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     

Homotopy Continuation Approach to Electrostatic Boundary-Value Problems of Nonlinear Media

The homotopy continuation method is employed to solve electrostatic boundaryvalue problems of nonlinear media. The difficulty associated with matching the inherently nonlinear boundary conditions on the interface is overcome by the mode expansion method, by which the nonlinear partial differential equations of the original problem are transformed into an infinite set of nonlinear ordinary differential equations. In this regard, the homotopy method has to be modified to handle the nonlinear boundary conditions. As an illustration, we study two cases:(a) nonlinear inclusion in linear host and (b) linear inclusion-in nonlinear host, both in two dimensions. The homotopy method is validated by comparing the results with the exact solution of case (a) and the results derived by perturbation method in case (b).     

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     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

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...     

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.     

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     

六组点堆中子动力学方程组的同伦分析解

鉴于目前六组点堆中子动力学方程仍然无法获得解析解,本文尝试将同伦分析方法应用于六组缓发中子动力方程组的求解,获得了它的级数解析解,并对级数解析解算法的有效性进行了检验.结果表明,该级数解析解算法从计算时间和精度上都能达到工程应用的要求,可适宜于反应堆中子动力学控制的设计分析和仿真计算.     

Analysis of a time fractional wave-like equation with the homotopy analysis method

The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when ?f=?g=−1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.     

离散修正KdV方程的解析近似解

将同伦分析法进行了推广,使之适用于求解离散修正KdV方程.获得了由指数函数表达的亮孤子解,该解析近似解与精确解符合很好.数值模拟结果说明了同伦分析法对求解复杂非线性问题的有效性和潜力.     

Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives KleinGordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.     

High-performance modeling acoustic and elastic waves using the parallel Dichotomy Algorithm

A high-performance parallel algorithm is proposed for modeling the propagation of acoustic and elastic waves in inhomogeneous media. An initial boundary-value problem is replaced by a series of boundary-value problems for a constant elliptic operator and different right-hand sides via the integral Laguerre transform. It is proposed to solve difference equations by the conjugate gradient method for acoustic equations and by the GMRES(k) method for modeling elastic waves. A preconditioning operator was the Laplace operator that is inverted using the variable separation method. The novelty of the proposed algorithm is using the Dichotomy Algorithm [26], which was designed for solving a series of tridiagonal systems of linear equations, in the context of the preconditioning operator inversion. Via considering analytical solutions, it is shown that modeling wave processes for long instants of time requires high-resolution meshes. The proposed parallel fine-mesh algorithm enabled to solve real application seismic problems in acceptable time and with high accuracy. By solving model problems, it is demonstrated that the considered parallel algorithm possesses high performance and efficiency over a wide range of the number of processors (from 2 to 8192).     

基于参数展开的同伦分析法在强非线性随机动力系统中的应用

将基于参数展开的同伦分析法(PE-HAM)进行了推广，使之适用于谐和激励与随机噪声联合作用下的强非线性随机动力系统. 通过构造合适的同伦映射，将对强非线性随机动力系统响应的求解转化为对一组线性随机微分方程的求解. 进一步研究了受到谐和与Gauss白噪声激励的强非线性Duffing振子，由PE-HAM得到了该系统的解过程和稳态概率密度的解析表达式. 数值模拟的结果说明了PE-HAM方法的精确性. 关键词： PE-HAM方法 强非线性随机动力系统 稳态概率密度 解过程 随机激励     

Analytical Solutions for the Two-Dimensional Gross-Pitaevskii Equation with a Harmonic Trap

Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation,a nonlinear Schrdinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential.The approximate analytical solutions are obtained successfully.Comparisons between the analytical solutions and the numerical solutions have been made.The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.     

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.     

Analytic Approximations for Soliton Solutions of Short-Wave Models for Camassa-Holm and Degasperis-Procesi Equations

In this paper, the short-wave model equations are investigated, which are associated with the Camassa- Holm （CH） and Degasperis Procesi （DP） shallow-water wave equations. Firstly, by means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformatioas back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems.     

Approximate solution of two-term fractional-order diffusion,wave-diffusion,and telegraph models arising in mathematical physics using optimal homotopy asymptotic method

This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems.     

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.