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.     

Solving systems of fractional differential equations by homotopy-perturbation method

In this Letter, approximate analytical solutions of systems of Fractional Differential Equations (FDEs) are derived by the Homotopy-Perturbation Method (HPM). The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that HPM is very effective and simple for obtaining approximate solutions of nonlinear systems of FDEs.     

Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy-perturbation and Adomian decomposition methods

In this work, two powerful analytical methods, called homotopy-perturbation method (HPM) and Adomian decomposition method (ADM) are introduced to obtain the exact solutions of linear and nonlinear Schrödinger equations. The main objective is to propose alternative methods of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results show that these methods are very efficient and convenient and can be applied to a large class of problems. The comparison of the methods shows that although the numerical results of these methods are the same, HPM is much easier, more convenient and efficient than ADM.     

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.     

Application of He's homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate

In this Letter, the problem of forced convection over a horizontal flat plate is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

In this research, two powerful analytical methods are introduced to handle nonlinear good Boussinesq, heat transfer and coupled Burgers' equations. One is the homotopy-perturbation method (HPM) and the other is the variational iteration method (VIM). VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. HPM converts a difficult problem into a simple one, which can be easily handled. The results attained in this paper confirm the idea that HPM and VIM are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.     

Numerical simulation of the regularized long wave equation by He's homotopy perturbation method

In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.     

Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this Letter; one is He's variational iteration method (VIM) and the other is the homotopy–perturbation method (HPM). Nonlinear convective–radiative cooling equations are used as examples to illustrate the simple solution procedures. These methods are useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. Comparison of the results obtained by both methods with exact solutions reveals that both methods are tremendously effective.     

基于同伦技术的Burgers方程的小波精细积分算法

以Burgers方程为例,结合区间小波精细积分方法,将同伦摄动方法的应用范围推广到多维非线性问题,给出一种求解非线性偏微分方程的新的小波精细积分方法,得到一种近似解析解的数值结果,对时间步长不敏感,更适合于求解非线性问题.     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

高功率微波在低电离层中的互作用效应分析

 高功率微波在低电离层中传输时会产生非线性效应。分析了低电离层中HPM的互作用效应，推导了HPM在低电离层中传输的互作用因子同初始场强和微波频率的关系，并对不同条件下振幅交调深度同两个脉冲的延迟时间之间的关系进行了理论研究与数值模拟。结果表明：改变脉冲的延迟时间，振幅交调深度的变化规律也随之改变；为了减少吸收衰减，低电离层中传输的HPM电波的干扰波电场初始振幅与特征等离子体电场振幅的比值应保持在10以下。     

高功率微波介质窗表面矩形槽抑制二次电子倍增

为深入研究高功率微波(HPM)作用下介质窗沿面击穿破坏的物理机制,探索提高闪络场强阈值的方法和途径,开展了介质窗表面矩形刻槽抑制电子倍增的理论与试验研究。首先根据动力学方程建立了介质窗表面电子倍增模型并分析了介质窗槽内电子运动轨迹,考虑了矩形槽结构对表面微波电场的影响,理论分析表明在闪络击穿的起始和发展阶段矩形槽可有效抑制电子倍增。在S波段(2.86 GHz,脉宽1μs)下开展了介质窗表面矩形刻槽的击穿破坏试验,试验结果发现表面矩形刻槽可大幅度提高微波传输功率,在槽深(1.0mm)一定时不同的刻槽宽度(0.5 mm和1.0 mm)对应的微波功率抑制范围不同。采用PIC-MC仿真模拟槽内倍增电子的时空演化,仿真结果很好地验证了试验现象。     

X波段长脉冲高功率微波产生的实验研究

 获得长脉冲高功率微波（HPM）输出是HPM源技术追求的重要目标之一。从物理机理上分析了影响慢波结构HPM器件实现长脉冲HPM输出的因素,并利用长脉冲脉冲功率源和过模慢波结构HPM器件,开展了X波段长脉冲HPM产生实验。实验中,采用介质-铜阴极,并在慢波结构表面镀Cr,在导引磁场约0.7 T、二极管电压约400 kV、电流约10 kA、束流脉宽200 ns的条件下,获得了功率500 MW、脉宽约100 ns、主模为TM₀₁的X波段长脉冲HPM输出。对实验结果的分析表明,脉冲功率源与HPM器件的阻抗不匹配,是导致HPM器件输出微波脉宽比电子束脉宽短、以及HPM器件输出微波功率效率较低的主要原因。     

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     

Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method

In this Letter, the He's homotopy perturbation method (HPM) to finding the soliton solutions of the two-dimensional Korteweg–de Vries Burgers' equation (tdKdVB) for the initial conditions was applied. Numerical solutions of the equation were obtained. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. The results reveal that the HPM is very effective and simple.     

引信高功率微波电磁辐射效应数据的FCM分析

高功率微波(HPM)电磁脉冲对引信辐照实验的数据分析是研究其电磁效应中的一个重要问题,数据分析的主要困难在于HPM电磁效应数据的高维复杂性。通过聚类算法设计,采用模糊C均值聚类(FCM)算法对某无线电引信的高功率微波电磁效应数据进行分析处理,利用其类内相似和类间相异的原则,经迭代运算,实现HPM对无线电引信效应数据的脉宽、峰值功率、功率密度等参数间的识别和分类。结果表明:采用FCM算法能够得到HPM对某引信的最佳聚类中心,即致使引信失效的最佳干扰阈值,证明了算法的有效性。     

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.     

Analytical solutions to heat transfer equations by homotopy-perturbation method revisited

In this Letter, the problem of determining the temperature distribution of a straight rectangular fin with power-law temperature-dependent surface heat flux is revisited. The solution of the corresponding nonlinear boundary-value problem (BVP) based on the homotopy-perturbation method (HPM) presented earlier is corrected and extended.     

Hydromagnetic Blood Flow of Sisko Fluid in a Non-uniform Channel Induced by Peristaltic Wave

In this paper, a smooth repetitive oscillating wave traveling down the elastic walls of a non-uniform twodimensional channels is considered. It is assumed that the fluid is electrically conducting and a uniform magnetic field is perpendicular to flow. The Sisko fluid is grease thick non-Newtonian fluid can be considered equivalent to blood. Taking long wavelength and low Reynolds number, the equations are reduced. The analytical solution of the emerging non-linear differential equation is obtained by employing Homotopy Perturbation Method(HPM). The outcomes for dimensionless flow rate and dimensionless pressure rise have been computed numerically with respect to sundry concerning parameters amplitude ratio ?, Hartmann number M, and Sisko fluid parameter b1. The behaviors for pressure rise and average friction have been discussed in details and displayed graphically. Numerical and graphical comparison of Newtonian and non-Newtonian has also been evaluated for velocity and pressure rise. It is observed that the magnitude of pressure rise is maximum in the middle of the channel whereas for higher values of fluid parameter it increases. Further, it is also found that the velocity profile shows converse behavior along the walls of the channel against multiple values of fluid parameter.     

Adaptation of homotopy-perturbation method for numeric-analytic solution of system of ODEs

A new reliable algorithm based on an adaptation of the standard Homotopy-Perturbation Method (HPM) is presented. The HPM is treated, for the first time, as an algorithm in a sequence of intervals (i.e., time step) for finding accurate approximate solutions of linear and nonlinear systems of ODEs. Numerical comparisons between the Multistage Homotopy-Perturbation Method (MHPM) and the available exact solution and the classical fourth-order Runge-Kutta (RK4) method reveal that the new technique is a promising tool for linear and nonlinear systems of ODEs.