Solitary-wave solutions of the Degasperis–Procesi equation by means of the homotopy analysis method

The homotopy analysis method (HAM) is applied to the Degasperis–Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the HAM is a powerful tool for finding excellent approximations to nonlinear solitary waves.     

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.     

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…     

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.     

Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.     

New analytical method for gas dynamics equation arising in shock fronts

This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving nonlinear homogeneous and nonhomogeneous time-fractional gas dynamics equations. The FHATM is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. 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 transform method. In this paper, it can be observed that the auxiliary parameter ?$?$, which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more qualitative difference in analysis between HATM and other methods. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving some numerical examples.     

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.     

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.     

A restarted iterative homotopy analysis method for two nonlinear models from image processing

Total variation (TV) minimization-based nonlinear models have been proven to be very useful and successful in image processing. A lot of effort has been devoted to overcome the nonlinearity of the model and at the same time to obtain fast numerical schemes. In this paper, we propose a restarted iterative homotopy analysis method (HAM) to improve the computational efficiency for the TV models and will show by experiments that this method demonstrates great potential for recovering the noise and with great speed in both image denoising and image segmentation models. The method modifies the existing HAM and makes it suitable to potentially solve other nonlinear partial differential equations arising from image processing models. In our examples, we will demonstrate the validity of a restarted HAM and that this method is efficient and robust even for images with large ratios of noise and with much less CPU time than other methods.     

An algorithm for solving multi-term diffusion-wave equations of fractional order

In this paper an algorithm, based on a new modified homotopy perturbation method (MHPM), is presented to obtain approximate solutions of multi-term diffusion-wave equations of fractional order. To illustrate the method some examples are provided. The results show the simplicity and the efficiency of the algorithm.     

Propagation of non-linear waves in a non-ideal relaxing gas

We have derived an evolution equation governing the far-field behaviour of small amplitude waves in a non-ideal relaxing gas for planar and converging flow. Asymptotic expansions of the flow variables for small amplitude waves have been used to derive the evolution equation. This equation turns out to be a generalized Burger's equation. The numerical solution of this equation is obtained by using the homotopy analysis method (HAM) proposed by Liao with two different initial conditions. Using the HAM, we have studied the effect of relaxation and nonlinearity. The convergence control parameter enables us to find a good approximate solution for such a complex flow problem. This method also confirms the capabilities and usefulness of convergence control parameter and HAM for complex and highly non-linear problems.     

An analytic algorithm for the space–time fractional advection–dispersion equation

Fractional advection–dispersion equation (FADE) is a generalization of the classical ADE in which the first order time derivative and first and second order space derivatives are replaced by Caputo derivatives of orders 0<α?1, 0<β?1 and 1<γ?2, respectively. We use Caputo definition to avoid (i) mass balance error, (ii) hyper-singular improper integral, (iii) non-zero derivative of constant, and (iv) fractional derivative involved in the initial condition which is often ill-defined. We present an analytic algorithm to solve FADE based on homotopy analysis method (HAM) which has the advantage of controlling the region and rate of convergence of the solution series via the auxiliary parameter ? over the variational iteration method (VIM) and homotopy perturbation method (HPM). We find that the proposed method converges to the numerical/exact solution of the ADE as the fractional orders α, β, γ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. Example 5 describes the intermediate process between advection and dispersion via Caputo fractional derivative.     

分数阶非线性Duffing振子方程的特性研究

将Caputo分数阶微分算子引入到非线性的Duffing振子方程中,运用同伦扰动变换法--一种同伦扰动法和Laplace变换相结合的方法来求解分数阶的非线性方程,借助Mathematica软件的符号计算功能得到了分数阶非线性Duffing振子方程的近似解,研究了振子运动过程与分数阶导数之间的关系。     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

同伦扰动法求解分数阶非线性扩散方程近似解

将Caputo分数阶微分算子引入到带有初值条件的扩散方程中,建立了时空分数阶方程。利用同伦扰动法并借助于Mathematica软件的符号计算功能,求解了分数阶非线性扩散方程的近似解,整数阶方程的结果作为特例被包含。     

Numerical comparison of methods for solving fractional differential–algebraic equations (FDAEs)

In this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are implemented to give approximate solutions for fractional differential–algebraic equations (FDAEs). Both methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. This paper presents a numerical comparison between these two methods and the homotopy analysis method (HAM) for solving FDAEs. Numerical results reveal that the VIM and the ADM are quite accurate and applicable.     

一种改进的同伦算法与H∞鲁棒控制器设计

提出了一种具有阶次限制的鲁棒控制器设计方法, 该算法将控制系统的性能指标转化为灵敏度函数问题, 并利用Nevanlinna-Pick插值算法进行求解. 提出了一种改进的同伦算法, 将其用于求解由灵敏度函数产生的非线性方程. 基于改进同伦算法设计的鲁棒控制器 不仅避免了传统H∞控制中加权函数的选择问题, 而且克服了鲁棒控制器阶次较高的缺陷. 最后,文章以4阶系统为例, 设计了具有阶次限制的H∞鲁棒控制器, 通过与传统鲁棒控制器的比较可以看出, 基于本文方法设计的控制器不仅具有较低的阶次, 而且其控制性能也具有明显的优越性.     

The quadratic method for computing the eigenpairs of a martix

In this paper we introduce a new method for computing the eigenpairs of a matrix. The quadratic method which has the advantage of working in parallel, is based on solving quadratic nonlinear systems. The starting values of Newton's method used to solve the systems, are obtained by using a homotopy method together with polynomial interpolation. The algorithm is described and several numerical examples are given.     

Homotopy analysis method for higher-order fractional integro-differential equations

In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM with the exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of series solution.