ALE Fractional Step Finite Element Method for Fluid-Structure Nonlinear Interaction Problem

A computational procedure is developed to solve the problems of coupled motion of a structure and a viscous incompressible fluid. In order to incorporate the effect of the moving surface of the structure as well as the free surface motion, the arbitrary Lagrangian-Eulerian formulation is employed as the basis of the finite element spatial discretization. For numerical integration in time, the fraction,step method is used. This method is useful because one can use the same linear interpolation function for both velocity and pressure. The method is applied to the nonlinear interaction of a structure and a tuned liquid damper. All computations are performed with a personal computer.

Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems

An integrated fluid-thermal-structural analysis approach is presented. In this approach, the heat conduction in a solid is coupled with the heat convection in the viscous flow of the fluid resulting in the thermal stress in the solid. The fractional four-step finite element method and the streamline upwind Petrov-Galerkin （SUPG） method are used to analyze the viscous thermal flow in the fluid. Analyses of the heat transfer and the thermal stress in the solid axe performed by the Galerkin method. The second-order semi- implicit Crank-Nicolson scheme is used for the time integration. The resulting nonlinear equations are lineaxized to improve the computational efficiency. The integrated analysis method uses a three-node triangular element with equal-order interpolation functions for the fluid velocity components, the pressure, the temperature, and the solid displacements to simplify the overall finite element formulation. The main advantage of the present method is to consistently couple the heat transfer along the fluid-solid interface. Results of several tested problems show effectiveness of the present finite element method, which provides insight into the integrated fluid-thermal-structural interaction phenomena.

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.

Evaluation of Full Experimental Design Method against Fractional Design Method and Taguchi Design Method in Machining Operation

DOE （design of experiments） is a systematic, rigorous approach to engineering problem-solving that applies principles and techniques at the data collection stage so as to ensure the generation of valid, defensible, and supportable engineering conclusions. This paper presents a comparison of three different experimental designs （full experimental design, fractional design and Taguchi design） aimed at studying the effects of cutting parameters variations on surface finish. The results revealed that the effects obtained by analyzing both fractional and Taguchi designs were comparable to the main effects and two-level interactions obtained by the full factorial design. Thus, we conclude that full factorial design appear to be reliable and more economical since they permit to reduce by a factor the amount of time and effort required to conduct the experimental design without losing valuable information. Thus, we conclude that full factorial design appear to be reliable and more economical and without losing valuable information.

Study on the wind field and pollutant dispersion in street canyons using a stable numerical method

A stable finite element method for the time dependent Navier-Stokes equations was used for studying the wind flow and pollutant dispersion within street canyons. A three-step fractional method was used to solve the velocity field and the pressure field separately from the governing equations. The Streamline Upwind Petrov-Galerkin(SUPG) method was used to get stable numerical results. Numerical oscillation was minimized and satisfactory results can be obtained for flows at high Reynolds numbers. Simulating the flow over a square cylinder within a wide range of Reynolds numbers validates the wind field model. The Strouhal numbers obtained from the numerical simulation had a good agreement with those obtained from experiment. The wind field model developed in the present study is applied to simulate more complex flow phenomena in street canyons with two different building configurations. The results indicated that the flow at rooftop of buildings might not be assumed parallel to the ground as some numerical modelers did. A counter-clockwise rotating vortex may be found in street canyons with an inflow from the left to right. In addition, increasing building height can increase velocity fluctuations in the street canyon under certain circumstances, which facilitate pollutant dispersion. At high Reynolds numbers, the flow regimes in street canyons do not change with inflow velocity.

Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.

Stochastic bifurcations of generalized Duffing–van der Pol system with fractional derivative under colored noise

The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.

An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

Regarding Deeper Properties of the Fractional Order Kundu-Eckhaus Equation and Massive Thirring Model

In this paper,the fractional natural decomposition method(FNDM)is employed to find the solution for the Kundu-Eckhaus equation and coupled fractional differential equations describing the massive Thirring model.Themassive Thirring model consists of a system of two nonlinear complex differential equations,and it plays a dynamic role in quantum field theory.The fractional derivative is considered in the Caputo sense,and the projected algorithm is a graceful mixture of Adomian decomposition scheme with natural transform technique.In order to illustrate and validate the efficiency of the future technique,we analyzed projected phenomena in terms of fractional order.Moreover,the behaviour of the obtained solution has been captured for diverse fractional order.The obtained results elucidate that the projected technique is easy to implement and very effective to analyze the behaviour of complex nonlinear differential equations of fractional order arising in the connected areas of science and engineering.

FULL DISCRETE NONLINEAR GALERKIN METHOD FOR THE NAVIER-STOKES EQUATIONS

This paper deals with the inertial manifold and the approximate inertialmanifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore,we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.

A Fractional Model for the Single Stokes Pulse from the Nonlinear Optics

In this paper we refer to equations of motion for the single Stokes pulse from the nonlinear optics, called the Stokes pulse system. A fractional-order model with Caputo derivative associated to Stokes pulse system (called the fractional Stokes pulse system) is proposed. The existence and uniqueness of solution of initial value problem for this fractional system are proved. The dynamic behavior for a special fractional Stokes pulse system is investigated, including: the fractional stability, the stabilization problem using suitable linear controls and the numerical integration based on fractional Euler method.

Proportional Fairness-Based Energy-Efficient Power Allocation in Downlink MIMO-NOMA Systems with Statistical CSI

In this paper, proportional fairness(PF)-based energy-efficient power allocation is studied for multiple-input multiple-output(MIMO) non-orthogonal multiple access(NOMA) systems. In our schemes, statistical channel state information(CSI) is utilized for perfect CSI is impossible to achieve in practice. PF is used to balance the transmission efficiency and user fairness. Energy efficiency(EE) is formulated under basic data rate requirements and maximum transmitting power constraints. Due to the non-convex nature of EE, a two-step algorithm is proposed to obtain sub-optimal solution with a low complexity. Firstly, power allocation is determined by golden section search for fixed power. Secondly total transmitting power is determined by fractional programming method in the feasible regions. Compared to the performance of MIMO-NOMA without PF constraint, fairness is obtained at expense of decreasing of EE.

New Exact Solutions of Fractional Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations Using Fractional Sub-Equation Method

In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.

Impact of pertinent parameters on foam behavior in the entrance region of porous media:mathematical modeling

Foam injection is a promising solution for control of mobility in oil and gas field exploration and development,including enhanced oil recovery,matrix-acidization treatments,contaminated-aquifer remediation and gas leakage prevention.This study presents a numerical investigation of foam behavior in a porous medium.Fractional flow method is applied to describe steady-state foam displacement in the entrance region.In this model,foam flow for the cases of excluding and including capillary pressure and for two types of gas,nitrogen(N2)and carbon dioxide(CO2)are investigated.Effects of pertinent parameters are also verified.Results indicate that the foam texture strongly governs foam flow in porous media.Required entrance region may be quite different for foam texture to accede local equilibrium,depending on the case and physical properties that are used.According to the fact that the aim of foaming of injected gas is to reduce gas mobility,results show that CO2 is a more proper injecting gas than N2.There are also some ideas presented here on improvement in foam displacement process.This study will provide an insight into future laboratory research and development of full-field foam flow in a porous medium.

IMPROVED FRACTIONAL SUB-EQUATION METHOD AND ITS APPLICATIONS TO FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

Based on an improved fractional sub-equation method involving Jumarie＇s mo- dified Riemann-Liouville derivative, we construct analytical solutions of space-time fractional compound KdV-Burgers equation and coupled Burgers＇ equations. These results not only reveal that the method is very effective and simple in studying solu- tions to the fractional partial differential equation, but also include some new exact solutions.

Quasinormal modes of a stationary axisymmetric EMDA black hole

The massless scalar quasiaormal modes （QNMs） of a stationary axisymmetric Einstein-Maxwell dilato-axioa （EMDA） black hole are calculated numerically using the continued fraction method first proposed by Leaver. The fundamental quasinormal frequencies （slowly damped QNMs） are obtained and the peculiar behaviours of them are studied. It is shown that these frequencies depend on the dilaton parameter D, the rotational parameter a, the multiple moment l and the azimuthal number m, and have the same values with other authors at the Schwarzschild and Kerr limit.

The Modified Upwind Finite Difference Fractional Steps Method for Compressible Two-phase Displacement Problem

For compressible two-phase displacement problem,the modified upwind finite difference fractionalsteps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplicationof difference operators,decomposition of high order difference operators,the theory of prior estimates and tech-niques are used.Optimal order estimates in L^2 norm are derived for the error in the approximate solution.Thismethod has already been applied to the numerical simulation of seawater intrusion and migration-accumulationof oil resources.

THE UPWIND FINITE DIFFERENCE FRACTIONAL STEPS METHOD FOR NONLINEAR COUPLED SYSTEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, trod two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.

Conformable Fractional Nikiforov–Uvarov Method

We introduce conformable fractional Nikiforov–Uvarov(NU) method by means of conformable fractional derivative which is the most natural definition in non-integer calculus. Since, NU method gives exact eigenstate solutions of Schr¨odinger equation(SE) for certain potentials in quantum mechanics, this method is carried into the domain of fractional calculus to obtain the solutions of fractional SE. In order to demonstrate the applicability of the conformable fractional NU method, we solve fractional SE for harmonic oscillator potential, Woods–Saxon potential, and Hulthen potential.

Solution of the time-fractional generalized Burger-Fisher equation using the fractional re duce d differential transform method

“The time-fractional generalized Burger-Fisher equation(TF-GBFE)”is used in various applied sciences and physical applications,including simulation of gas dynamics,financial mathematics,fluid mechan-ics,and ocean engineering.This equation represents a concept for the coordination of reaction systems,as well as advection,and conveys the understanding of dissipation.The Fractional Reduced Differential Transform Method(FRDTM)is used to evaluate“the time-fractional generalized Burger-Fisher equation(TF-GBFE).”Todeterminethemethod’s validity,whenthesolutionsareobtained,theyarecorrelatedto exact solutions ofα=1 order,and even for various values ofα.Three-dimensional graphs are used to depict the solutions.Additionally,the analysis of exact and FRDTM solutions indicates that the proposed approach is very accurate.