微重力环境下无限长柱体内角毛细流动解析近似解研究

应用同伦分析法研究无限长柱体内角毛细流动解析近似解问题,给出了级数解的递推公式.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,同伦分析法为内角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.     

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     

Solutions of Heat-Like and Wave-Like Equations with Variable Coefficients by Means of the Homotopy Analysis Method

We employ the homotopy analysis method （HAM） to obtain approximate analytical solutions to the heat-like and wave-like equations. The HAM contains the auxiliary parameter h, which provides a convenient way of controlling the convergence region of series solutions. The analysis is accompanied by several linear and nonlinear heat-like and wave-like equations with initial boundary value problems. The results obtained prove that HAM is very effective and simple with less error than the Adomian decomposition method and the variational iteration method.     

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.     

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.     

The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation

Here, an analytic technique, namely the homotopy analysis method (HAM), is applied to solve a generalized Hirota–Satsuma coupled KdV equation. HAM is a strong and easy-to-use analytic tool for nonlinear problems and dose not need small parameters in the equations. Comparison of the results with those of Adomian's decomposition method (ADM) and homotopy perturbation method (HPM), has led us to significant consequences. The homotopy analysis method contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series.     

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.     

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.     

Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface

This Letter looks at the rotating flow of a second grade fluid past a porous shrinking surface. The fluid is electrically conducting in the presence of a constant applied magnetic field. The governing partial differential equations are first reduced into ordinary differential equations and then solved by homotopy analysis method (HAM). Convergence of the series solution is shown explicitly. In addition, the obtained results are illustrated graphically to indicate the effects of the pertinent physical parameters.     

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.     

外加直流电场作用下高阶弱非线性复合介质的电势分布

利用同伦分析方法, 研究了一类由柱形杂质随机嵌入基质所形成的、电场和电流密度满足J = σ E + χ |E|²E + η|E|⁴E 形式本构关系的高阶弱非线性复合介质在外加直流电场作用下的电势分布问题. 首先利用模函数展开法, 将本构方程及边界条件化成了一系列非线性常微分方程的边值问题; 再利用同伦分析方法进行求解, 给出了电势在基质和杂质区域的渐近解析解. 关键词： 高阶弱非线性复合介质 模函数展开法 同伦分析方法 电势分布     

A short comment on the effect of a shear layer on nonlinear water waves

In this paper we study nonlinear periodic deep water waves propagating on a background shear current,which decays exponentially with depth.We extend the study of Cheng,Cang and Liao(2009) by introducing a second parameter which measures the depth of the shear current.A high-order convergent analytical series solution is obtained by the homotopy analysis method(HAM).A detailed analysis of the impact of the depth parameter is given.We find that increasing this parameter so that the shear current is thinner re...     

Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet

The steady two-dimensional mixed convection flow of a micropolar fluid over a non-linear stretching sheet is investigated. The governing non-linear equations and their associated boundary conditions are transformed into coupled non-linear ordinary differential equations. The series solution of the problem is obtained by utilizing the homotopy analysis method (HAM). The convergence of the obtained series solutions is carefully checked. The physical significance of interesting parameters on the flow and the thermal fields are shown through graphs and discussed in detail. The values of wall shear stress, couple wall stress and the local Nusselt number are tabulated. Comparison is also made with the corresponding results of viscous fluid with no mixed convection and an excellent agreement is noted.     

同伦分析法在求解非线性演化方程中的应用

利用同伦分析法求解了(2+1)维改进的 Zakharov-Kuznetsov方程, 得到了它的近似周期解,该解与精确解符合很好. 结果表明，同伦分析法在求解高维非线性演化方程时, 仍然是一种行之有效的方法. 同时,还对该方法进行了一定的扩展， 经过扩展后的方法能够更方便地求解更多非线性演化方程的高精度近似解析解. 关键词： 同伦分析法 改进的 Zakharov-Kuznetsov方程 周期解     

Heat transfer analysis in the flow of Walters' B fluid with a convective boundary condition

Radiative heat transfer in the steady two-dimensional flow of Walters' B fluid with a non-uniform heat source/sink is investigated. An incompressible fluid is bounded by a stretching porous surface. The convective boundary condition is used for the thermal boundary layer problem. The relevant equations are first simplified under usual boundary layer assumptions and then transformed into a similar form by suitable transformations. Explicit series solutions of velocity and temperature are derived by the homotopy analysis method(HAM). The dimensionless velocity and temperature gradients at the wall are calculated and discussed.     

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     

同伦分析法在求解耗散系统中的应用

利用同伦分析法求解了KdV-Burgers方程,得到了它的解析近似解,该解与精确解符合得非常好.结果表明同伦分析法在求解某些耗散系统时,仍然是一种行之有效的方法.     

Impact of Linear Operator on the Convergence of HAM Solution: A Modified Operator Approach

The linear operator plays an important role in the computational process of Homotopy Analysis Method (HAM). In HAM frame any kind of linear operator can be chosen to develop a solution. Hence, it is easy to introduce the modified/physical parameter dependent linear operators. The effective use of these operators has been judged through solving fluid flow problems. Modification in linear operators affects the solution and improves the computational efficiency of HAM for larger values of parameters. The convergence rate of the solution is rapid and several times higher resulting in less computational time.     

Analytic and numerical solutions to the Lane-Emden equation

In this Letter, we present analytical solutions to the Lane-Emden equation describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solutions are obtained by using the traditional power series approach and by using the Homotopy Analysis Method (HAM). We show that the series solutions obtained by the Homotopy Analysis Method converge in a larger interval than in the case of the corresponding traditional series solutions. Furthermore, we obtained numerical solutions (using Runge-Kutta-Fehlberg 4-5 technique) which are used to validate the analytical solutions.