Numerical method for non‐linear wave and diffusion equations by the variational iteration method

This paper applies He's variational iteration to the wave equations in an infinite one‐dimensional medium and some non‐linear diffusion equations. A suitable choice of an initial solution can lead to the needed exact solution by a few iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

A comparison of two Trefftz-type methods: the ultraweak variational formulation and the least-squares method,for solving shortwave 2-D Helmholtz problems

Trefftz methods for the numerical solution of partial differential equations (PDEs) on a given domain involve trial functions which are defined in subdomains, are generally discontinuous, and are solutions of the governing PDE (or its adjoint) within each subdomain. The boundary conditions and matching conditions between subdomains must be enforced separately. An interesting novel result presented in this paper is that the least-squares method (LSM) and the ultraweak variational formulation, two methods already established for solving the Helmholtz equation, can be derived in the framework of the Trefftz-type methods. In the first case, the boundary conditions and interelement continuity are enforced by means of a least-squares procedure. In the second, a Galerkin-type weighted residual method is used. Another goal of the work is to assess the relative efficiency of each method for solving shortwave problems in acoustics and to study the stability of each method. The numerical performance of each scheme is assessed with reference to two 2-D test problems; acoustic propagation in an uniform soft-walled duct, and propagation in an L-shaped domain, the latter involving singular behaviour at a sharp corner. Copyright © 2006 John Wiley & Sons, Ltd.     

A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: the Dirac Coulomb type-equation

Exact eigenspectra and eigenfunctions of the Dirac quantum equation are established using the semi-inverse variational method. This method improves of a considerable manner the efficiency and accuracy of results compared with the other usual methods much argued in the literature. Some applications for different state configurations are proposed to concretize the method.     

Evaluation of the T-stress and stress intensity factor for a cracked plate in general case using eigenfunction expansion variational method

This paper investigates the T-stress and stress intensity factor for a cracked plate in general case. In the general case, the shape of boundary and the applied loading are arbitrary. The eigenfunction expansion variational method (EEVM) is developed to evaluate the T-stress and stress intensity factor. For the traction boundary value problem, the EEVM is equivalent to the theorem of least potential energy in elasticity. Therefore, the EEVM possesses a clear physical meaning and it does not depend on any boundary collocation scheme. Several numerical examples are presented, which include: (1) a line crack in circular plate and (2) a line crack in rectangular plate. Numerical examination for convergence in an example is carried out.     

相干结构中特征值积分方程求解的一种新方法

相干结构在湍流研究中起着非常重要的作用。文中采用最优正交分解(POD)定义相干结构,将其表示为使得内能极大的空间变量的函数。将极大值问题转化求解特征值积分方程。利用变分原理和Galerkin方法的思想进行求解。文中的算例说明算法是有效可行的。算法可用于流体的优化控制和高效率算法的构造。     

Overview of the discontinuous enrichment method,the ultra‐weak variational formulation,and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons

The efficient finite element discretization of the Helmholtz equation becomes challenging in the medium frequency regime because of numerical dispersion, or what is often referred to in the literature as the pollution effect. A number of FEMs with plane wave basis functions have been proposed to alleviate this effect, and improve on the unsatisfactory preasymptotic convergence of the polynomial FEM. These include the partition of unity method, the ultra‐weak variational formulation, and the discontinuous enrichment method. A previous comparative study of the performance of such methods focused on the first two aforementioned methods only. By contrast, this paper provides an overview of all three methods and compares several aspects of their performance for an acoustic scattering benchmark problem in the medium frequency regime. It is found that the discontinuous enrichment method outperforms both the partition of unity method and the ultra‐weak variational formulation by a significant margin. Copyright © 2011 John Wiley & Sons, Ltd.     

A homotopy analysis method for the nonlinear partial differential equations arising in engineering

In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.