面向欠约束几何系统的一种同伦求解方法

针对几何约束系统的数值求解过程中，经常发生的数值不稳定性问题，构造了一种面向欠约束系统的同伦方法，并将其与现有的求解与分解方法有机地结合起来，提出了一种牛顿－同伦混合方法，在牛顿迭代失败的位置自动调用欠约束同伦法，既提高了几何约束求解器的效率，同时又保证了求解的效率。     

改进同伦分析方法及推广Kuramoto Sivashinsky方程的近似解

首先介绍了带有两个辅助参数的改进同伦分析方法,然后用该方法得到了推广Kuramoto-Sivashin-sky方程的同伦近似解.所得近似解与精确孤立波解进行比较,发现本文得到的近似解更有效地逼近真实解.因为该解包含了两个辅助参数,所以能够更有效地调节和控制其收敛区域和速度.研究表明带有两个辅助参数的改进同伦分析方法对复杂非线性系统的研究更有它的优点.     

Approximate analytical solutions to thermo-poroelastic equations by means of the iterated homotopy analysis method

A new approach of the homotopy analysis method (HAM), named iterated homotopy analysis method (I-HAM) is used to find approximate analytical solutions to thermo-poroelastic equations. The homotopy analysis method contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of the series solution. This method is reliable and manageable. The I-HAM solutions are compared with numerical solutions. To the best of our knowledge, this kind of analytic solutions has been never reported. Also, in each example, the comparison between I-HAM and traditional HAM is done, which shows the efficiency of I-HAM.     

A new approach to modified regularized long wave equation

In this paper, a homotopy analysis method employed to obtain approximate numerical solution of the modified regularized long wave (MRLW) equation with some specified initial conditions. The results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials, which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation method. Numerical results are presented graphically, and tabular form reveals that the homotopy analysis method is an effective and convenient method to solve MRLW equation.     

A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem

In this paper a novel hybrid spectral-homotopy analysis technique developed by Motsa et al. (2009) and the homotopy analysis method (HAM) are compared through the solution of the nonlinear equation for the MHD Jeffery-Hamel problem. An analytical solution is obtained using the homotopy analysis method (HAM) and compared with the numerical results and those obtained using the new hybrid method. The results show that the spectral-homotopy analysis technique converges at least twice as fast as the standard homotopy analysis method.     

Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method

In this study, the homotopy perturbation method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth-order integro-differential equations. The analysis is accompanied by numerical examples. The results show that the homotopy perturbation method is of high accuracy, more convenient and efficient for solving integro-differential equations.     

Robustness of declarative modeling languages: Improvements via probability-one homotopy

Robustness issues with steady-state initialization remain a barrier in the practical use of declarative modeling languages for multi-domain modeling of large, complex, and heterogeneous technical systems. The objective of this paper is to illustrate how probability-one homotopy, an established method from topology, can solve this issue. This is achieved by establishing a framework for application-specific probability-one homotopy in declarative modeling languages. The analysis is based on domain-specific probability-one homotopy maps, which were reformulated in a declarative fashion. Additionally, a novel probability-one homotopy map and associated coercivity proof is introduced for a class of thermo-fluid dynamics problems. It was found that the approach enables robust initialization for declarative modeling languages on several test cases and leads to a concise declarative problem formulation.     

Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation

In this paper, the approximate analytical solutions of the mathematical model of vibration equation with fractional-order time derivative β (1<β≤2) for very large membranes are obtained with the help of powerful mathematical tools like homotopy perturbation method and homotopy analysis method. Both the methods perform extremely well in terms of efficiency and simplicity. The validity and applicability of the techniques are shown for obtaining approximate numerical solutions for different particular cases which are presented through figures and tables.     

Homotopy perturbation method for solving sixth-order boundary value problems

In this paper, we apply the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations. This equivalent formulation is obtained by using a suitable transformation. The analytical results of the integral equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method. Comparisons are made to confirm the reliability of the homotopy perturbation method. We have also considered an example where the homotopy perturbation method is not reliable.     

用近似同伦法确定空间7H连杆机构的装配构形

对于含螺旋副（Ｈ）的空间连杆机构或机器人机构,无法用有理化法将其分析与综合方程组化成多项式方程组,因而不能用精确同伦法求解这些机构的多解问题．本文提出近似同伦法并用该法首次解决了空间７Ｈ连杆机构的装配构形问题．本文方法适用于求解任何含Ｈ副的空间连杆机构或机器人机构的多解问题．     

PHoMpara – Parallel Implementation of the Polyhedral Homotopy Continuation Method for Polynomial Systems

The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral homotopy functions, tracing the solution curves of the homotopy equations, and verifying the obtained solutions. A software package PHoMpara parallelizes PHoM to solve a polynomial system of large size. Many characteristics of the polyhedral homotopy continuation method make parallel implementation efficient and provide excellent scalability. Numerical results include some large polynomial systems that had not been solved.     

He’s homotopy perturbation method: A strongly promising method for solving non-linear systems of the mixed Volterra–Fredholm integral equations

In this paper, He’s homotopy perturbation method is applied to solve non-linear systems of mixed Volterra–Fredholm integral equations. Two examples are presented to illustrate the ability of the method. Also comparisons are made between the Adomian decomposition method and the homotopy perturbation method. The results reveal that He’s homotopy perturbation method is very effective and simple and in these examples leads to the exact solutions.     

The multistage homotopy analysis method: application to a biochemical reaction model of fractional order

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge–Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order.     

一种基于同伦函数的迭代法—同伦迭代法

§１．引言 工程中的许多问题常常最后可归结为求解一组非线性多项式代数方程．对非线性多项式方程组的求解,可采用符号求解和数值求解两种方式．符号求解可求出问题的封闭形式的解析解,当然是最理想的,但其难度往往也是很大的,随着问题数学模型的增大,消元过程变得愈加复杂,使得即使采用计算机也无法进行下去．因此,对于复杂的大规模问题,仍只能采用数值迭代法进行数值求解． 传统的数值迭代法存在的最大问题是方法的有效性依赖于初值的选取．初值选取不当常导致迭代过程不收敛,而且一次只能求出问题的一个数值解．山提出的区间分…     

Homotopy perturbation technique for solving two-point boundary value problems - comparison with other methods

The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.     

The homotopy model: a generalized model for smooth surface generation from cross sectional data

A generalized model, called the homotopy model, is presented to reconstruct surfaces from cross-sectional data of objects using a homotopy to generate surfaces connecting consecutive contours. The homotopy model consists of continuous toroidal graph representation and homotopic generation of surfaces from the representation. It is shown that the homotopy model includes triangulation as a special case and generates smooth parametric surfaces from contour-line definitions using homotopy. The model can be applied to contours represented by parametric curves as well as linear line segments. First, a heuristic method that finds the optimal path on the toroidal graph is presented. Then the toroidal graph is expanded to a continuous version. Finally, homotopy is used for reconstructing parametric surfaces from the toroidal graph representation. A loft surface is also a special case of homotopy, a straight-line homotopy. Homotopy that corresponds to the cardinal spline surface is also introduced. Three-dimensional surface reconstruction of human auditory surface reconstruction of human auditory ossicles illustrates the advantages of the homotopy model over the others.     

The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets

In this study, by means of homotopy perturbation method (HPM) an approximate analytical solution of the magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous stretching sheet 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 of 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 solution. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems. Also it is shown that this method coincides with homotopy analysis method (HAM) for the studied problem.     

Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.     

An analytic solution for the space–time fractional advection–dispersion equation using the optimal homotopy asymptotic method

We present an analytic algorithm to solve the space–time fractional advection–dispersion equation (FADE) based on the optimal homotopy asymptotic method (OHAM), which has the advantage of controlling the region and rate of convergence of the solution series via several auxiliary parameters over the traditional homotopy analysis method (HAM) having only one auxiliary parameter. Furthermore, our proposed algorithm gives better results compared to the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) in the sense that fewer iterations are required to get a sufficiently accurate solution and the solution has a greater radius of convergence. We find that the iterations obtained by the proposed method converge to the numerical/exact solution of the ADE as the fractional orders $\alpha ,\beta ,\gamma$ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. The figures and tables show the superiority of the OHAM over the HAM.