On numerical solution of Burgers' equation by homotopy analysis method

In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions.     

交通拥堵相变问题的同伦分析法

利用同伦分析法研究了一类基于洛伦兹系统的交通拥堵相变问题的非线性方程. 通过选取不同的初始解和不同的线性算子,分别得到了问题的近似解和相应的残留误差. 通过与前人结果的比较得出,在研究该类问题时同伦分析法优于微分变换法; 在应用同伦分析法时,要选取尽可能接近原算子线性部分作为线性算子. 本文还给出了一种新的初始解选取方法(双同伦分析法). 数值模拟的结果证实了理论分析的正确性. 关键词： 同伦分析法 交通拥堵 近似解 残留误差     

On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger’s equations using homotopy analysis transform method

In this paper we consider the homotopy analysis transform method (HATM) to solve the time fractional order Korteweg-de Vries (KdV) and Korteweg-de Vries-Burger’s (KdVB) equations. The HATM is a combination of the Laplace decomposition method (LDM) and the homotopy analysis method (HAM). The fractional derivatives are defined in the Caputo sense. This method gives the solution in the form of a rapidly convergent series with h-curves are used to determine the intervals of convergent. Averaged residual errors are used to find the optimal values of h. It is found that the optimal h accelerates the convergence of the HATM, with the rate of convergence depending on the parameters in the KdV and KdVB equations. The HATM solutions are compared with exact solutions and excellent agreement is found.     

Magnetic stabilization,transition and energy analysis in the Marangoni driven Full-Zone at low Prandtl numbers

The linear stability of the Marangoni-driven Full-Zone is investigated for low Prandtl number fluids. A constant and uniform magnetic field is applied along the axial axis of the liquid bridge to stabilize the axisymmetric base state. Dramatic contraction of the flow circulation in both radial and azimuthal directions is observed with moderate magnetic fields. The numerical solution utilizes a vorticity transport formulation and high resolution spectral collocation scheme with Chebyshev polynomial basis functions. Critical transitions to three-dimensional, stationary flows are observed up to Ha = 300 for Pr = 0.02 and Ha = 500 for Pr = 0.001. A hydrodynamically driven instability is suggested by the perturbation flows and confirmed through an energy analysis.     

The application of homotopy analysis method to nonlinear equations arising in heat transfer

Here, the homotopy analysis method (HAM), which is a powerful and easy-to-use analytic tool for nonlinear problems and dose not need small parameters in the equations, is compared with the perturbation and numerical and homotopy perturbation method (HPM) in the heat transfer filed. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series.     

A novel Galerkin-like weakform and a superconvergent alpha finite element method (SαFEM) for mechanics problems using triangular meshes

A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter α. A novel Galerkin-like weakform derived from the Hellinger–Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (SαFEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the SαFEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the SαFEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The SαFEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an α. In addition, a preferable-α approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined SαFEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the SαFEM.     

Lattice Boltzmann method for nanofluid flow in a porous cavity with heat sources and magnetic field

In this article, Lattice Boltzmann method (LBM) has been applied to investigate the influences of magnetic field and heat sources on water based nanofluid natural convection inside a porous cavity with three square heat sources. Koo–Kleinstreuer–Li (KKL) model is applied to study Brownian motion impact on nanofluid flow. Effects of Rayleigh number (Ra), Darcy number (Da), nanofluid volume fraction (ϕ), and Hartmann number (Ha) on heat transfer characteristics are analyzed. From the obtained results we observe a decrease in the temperature gradient with increasing Ha; while quite the opposite effect is true with increasing Da and Ra. In the absence of magnetic field, for higher values of Darcy and Rayleigh numbers, thermal plumes are generated and the temperature gradient is enhanced. Moreover, small eddies are generated near the vertical centerline. However, in the presence of magnetic field, the number of thermal plumes decreases.     

Space–time discontinuous Galerkin finite element method for two-fluid flows

A novel numerical method for two-fluid flow computations is presented, which combines the space–time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space–time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local hp-refinement. The front tracking is incorporated via cut-cell mesh refinement to ensure a sharp interface between the fluids. To compute the interface dynamics the level set method (LSM) is used because of its ability to deal with merging and breakup. Also, the LSM is easy to extend to higher dimensions. Small cells arising from the cut-cell refinement are merged to improve the stability and performance. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. The numerical method is applied to one and two dimensional two-fluid test problems using the Euler equations.     

The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels

Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). In this Letter, similarity transformation for the Navier-Stokes equation for 2-dimensional steady slip flow in microchannels is given. We provide an analytical solution for the slip flow using a powerful, easy-to-use analytic technique for non-linear problems, that is, the homotopy analysis method (HAM). The analytical solution is presented in the form of an infinite series. The effects of the Knudsen number (Kn) is discussed on the velocity profiles. It is found that the results are in excellent agreement with the existing results in the literature for the case of laminar developed flow.     

The first principles investigation of ferrite magnetic response with mismatch stress

The quasi-ferrite model is proposed and an appropriate PBE exchange functional with the spin density functional theory(SDFT) is selected for the calculation of the relation between magnetic moment and residual stress in ferrite using a quantum mechanics code. The relationship between ferrite magnetism and the carbon content is determined,and then a ferrite interstitial solid solution(ISS) model in a low carbon concentration state is replaced with an α- Fe model in the case of majority magnetic calculation. The band structure of the loaded-Fe is compared with that of the unloaded α-Fe. The comparison shows that the energy of Fe atomic 3d orbital changes a little,while the energy of electron orbital of iron core below 3d almost keeps unchanged. The relationship between the magnetic moment and the stress appears intermittent due to the Bragg total reflection. The change in the magnetic moment due to lattice mismatch is much larger than that caused by mechanical loading.     

On the analytical solution of viscous fluid flow past a flat plate

In this Letter, we propose a simple approach using HAM to obtain accurate totally analytical solution of viscous fluid flow over a flat plate. First, we show that the solution obtained using HPM is not a reliable one; moreover, we show that HPM is only a special case of HAM and its basic assumptions are restrictive rather than useful. We set ?=−1 for the case of comparison of our results to those obtained using HPM. Afterwards, we introduce an extra auxiliary parameter and a straightforward approach to find best values of this auxiliary parameter which plays a prominent role in the frame of our solution and makes it more convergent in comparison to previous works.     

A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].      

Homotopy Solution for Stagnation-Point Flow

This article reports the homotopy solution for stagnation point flow of a non-Newtonian fluid.An incompressible second grade fluid impinges on the wall either orthogonally or obliquely.The resulting nonlinear problems have been solved by a homotopy analysis method(HAM).Convergence of the series solutions is checked.Such solutions are compared with the numerical solutions presented in a study [Int.J.Non-Linear Mech.43(2008) 941].Excellent agreement is noted between the numerical and series solutions.     

Numerical investigation of incompressible fluid flow and heat transfer around a rotating circular cylinder

The heat transfer and air flow around an unconfined heated rotating circular cylinder is investigated numerically for varying rotation rates (α = 0–6) in the Reynolds number range of 20–200. The numerical calculations are carried out by using a finite volume method based commercial computational fluid dynamics solver FLUENT. The successive changes in the flow pattern are studied as a function of the rotation rate. Suppression of vortex shedding occurs as the rotation rate increases (α > 2). A second kind of instability appears for higher rotation speed where a series of counter-clockwise vortices is shed in the upper shear layer. The rotation attenuates the secondary instability and increases the critical Reynolds number for the appearance of this instability. Besides, time-averaged (lift and drag coefficients and Nusselt number) results are obtained and compared with the literature data. A good agreement has been obtained for both the local and averaged values.     

微重力下圆管毛细流动解析近似解研究

应用同伦分析法研究微重力环境下圆管毛细流动解析近似解问题, 给出了级数解的表达公式. 不同于其他解析近似方法, 该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 适用范围广. 同伦分析法提供了选取基函数的自由, 可以选取较好的基函数, 更有效地逼近问题的解, 通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度, 同伦分析法为圆管毛细流动问题的解析近似求解开辟了一个全新的途径. 通过具体算例, 将同伦分析法与四阶龙格库塔方法数值解做了比较, 结果表明, 该方法具有很高的计算精度. 关键词： 圆管 微重力 毛细流动 同伦分析法     

Remarks on optimized predictions for the Drell-Yan process

We discuss the optimization of the Drell-Yan pair production cross section calculated atO(α _s ) and compare it with theO(α _s ² ) results of van Neerven et al. It is shown that the optimized predictions do agree analytically with the latter calculation near the phase space boundary. They also provide a good numerical approximation wherever the stability equations have a solution.     

Series Solution for Rotating Flow of an Upper Convected MaxwellFluid over a Stretching Sheet

The equations for two-dimensional flow of an upper convected Maxwell (UCM) fluid in a rotating frame are modeled. The resulting equations are first simplified by a boundary layer approach and then solved by a homotopy analysis method (HAM). Convergence of series solution is discussed through residual error curves. The results of the influence of viscoelastic and rotation parameters are plotted and discussed.     

Solutions to the 1d Klein-Gordon equation with cut-off Coulomb potentials

In a recent paper by Barton [G. Barton, J. Phys. A: Math. Gen. 40 (2007) 1011], the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V₁(|x|)=−α/(|x|+a). In the present Letter, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cut-off Coulomb potential V₂(|x|)=−α/|x|, |x|>a, and otherwise V₂(|x|)=−α/a.     

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.