An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

Modified Variational Iteration Method for Heat and Wave-Like Equations

In this paper, we apply the modified variational iteration method (MVIM) for solving the heat and wave-like equations. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that proposed technique solves nonlinear problems without using the Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

Shift-and-Invert Krylov Methods for Time-Fractional Wave Equations

The article deals with evolution problems involving time derivatives of fractional order α, with 1 < α ≤2. The solutions are expressed in terms of operator Mittag-Leffler functions. The action of such operator functions is approximated by rational Krylov methods whose convergence features are investigated.     

Improving Variational Iteration Method with Auxiliary Parameter for Nonlinear Time-Fractional Partial Differential Equations

In this research, we present a new approach based on variational iteration method for solving nonlinear time-fractional partial differential equations in large domains. The convergence of the method is shown with the aid of Banach fixed point theorem. The maximum error bound is specified. The optimal value of auxiliary parameter is obtained by use of residual error function. The fractional derivatives are taken in the Caputo sense. Numerical examples that involve the time-fractional Burgers equation, the time-fractional fifth-order Korteweg–de Vries equation and the time-fractional Fornberg–Whitham equation are examined to show the appropriate properties of the method. The results reveal that a new approach is very effective and convenient.     

Legendre Wavelets Direct Method for the Numerical Solution of Time-Fractional Order Telegraph Equations

In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann–Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.     

Symmetry Analysis of Variable-Coefficient Time-Fractional Nonlinear Systems of Partial Differential Equations

Theoretical and Mathematical Physics - We investigate some well-known variable-coefficient time-fractional nonlinear systems of partial differential equations using the Lie symmetry method and...     

Homotopy Iterative Methods for Polynomial Equations

Globally convergent iterative methods for polynomial equationsf(z)= 0 are obtained. They are derived by modifying iterative functionssuch as Newton, Steffensen, etc., so as to avoid the criticalpoints and ensure the convergence. The modification is realized by the use of a one-parameter embeddingoperator (homotopy) associated with the polynomial.     

The Homotopy Perturbation Method and the Adomian Decomposition Method for the Nonlinear Coupled Equations

In this paper, we have used the homotopy perturbation and the Adomian decomposition methods to study the nonlinear coupled Kortewge-de Vries and shallow water equations. The main objective of this paper is to propose alternative methods of solutions, which do not require small parameters and avoid linearization and physical unrealistic assumptions. The proposed methods give more general exact solutions without much extra effort and the results reveal that the homotopy perturbation and the Adomian decomposition methods are very effective, convenient and quite accurate to the systems of coupled nonlinear equations.     

Homotopy Analysis Method for Solving(2+1)-dimensional Navier-Stokes Equations with Perturbation Terms

In this paper Homotopy Analysis Method (HAM) is implemented for obtaining approximate solutions of (2+1)-dimensional Navier-Stokes equations with perturbation terms. The initial approximations are obtained using linear systems of the Navier-Stokes equations;by the iterations formula of HAM,the first approxima-tion solutions and the second approximation solutions are successively obtained and Homotopy Perturbation Method(HPM)is also used to solve these equations;finally, approximate solutions by HAM of (2+1)-dimensional Navier-Stokes equations with-out perturbation terms and with perturbation terms are compared. Because of the freedom of choice the auxiliary parameter of HAM,the results demonstrate that the rapid convergence and the high accuracy of the HAM in solving Navier-Stokes equa-tions;due to the effects of perturbation terms,the 3rd-order approximation solutions by HAM and HPM have great fluctuation.     

The Maximum Principle for Time-Fractional Diffusion Equations and Its Application

Using an equivalent reformulation of the definition of the Caputo fractional derivative, we present the stability and convergence analysis for a finite difference scheme that we use to solve a time-fractional diffusion equation. The analysis is based on a maximum principle for such equations.     

The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models

This paper presents comparison between Homotopy Analysis Method (HAM) and Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear age-structured population models. Three examples have been presented to illustrate and compare these methods. In OHAM the convergence region can be easily adjusted and controlled. Comparison between our solution and the exact solution shows that the both methods are effective and accurate in solving nonlinear age-structured population models with HAM being the more accurate for the same number of terms. It was also found that OHAM require more CPU time.     

Finding Periodic Solutions of High Order Duffing Equations via Homotopy Method

This paper presents a detailed analysis of finding the periodic solutions for the high order Duffing equation x^（2n） ＋ g（x） = e（t） （n ≥ 1）. Firstly, we give a constructive proof for the existence of periodic solutions via the homotopy method. Then we establish an efficient and global convergence method to find periodic solutions numerically.     

求常微分方程周期解的同伦方法

As an important aspect of applications, it is discussed how to find periodic solutions for ordinary differential equations. By using the homotopy method, a global method for finding those solutions is proposed.     

应用同伦方法的奇数阶常微分方程周期解的存在性

This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form     

同伦分析方法：一种新的求解非线性问题的近似解析方法

本文描述了一种称为“同伦分析方法”（ＨＡＭ）的新的求解非线性问题的近似解析方法之基本思想·不同于摄动展开方法，“同伦分析方法”的有效性不依赖于所研究的非线性方程中是否含有小参数·因此，该方法提供了一个强有力的分析非线性问题的新工具·作为示例，我们应用一个典型的非线性问题来说明该方法的有效性及其巨大潜力·     

Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations

In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are applied to solve nonlinear oscillator differential equations. Illustrative examples reveal that these methods are very effective and convenient for solving nonlinear differential equations. Moreover, the methods do not require linearization or small perturbation. Comparisons are also made between the exact solutions and the results of the homotopy perturbation method and variational iteration method in order to prove the precision of the results obtained from both methods mentioned.     

Homotopy Method for Variational Inequalities

Solving a finite-dimensional variational inequality is to find a vector x ∈X Rn such that(x - x)TF(x) ≥0, x ∈X, (1)where X is a nonempty, closed and convex subset of Rn and F is a mapping from Rn to itself,denoted by Ⅵ(X, F).The variational inequality problem (VIP) has had many successful practical applications inthe last three decades. It has been used to formulate and investigate equilibrium models arisingin economics, transportation, regional science and operations research. So fa…     

一类海-气振子ENSO模型的同伦分析解法

研究了一个ENSO海-气时滞振子模型.利用同伦分析方法,得到了该模型解的近似展开式,通过与特殊情况下的精确解比较,得到的二级近似解具有较高的精度.