B-spline collocation and quasi-interpolation methods for boundary layer flow and convection heat transfer over a flat plate

The B-spline collocation methods and a new ODEs solver based on B-spline quasi-interpolation are developed to study the problem of forced convection over a horizontal flat plate, numerically. The problem is a system of nonlinear ordinary differential equations which arises in boundary layer flow. A more accurate value of \(\sigma =f^{\prime \prime }(0)\) obtained by applying quartic B-spline collocation method and utilized to solve the system of ODE. The results are shown to be precise as compared to the corresponding results obtained by Howarth.     

An optimization method based on the Legendre wavelets for 3D rotating,squeezing and stretching magnetohydrodymanic flow in a channel with porous wall

In this paper, the 3D flow in a rotating channel with the lower permeable stretching wall is investigated. The unsteady squeezing flow in the presence of transverse magnetic flux is mathematically modeled by utilizing Navier-Stokes equations. A new optimal wavelet method based on coupling the Legendre wavelets and the hat functions with some auxiliary convergence-control parameters is proposed to solve the obtained fourth-order system of nonlinear differential equations. In the proposed method, the residual function and its error of norm two are employed for converting the main problem into an optimization one to optimally choose the auxiliary convergence-control parameters. The main idea behind this approach is to reduce solving such problems to those of solving optimization ones which greatly simplifies the problem and also leads to obtaining a good approximate solution for the problem under consideration.

     

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.     

Enhancement of heat transfer in a convergent/divergent channel by using carbon nanotubes in the presence of a Darcy–Forchheimer medium

This article explores the influence of thermal radiation on the flow and heat transfer of single-walled carbon nanotubes over both a convergent and divergent channel. Flow is induced due to a Darcy–Forchheimer medium. Further, the heat transfer mechanism is analyzed in the presence of a thermal radiation process. Guided by some appropriate similarity transformations, the fundamental PDEs are converted into a self-similar system of coupled non-linear ODEs. The findings are obtained with the help of the Runge–Kutta-45-based shooting method. The roles of the Reynolds number, porosity parameter, inertia coefficient parameter, Prandtl number and radiation parameter are presented graphically. Results are displayed and show that the rate of heat transfer is higher in a divergent channel as compared to a convergent channel.

     

多模型小波网络非线性动态系统辨识

由于许多复杂的工业系统具有非线性特性，难以建立确切的数学模型，因此提出用 多模型小波网络辨识非线性动态系统，并给出了辨识结构和训练算法．仿真实验比较了多模型小波网络与单小波网络在辨识非线性系统时性能上的差异，验证了该方法收敛速度快，抗干扰能力强，具有较高的逼近精度．     

Instability of channel flow with fluid injection and parietal vortex shedding

This paper is devoted to the computation of the stability properties and the aeroacoustic interaction in a planar channel flow driven by air injection through porous wall. The purpose is to establish a simulation method which estimates the vortex-shedding and acoustic-wave interaction. To perform these unsteady computations, turbulent motion is taken into account by a first-order model based on a nonlinear relationship between time-dependent Reynolds stresses and velocity gradients. Model coefficients are explicit functions of both strain and rotation. This model is applied to the computation of parietal vortex shedding in a simplified configuration. It is a channel flow with fluid injection. Two cases are computed here. The first one is stable and the second one is unstable. To improve the evaluation of the turbulent effects, computations of the second case are presented with and without a turbulence model. In first configuration, a good agreement with experimental data is found. In the second case both with and without turbulence model the natural unsteadiness of the flow is captured. The parietal vortex shedding is described and the turbulence effects are characterized.     

A new model for Brownian force and the application to simulating nanofluid flow

A new expression is proposed to simulate Brownian force based on the experimental measurement results of Brownian motion, which follows white Gaussian noise process. As the time t → 0 and the particle density is equal to the fluid density, the new expression approaches the classical formula of the model used by many researchers. The modified model is validated by theoretical and experimental data. On the other hand, as it origins from the unbalanced force exerted by surrounding fluid molecules, the drag analogy force model is constructed describing the Brownian force, which depends on size-related statistical velocity. Thus, a different expression for the Langevin equation is presented. The present model is applied in simulating flow and heat transfer in a channel utilizing alumina–water nanofluid. Navier–Stokes equations with modified source terms for the continuous flow have been discretized using finite element method. The velocities and temperatures of nanoparticles are determined in the Lagrangian reference frame. The simulation results show that the distribution of nanoparticles inside the channel is obviously unsteady and nonuniform. The fluid velocity and temperature profiles show significant fluctuation feature at low Reynolds numbers (Re). The impact of Brownian motion on the fluid flow is analyzed quantitatively. We have found that for Re < 0.06, the affected intensity increases rapidly.     

Chebyshev wavelet collocation method for magnetohydrodynamic flow equations

This study proposes Chebyshev wavelet collocation method for partial differential equation and applies to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Approximate solutions of velocity and induced magnetic field are obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. Numerical results of the MHD flow problem show that the accuracy of proposed method is quite good even in the case of a small number of grid points. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H_a ≤ 1000.

     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.     

基于小波神经网络的非线性误差校正模型及其预测

针对非线性系统的预测问题，在线性和非线性协整理论涵义的基础上，提出利用小波神经网络进行非线性协整系统的非线性误差校正模型的研究，并给出该模型的建模方法．对沪深股市进行实证研究，与线性向量自回归模型进行比较．研究证明，小波神经网络所建立的非线性误差校正模型有较好的预测效果，能够有效地预测非线性经济系统．     

An improved model along with a spectral numerical simulation for fractional predator–prey interactions

Predator–prey models appear in various fields of bio-mathematics used for the analysis of interactions of biological systems. Due to the complexities of the physical context for the real-world problems of food chain dynamics, introducing new models compatible with experimental results stays ongoing research. Many models have been proposed and analyzed for these systems in recent years. In this paper, we propose a new fractional-order predator–prey model with negative feedback on both species with memory-dependent effects, which increases the compatibility level of the model. Then we present a novel Laguerre spectral numerical simulation for the proposed model by introducing Laguerre modal basis functions with collocation and Galerkin techniques. We then transfer the nonlinear model into a system of algebraic equations, which is solved by efficient numerical solvers. Finally, we provide some test problems to show the efficiency of the proposed model and the computational method.

     

Numerical study of asymmetric laminar flow of micropolar fluids in a porous channel

A numerical study is presented for the two-dimensional flow of a micropolar fluid in a porous channel. The channel walls are of different permeability. The fluid motion is superimposed by the large injection at the two walls and is assumed to be steady, laminar and incompressible. The micropolar model due to Eringen is used to describe the working fluid. The governing equations of motion are reduced to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form by using an extension of Berman’s similarity transformations. A numerical algorithm based on finite difference discretization is employed to solve these ODEs. The results obtained are further improved by Richardson’s extrapolation for higher order accuracy. Comparisons with the previously published work are performed and are found to be in a good agreement. It has been observed that the velocity and microrotation profiles change from the most asymmetric shape to the symmetric shape across the channel as the parameter R or the permeability parameter A are varied between their extreme values. The results indicate that larger the injection velocity at a wall relative to the other is, smaller will be the shear stress at it than that at the other. The position of viscous layer has been found to be more sensitive to the permeability parameter A than to the parameter R. The micropolar fluids reduce shear stress and increase couple stress at the walls as compared to the Newtonian fluids.     

Exact linearization of nonlinear ordinary autonomous differential equations

Many methods for finding exact solutions to nonlinear ordinary differential equations (ODE) are based on certain euristic rules. The author suggested a newexact linearization method that provides an algorithmic procedure for constructing exact solutions for some important classes of ODEs [1].     

基于ARMA和小波变换的交通流预测模型研究

基于小波变换理论和自回归滑动平均(ARMA)时间序列模型的相关知识,研究了智能交通优化控制系统中的交通流量的预测问题。首先,在对实际监测的交通流量数据进行小波变换处理的基础上,建立交通流量的预测模型;然后,利用最小二乘法理论对ARMA模型的参数结构进行了详细地分析;同时给出基于小波变换和ARMA模型的交通流优化控制系统的运行机理并设计出相应的网络拓扑模型和数据传输模型;最后,用某交通观测站的实测数据对模型进行实际仿真。仿真结果表明,文中所设计的模型和算法是有效的。     

Precipitation phenomenon of nanoparticles in power-law fluids over a rotating disk

The steady flow and mass transfer of nanofluids with power-law type base fluids over a free-rotating disk are investigated. Previously, we have modeled the volume fraction of nanoparticles and verified the experimental conclusion through the numerical simulation of particle distribution in nanofluid in a Petri dish under the influence of movement using a power-law model of mass diffusivity. We further this study by a similar model of the mass diffusivity following a power-law type to consider the laminar non-Newtonian power-law flow in a rotating infinite disk with angular velocity about the z-axis. The coupled governing equations are transformed into ODEs. Homotopy analysis method (HAM) is applied to solve the ODEs while special attention is paid to deal with the nonlinear items in the ODEs. In the last section, we provide images of nanoparticles suspended in power-law fluids in a rotating disk as obtained using the laser speckle method. When they are compared with the analytical results gained by the HAM, they qualitatively matched the solutions of the concentration equation of nanofluids.     

Numerical solution of variable-order space-time fractional KdV–Burgers–Kuramoto equation by using discrete Legendre polynomials

In this paper, a new version of the nonlinear space-time fractional KdV–Burgers–Kuramoto equation has been generated via the variable-order (VO) fractional derivatives defined in the Caputo type. A numerical method has been developed based on the discrete Legendre polynomials (LPs) and the collocation scheme for solving this equation. First, the solution of the problem is expanded in terms of the shifted discrete LPs. Then, this expansion and its derivatives, including the classical partial derivatives and the VO fractional partial derivatives are replaced in the equation. Eventually, the operational matrices of the shifted discrete LPs, including the classical derivatives and the VO fractional derivatives (which are derived in this study), and the collocation method are employed to convert the approximated problem into an algebraic system of equations. Some numerical results are given to illustrate the accuracy of the method.

     

求解非线性期权定价模型的自适应小波同伦摄动技术

Leland 模型是在考虑交易费用的情况下,对 Black - Scholes 模型进行修改得到的非线性期权定价模型. 本文针对 Leland 模型,提出了一种求解非线性动力学模型的自适应多尺度小波同伦摄动法. 该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性期权定价模型方程自适应离散为非线性常微分方程组; 然后将用于求解非线性常微分方程组的同伦摄动技术和小波变换的动态过程相结合,构造了求解 Leland 模型的自适应数值求解方法. 数值模拟结果验证了该方法在数值精度和计算效率方面的优越性.     

飞机超机动状态动力学特征及对控制系统的挑战

超机动能力作为第四代战斗机的重要标志性特征,在近距空战中具有极其重要的作用.针对非线性非定常气动效应、强耦合和操纵器异构冗余等超机动状态飞机的基本动力学特征,分析了其非线性气动力/力矩、非定常迟滞效应、参数快时变、惯性耦合和推力矢量等因素给控制系统带来的挑战,总结了超机动飞机的非线性非定常气动力建模、飞行控制和控制分配的研究现状,给出了一种基于非线性补偿和气动力模型数据库的鲁棒解耦控制策略,为超机动飞行控制方法的工程应用提供了参考.     

A comparison of numerical models for evaporative two-phase flow in a self-heated porous medium

Two different numerical models using the finite difference method (FDM) for one-component time-dependent two-phase flows in a porous medium are investigated: the iterative four-variable model (I4VM) and the direct three-variable model (D3VM). The former includes the pressure gradient and uses the iterative method to solve a system of flow equations, whereas for the latter, the formulation without the pressure gradient is simultaneously solved using the algorithm for tri-tridiagonal equations of three dependent variables. The steady-state solution as well as the unsteady results obtained by two models are compared only for the low heat generation rate below the dryout limit. For the high heat generation rate the effects of two numerical models on the time-dependent flow and dryout behavior up to incipient dryout are discussed in terms of liquid volumetric fraction and liquid superficial velocity distributions. It was found that the I4VM is numerically more stable for the case of strongly nonlinear physical models (e.g. the Ergun constants model of Fand, R. M., Kim, B. Y. K., Lam, A. C. and Phan, R. T., Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J. Fluids Engng, 1987, 109, 268–274) and enables us to analyze those, whereas the D3VM is advantageous for fast analysis of the weakly nonlinear model (e.g. the Ergun constants model of Macdonald, I. F., El-Sayed, M. S., Mow, K. and Dullien, F. A. L., flow through porous media—the Ergun equation revisited. Ind. Engng Chem. Fundam., 1979, 18, 199–208). Finally, a comparative evaluation of both numerical models is presented.     

Stability analysis for the Whipple bicycle dynamics

It has been known that bicycle stability is closely linked to a pair of ordinary differential equations (ODEs). The linearization technique used to derive these ODEs, nevertheless, has yet to be thoroughly examined. For this purpose, we conduct an analysis of the dynamics of the Whipple bicycle, starting with the contact kinematics, using the Gibbs–Appell method. The effort results in a complete nonlinear model with minimal dimensions, from which equilibrium points during the bicycle’s straight and circular motions can be determined. The model can be linearized around these points via a perturbation analysis under no additional assumptions. Given the non-hyperbolic nature of the equilibria, we apply the center manifold theorem to analyze their stability, offering a rigorous derivation of the (well-know) exponential stability of the bicycle in its leaning and steering motions. Finally, a dimensionless index is defined to characterize the influence of physical parameters on the bicycle stability.