Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions

The generalized fractional element networks are presented in this paper. In order to extend the structure of the model solutions to the generalized function space and make it contain more physical meanings, the restriction on the parameters of the fractional element proposed by Schiessel et al. is eliminated and a "compatibility equation" is added. The discretization method for solving the inverse Laplace transform is used and developed. The generalized solutions of the model equations are given. At the same time the generalized fractional element network--Zener and Poyinting-Thomson models are discussed in detail. It is shown that all the results obtained previously about the models of single parameter with fractional order and the classical models with integer order can be contained as the special cases of the results of this paper.     

分数阶Oldroyd-B黏弹性Poiseuille流的Laplace数值反演分析

采用Laplace数值反演的Stehfest算法研究了分数阶Oldroyd-B粘弹性流体在两平板间非定常的Poiseuille流动问题.首先,通过数值解与近似解析解的比较验证了Stehfest算法的有效性.其次,运用Stehfest算法对平板Poiseuille流动进行了研究,揭示了分数阶黏弹性平板流的速度过冲和应力过冲现象,指出这些现象对分数导数的阶数存在明显的依赖性.同时,数值结果表明,整数阶本构方程仅仅是分数阶本构方程的特例,分数阶本构方程较整数阶本构方程具有更广泛的适用性。     

Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress is also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.     

Maxwell fluid flow between vertical plates with damped shear and thermal flux: Free convection

In this article, we studied free convection flow of Maxwell fluid between two parallel plates a distance d apart from each other. The Caputo time-fractional derivative is used in model and the model is fractionalized through mechanical laws (generalized shear stress constitutive equation and generalized Fourier's law). Closed form solutions are found by means of Laplace and sine-Fourier transforms which are suitable for our boundary conditions. The solutions are expressed in the form of Mittag–Leffler function and generalized G–function of Lorenzo and Hartley. The viscous fractional and ordinary Maxwell and fractional model are presented as special cases. The effects of fractional and physical parameters are graphically illustrated.     

Generalized classical mechanics

Generalized classical mechanics has been introduced and developed as a classical counterpart of the fractional quantum mechanics. The Lagrangian of generalized classical mechanics has been introduced, and equation of motion has been obtained. Lagrange, Hamilton and Hamilton-Jacobi frameworks have been implemented. Oscillator model has been launched and solved in 1D case. A new equation for the period of oscillations of generalized classical oscillator has been found. The interplay between the energy dependency of the period of classical oscillations and the non-equidistant distribution of the energy levels for fractional quantum oscillator has been discussed. We discuss as well, the relationships between new equations of generalized classical mechanics and the well-known fundamental equations of classical mechanics.     

Quantum Maps with Memory from Generalized Lindblad Equation

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.     

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.     

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

In both the oil reservoir engineering and seepage flow mechanics, heavy oil with relaxation property shows non-Newtonian rheological characteristics. The relationship between shear rate g& and shear stress t is nonlinear. Because of the relaxation phenomena of heavy oil flow in porous media, the equation of motion can be written as[1] 2,rrvpqkppqtrrtll秏骣+=-+琪抖桫 (1) where lv and lp are velocity relaxation and pressure retardation times. For most porous media, the above motion equation (1)…     

General Non-Markovian Quantum Dynamics

A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.     

A Tutorial Review on Fractal Spacetime and Fractional Calculus

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz’s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie’s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.     

A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.     

New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves

This treatise analyzes the coupled nonlinear system of the model for the ion sound and Langmuir waves.The modified(G'/G)-expansion procedure is utilized to raise new closed-form wave solutions.Those solutions are investigated through hyperbolic,trigonometric and rational function.The graphical design makes the dynamics of the equations noticeable.It provides the mathematical foundation in diverse sectors of underwater acoustics,electromagnetic wave propagation,design of specific optoelectronic devices and physics quantum mechanics.Herein,we concluded that the studied approach is advanced,meaningful and significant in implementing many solutions of several nonlinear partial differential equations occurring in applied sciences.     

Multi-layer flows of immiscible fractional Maxwell fluids with generalized thermal flux

Unsteady laminar flows and heat transfer of n-immiscible fractional Maxwell fluids in a channel are investigated under influence of time-dependent pressure gradient. The isothermal channel walls have translational motions in their planes with time-dependent velocities. Governing equations of the mathematical model are based on the generalized constitutive equations for shear stress and thermal flux described by the time-fractional Caputo derivative. Analytical and semi-analytical solutions for velocity, shear stress, and temperature fields are obtained by using finite sine-Fourier and Laplace transforms. In the case of semi-analytical solutions, the inverse Laplace transforms are obtained numerically by employing the Talbots algorithms. Using the software Mathcad, numerical calculations have carried out and results are presented in graphical illustrations in order to analyze the memory effects on the fluid temperature and motion. It is found that in fluids with thermal memory the heat transfer is slower compared with the ordinary fluid, while the fractional velocity parameters act as braking/accelerating factors of the fluids.     

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.     

Symmetries,Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations

In this paper, the Lie group classification method is performed on the fractional partial differential equation (FPDE), all of the point symmetries of the FPDEs are obtained. Then, the symmetry reductions and exact solutions to the fractional equations are presented, the compatibility of the symmetry analysis for the fractional and integer-order cases is verified. Especially, we reduce the FPDEs to the fractional ordinary differential equations (FODEs) in terms of the Erdélyi-Kober (E-K) fractional operator method, and extend the power series method for investigating exact solutions to the FPDEs.     

Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe

Generally speaking, rheological properties of materials are specified by their so-called constitutive equations. The simplest constitutive equation for a fluid is a Newtonian one, on which the classical Navier-Stokes theory is based. The mechanical behavior of many fluids is well described by this theory. However, there are many rheologically compli- cated fluids such as polymer solutions, blood and heavy oils which are inadequately de- scribed by a Newtonian constitutive equation that does …     

Toward a New Algorithm for Nonlinear Fractional Differential Equations

This paper is concerned with the development of an efficient algorithm for the analytic solutions of nonlinear fractional differential equations. The proposed algorithm Laplace homotopy analysis method (LHAM) is a combined form of the Laplace transform method with the homotopy analysis method. The biggest advantage the LHAM has over the existing standard analytical techniques is that it overcomes the difficulty arising in calculating complicated terms. Moreover, the solution procedure is easier, more effective and straightforward. Numerical examples are examined to demonstrate the accuracy and efficiency of the proposed algorithm.     

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

In this paper, the famous Klein–Gordon–Zakharov (KGZ) equations are first generalized, and the new special types of KGZ equations with the positive fractional power terms (gKGZE) are presented. In order to derive exact solutions of the new special gKGZE, subsidiary higher-order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aid of the sub-ODE, exact solutions of four special types of the gKGZE are derived, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal travelling wave solution, provided that the coefficients of gKGZE satisfy certain constraint conditions.     

A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration

In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.