problem arising in the theory of linear thermoviscoelasticity with hereditary heat conduction. Depending on the istantaneous conductivity K₀, both Coleman-Gurtin (K₀>0) and Gurtin-Pipkin (K₀=0) heat flow theories are involved. In any case, the exponential stability of the corresponding semigroup is proved for a class of memory functions including weakly singular kernels. In order to achieve the exponential decay of the energy, we assume that mechanical and thermal memory kernels decay exponentially for large time. Entrata in Redazione il 23 luglio 1998.