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.     

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.     

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.     

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     

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.     

Analytic approximations of Von Krmn plate under arbitrary uniform pressure—equations in integral form

Analytic approximations of the Von Krmn's plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method(HAM), an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure Q or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters c_1 and c_2 in the frame of the HAM. Besides, it is found that the HAMbased iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure Q when c_1 =.θ and c_2 =-1, where θ denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case c_1 =.θ and c_2 =-1. In addition, we prove that the HAM approach for the Von Krmn's plate equations in differential form is just a special case of the HAM for the Von Krmn's plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems,and its superiority over perturbation techniques.     

Two dimensional fractional projectile motion in a resisting medium

In this paper we propose a fractional differential equation describing the behavior of a two dimensional projectile in a resisting medium. In order to maintain the dimensionality of the physical quantities in the system, an auxiliary parameter k was introduced in the derivative operator. This parameter has a dimension of inverse of seconds (sec)^?1 and characterizes the existence of fractional time components in the given system. It will be shown that the trajectories of the projectile at different values of γ and different fixed values of velocity v ₀ and angle θ, in the fractional approach, are always less than the classical one, unlike the results obtained in other studies. All the results obtained in the ordinary case may be obtained from the fractional case when γ = 1.     

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.     

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.     

Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method

In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM soluti...     

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.     

Coupled effects of nano-size,stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method

In this paper, natural frequency and nonlinear response of carbon nano-tube (CNT) conveying fluid based on the coupling of nonlocal theory and von Karman's stretching have been obtained. The homotopy analysis method (HAM) has been used for solving nonlinear differential equation of system and convergence region of approach presented. Effects of mid-plane stretching, nonlocal parameter and their coupling in the model have been investigated. It has been concluded that stretching effect is significant only for higher-amplitude initial excitations and lower beam aspect ratios. Moreover, by including the slip boundary condition, the effect of nano-size flow has been revealed in the nonlinear vibration model. We have concluded that small-size effects of nano-tube and nano-flow have impressed critical velocity of fluid significantly specially for gas fluid. Analytical results obtained from HAM solution show satisfactory agreement with numerical solutions such as Runge–Kutta. Having an analytical approach, we have been able to investigate the unbounded growth of amplitude of vibrations for flow velocities near the critical value. Moreover, by employing the second-order approximation of Galerkin's method, the estimated natural frequency of the first mode is verified. The obtained results would indicate that the effects of higher mode on the first natural frequency are negligible for the doubly-clamped CNT.     

Electrohydrodynamic flow analysis in a circular cylindrical conduit using Least Square Method

In this article, Electrohydrodynamic flow (EHD flow) in a circular cylindrical conduit is studied by a semi-exact and high efficient weighted residual method called Least Square Method (LSM). A principle of LSM is briefly introduced and later is employed to solve the described problem. Furthermore, the effects of the Hartmann electric number (Ha) and the strength of nonlinearity (α) on velocity profiles are discussed and presented graphically. Results are compared with numerical solution and obtained residuals are compared with those of HAM which previously were done by Mastroberardino in Ref. [3]. Outcomes reveal that LSM has an excellent agreement with numerical solution; also depicted residual functions showed that LSM is more acceptable than HAM especially for large values of Ha and α numbers, also it is simpler and needs fewer computations.     

Analytic approach to the orientational ordering in the high pressure phase II of o-D2 and p-H2

Possible kinds of orientational order in the high pressure phase II of o-D₂ and p-H₂ are considered as bifurcations of solutions of some nonlinear equations. Before, we found this approach to be fruitful in the case of orientational ordering in o-H₂ and p-D₂. A solution with four sublattices in the hcp structure is obtained. The order parameter of this solution abruptly decreases with temperature and becomes zero in phase I. The curve of the phase boundary T_c(P) is calculated.     

Effect of magnetic and boundary parameters on flow characteristics analysis of micropolar ferrofluid through the shrinking sheet with effective thermal conductivity

Ferrofluids have many applications in mechanical and electrical engineering. In this paper, characteristics of ferrofluid over a shrinking sheet with effective thermal conductivity model are studied by the homotopy perturbation method (HPM) and Akbari-Ganji's method (AGM). Also, the Finite Element Method (FEM) has been applied for numerical solution. The governing equations formulated by the Tiwari-Das model. It is supposed that base fluid and nanoparticles are water and Fe₃O₄respectively. Effect of related parameters of micro-rotation velocity and dimensionless velocity have taken for suction and injection of mass transfer parameter. Results show that the magnetic and boundary parameters, in contrast to the micro-rotation parameter, have the same impact on velocity. Moreover, a comparison has been made between the results of this study with other researchers shows the impressive accuracy and efficiency of these methods.     

Entropic uncertainty for two atoms interacting with a cavity field under the influence of two photons (off-resonance case)

In this paper, we introduce a Hamiltonian model describing the interaction of two photons with two two-level atoms and a degenerate parametric amplifier. In the near-resonance case, we obtain an analytic solution of the evolution equation for the wave function in the Schr¨odinger picture and use the obtained result for discussing the atomic inversion, the purity, and the phenomenon of squeezing. We show that the phenomenon of superstructure appears in the atomic inversion in the presence of detuning (with parameter ??) and coupling (with parameter ??₃). Our study of the purity shows that the system is always in a mixed state, and the maximum value of entanglement occurs around ~0.6. Also we show that detuning leads to a reduction in the value of squeezing for all quadrature variances. In contrast, the coupling parameter leads to an increase in the value of squeezing. However, for the usual single-mode squeezing (of quadratures) the effect of detuning consists in increase in the squeezing period.     

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...     

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.     

A matter-dominated cosmological model with variable G and Λ and its confrontation with observational data

In the framework of renormalization-group improved cosmologies, we analyze both theoretically and observationally the exact and general solution of the matter-dominated cosmological equations, by using the expression of the cosmological term as a function of the Newton parameter already determined by the integration method employed in a previous paper. A rough comparison between such a model and the concordance ΛCDM model from the point of view of the magnitude-redshift relationship has been already considered, without showing any appreciable differences. Here we test our model by using astrophysical data (the Union2 type Ia supernovae (SNIa) dataset, the Hubble diagram constructed from some gamma ray bursts luminosity distance indicator), to constrain its parameters. We also apply a cosmographic approach to our cosmological model. In order to estimate the cosmographic parameters we fit a large dataset, including not only the Hubble diagram, as traced by SNIa and gamma ray bursts, but also the H(z) measurements from passively evolving galaxies, baryon acoustic oscillations and the distance priors from the cosmic microwave background radiation anisotropy spectrum. We show that this matter-dominated cosmological model with variable Newton parameter and variable cosmological term is indeed compatible with the observations above. The cosmographic approach adopted confirms such conclusions. Last, it seems possible to include radiation into the model, since numerical integration of the equations derived by the presence of both radiation and matter shows that, after inflation, the total density parameter is initially dominated by the radiation contribution and later by the matter one.