Wave equation for generalized Zener model containing complex order fractional derivatives

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.     

Efficient solution of a vibration equation involving fractional derivatives

Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980s, and they grow more and more popular nowadays. However, their efficient numerical calculation is non-trivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.     

Relaxation and retardation functions of the Maxwell model with fractional derivatives

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type.The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles.This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value.In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.     

An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation

In this paper, an optimization method based on a new class of basis functions, namely generalized polynomials (GPs), is proposed for nonlinear variable-order time fractional diffusion-wave equation. Variable-order time fractional derivative is expressed in the Caputo sense. In the proposed method, solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. In this way, some new operational matrices of the ordinary and fractional derivatives are derived for these basis functions. The residual function and its 2-norm are employed for converting the problem under study to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and effectiveness of the method are demonstrated by solving some numerical examples. The results demonstrate that the proposed method is a powerful algorithm with good accuracy for solving such kind of problems.     

Darboux problem for a differential equation with fractional derivative

We find conditions for the unique solvability of the problem u _xy (x, y) = f(x, y, u(x, y), (D ₀ ^r u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D ₀ ^r u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r ₁, r ₂), 0 < r ₁, r ₂ < 1, in the class of functions that have the continuous derivatives u _xy (x, y) and (D ₀ ^r u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.     

Analytical approach to Boussinesq equation with space‐ and time‐fractional derivatives

In this paper, the homotopy perturbation method (HPM) is developed to obtain approximate analytical solutions of a fractional Boussinesq equation with initial condition. The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made, the comparisons show that the HPM is very effective and convenient and overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to space‐ and time‐fractional equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.Dedicated to professor Hari M. Srivastava on the occasion of his 65th birthday     

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional fractional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seepage flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.     

Study on the constitutive equation with fractional derivative for the viscoelastic fluids – Modified Jeffreys model and its application

Based on the classic linear viscoelastic Jeffreys model, a modified Jeffreys model is suggested. The corresponding five-parameter equation with fractional derivatives of different orders of the stress and rate of strain is stated and the characteristic material functions of the linear viscoelasticity theory, such as the dynamic moduli, are derived. The comparison between the measured dynamic moduli of Sesbania gel and xanthan gum and the theoretical predictions of the proposed phenomenological model shows an excellent agreement. Received: 26 August 1997 Accepted: 26 May 1998     

Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced. Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus. The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates. In addition, the solutions of the shear stresses at the plates are also determined. The project supported by the National Natural Science Foundation of China (10372007, 10002003) and CNPC Innovation Fund     

Period bounds for generalized Rayleigh equation

Simple upper and lower bounds are obtained for the least period T of any non-constant periodic solution x(t) of the differential equation x″ − F(x') + g(x) = 0.     

Multi-symplectic method for generalized Boussinesq equation

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.     

An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index

In this paper, we consider the main problem of variational calculus when the derivatives are Riemann?CLiouville-type fractional with incommensurate orders in general. As the most general form of the performance index, we consider a fractional integral form for the functional that is to be extremized. In the light of fractional calculus and fractional integration by parts, we express a generalized problem of the calculus of variations, in which the classical problem is a special case. Considering five cases of the problem (fixed, free, and dependent final time and states), we derive a necessary condition which is an extended version of the classical Euler?CLagrange equation. As another important result, we derive the necessary conditions for an optimization problem with piecewise smooth extremals where the fractional derivatives are not necessarily continuous. The latter result is valid only for the integer order for performance index. Finally, we provide some examples to clarify the effectiveness of the proposed theorems.     

Chaos transition of the generalized fractional duffing oscillator with a generalized time delayed position feedback

Nonlinear Dynamics - This article is concerned with the study of chaos transition of the duffing oscillator in the presence of fractional-order derivative and third-order polynomial delayed...     

Constitutive equations for polymer fluids based on the concept of configuration-dependent molecular mobility: a generalized mean-configuration model

After a brief outline of the concept of configuration-dependent molecular mobility for the particular case of the one-mode mean-configuration theory, a generalized model is introduced in which the dependence of the mobility tensor on the configuration tensor is given by a relaxation-type functional. This model is analysed for steady and transient extensional and shear flows. In extensional flow it predicts a maximum in the steady-state uniaxial viscosity curves and stress overshoot in the stressing curves, and in shear flow it predicts even larger stress overshoot in the stressing curves. This model bridges the gap between the current molecular models and the most elaborate network models. In an appendix it is shown that for the relaxation-type dependence of the mobility it is only by using the upper Oldroyd derivative that physically acceptable results are predicted.     

A generalized model of elastic foundation based on long-range interactions: Integral and fractional model

The common models of elastic foundations are provided by supposing that they are composed by elastic columns with some interactions between them, such as contact forces that yield a differential equation involving gradients of the displacement field. In this paper, a new model of elastic foundation is proposed introducing into the constitutive equation of the foundation body forces depending on the relative vertical displacements and on a distance-decaying function ruling the amount of interactions. Different choices of the distance-decaying function correspond to different kind of interactions and foundation behavior. The use of an exponential distance-decaying function yields an integro-differential model while a fractional power-law decay of the distance-decaying function yields a fractional model of elastic foundation ruled by a fractional differential equation. It is shown that in the case of exponential-decaying function the integral equation represents a model in which all the gradients of the displacement function appear, while the fractional model is an intermediate model between integral and gradient approaches. A fully equivalent discrete point-spring model of long-range interactions that may be used for the numerical solution of both integral and fractional differential equation is also introduced. Some Green’s functions of the proposed model have been included in the paper and several numerical results have been also reported to highlight the effects of long-range forces and the governing parameters of the linear elastic foundation proposed.     

Constitutive equation of co-rotational derivative type for anisotropic-viscoelastic fluid

A constitutive equation theory of Oldroyd fluid B type, i.e. the co-rotational derivative type, is developed for the anisotropic-viscoelastic fluid of liquid crystalline (LC) polymer. Analyzing the influence of the orientational motion on the material behavior and neglecting the influence, the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation, retardation times and anisotropic viscosities are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid. Using the equation for the shear flow of LC polymer fluid, the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al. For the fiber spinning flow of the fluid, the analytical expression of the extensional viscosity is given. The project supported by the National Natural Science Foundation of China (19832050 and 10372100)     

Exponential attractors for a generalized ginzburg-landau equation

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Couette problem for the generalized Krook equation stress-peak effect

The Couette problem is the simplest problem of steady shear flow of rarefied gas in a region bounded by solid surfaces. This problem has been examined in the linear formulation by many authors, using either the linearized Krook equation or the moment methods (see [1]). It has recently been solved by the Monte Carlo method [2].The nonlinear problem of Couette flow with heat transfer for the Krook equation has been solved by reducing the problem to a system of integral equations [3] over a wide range of flat-plate velocities and temperature ratios and by the discrete-velocity method [4] for moderate plate velocities. In this article we solve the same problem for the generalized Krook equation [5] which approximates the Boltzmann equation for a pseudo-Maxwellian gas in accordance with the method suggested by the author [6, 7]. The generalized Krook equation was solved numerically by a modified discrete-velocity method which has been used by the author previously to solve the problem of shock wave structure [8].The primary case examined is that of pseudo-Maxwellian molecules, in which the viscosity is proportional to the temperature. The computations were made for Prandtl numbers of 1 and 2/3 over a wide range of Mach and Knudsen numbers as well as flat-plate temperature ratios. As we would expect, the Prandtl number effect is greatest for small Knudsen numbers. The flow velocity profiles are not very sensitive to variation of the Prandtl number (at least for pseudo-Maxwellian molecules).However, the most interesting result of the study is independent of the Prandtl number. Specifically, it was found that for any sufficiently high flat-plate velocities the friction stress, referred to the corresponding free molecular value, does not change monotonically with variation of the Knudsen number; instead, there is a peak. As far as the author is aware, this nonlinear effect has not been discussed previously in the literature (including articles [3, 4]).