一类二维对偶积分方程的解及其应用

二维对偶积分方程的理论与方法,在数学上尚未建立,因而完全的分析解不可能得到,从而使一些力学、物理与工程问题无法求解．利用双重展开和边界配置方法,得到了在数学和物理学上有着广泛应用的一类二维对偶积分方程的解答．把二维对偶积分方程化简成无限代数方程组,此方法的精确度取决于计算点的配置（即所谓边界配置）．通过对固体力学中某些复杂的初值-边值问题的应用说明此是方法有效的．     

Efficient mesh selection for collocation methods applied to singular BVPs

We describe a mesh selection strategy for the numerical solution of boundary value problems for singular ordinary differential equations. This mesh adaptation procedure is implemented in our MATLAB code sbvp which is based on polynomial collocation. We prove that under realistic assumptions our mesh selection strategy serves to approximately equidistribute the global error of the collocation solution, thus enabling to reach prescribed tolerances efficiently. Moreover, we demonstrate that this strategy yields a favorable performance of the code and compare its computational effort with other implementations of polynomial collocation.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

自适应快速多极正则化无网格法求解大规模三维位势问题

正则化无网格法（regularized meshless method, RMM）是一种新的边界型无网格数值离散方法．该方法克服了近年来引起广泛关注的基本解方法（method of fundamental solutions, MFS）的虚假边界缺陷,继承了其无网格、无数值积分、易实施等优点．另一方面,RMM方法同MFS方法的插值方程都涉及非对称稠密系数矩阵,运用常规代数方程的迭代法求解时都要求O(N2)量级的乘法计算量和存储量．随着问题自由度的增加,该方法的计算量增加极快,效率较低,一般难以计算大规模问题．为了克服这个缺点,利用对角形式的快速多级算法（fast multipole method, FMM）来加速RMM方法,发展了快速多级正则化无网格法（fast multipole regularized mesheless method, FM-RMM）．该方法无需数值积分并且具有O(N)量级的计算量和存储量,可有效地求解大规模工程问题．数值算例表明,FM-RMM算法可成功在内存为4GB的Core(TM)Ⅱ台式机上求解高达百万级自由度的三维位势问题．     

On the finite element approximation for maxwell’s problem in polynomial domains of the plane

The time-harmonic Maxwell boundary value problem in polygonal domains of R² is considered. The behaviour of the solution in the neighbourhood of nonregular boundary points is given and asymptotic error estimates in L₂- and in curl-div-norm for a finite element approximation of the solution are derived     

Applications of RBF collocation method to elliptic PDEs in an arbitrary domain by fictitious domain extensions

A fictitious domain extension approach is introduced to study elliptic PDE's defined in arbitrary domains by the radial basis function (RBF) collocation method. In this approach, arbitrary physical geometries are extended to domains which are topologically rectangular. The solution domain is also extended to the fictitious area and assumed to satisfy the same governing equation in it and on its extended boundaries. The boundary conditions are still specified on the boundaries of the original physical domain. The problem in the extended domain becomes ill-posed. However, it can be easily circumvented by the collocation method. We demonstrate that the solution can be directly obtained without domain decompositions and iterations. The new approach is simple, efficient and accurate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient numerical solution of the generalized Dirichlet-Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.     

Properties of discrete Chebyshev collocation differential operators in curvilinear geometries

The properties of discrete systems resulting from spectral Chebyshev collocation discretizations are investigated with respect to the solution efficiency of corresponding solvers. Complex geometries are encountered by a mapping technique to connect computational and physical domains. Several representative transformation techniques are considered. The influences of the differential operators, the boundary conditions, the geometry, and the number of grid points are systematically studied. The convergence properties of the BiCGSTAB method when iteratively solving the discrete systems are investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation

In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor-corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations.     

An adaptive boundary collocation method for linear PDEs

The article proposes an adaptive algorithm based on a boundary collocation method for linear PDEs satisfying the maximal principle with possibly nonlinear boundary conditions. Given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficients by collocation of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error tolerance, additional expansion terms and boundary collocation points are added and the process repeated until the tolerance is satisfied. The performance of the algorithm is illustrated by an example of the potential flow past a cylinder placed between parallel walls. © 1995 John Wiley & Sons, Inc.     

Boundary collocation method for a cracked rectangular plate with double external tension

In this article, the boundary collocation method is employed to investigate the problems of a central crack in a rectangular plate which applied double external tension on the outer boundary under the assumption that the dimensions of the plate are much larger than that of the crack. A set of stress functions has also been proposed based on the theoretical analysis which satisfies the condition that there is no external force on the crack surfaces. It is only necessary to consider the condition on the external boundary. Using boundary collocation method, the linear algebra equations at collocation points are obtained. The least squares method is used to obtain the solution of the equations, so that the unknown coefficients can be obtained. According to the expression of the stress intensity factor at crack tip, we can obtain the numerical results of stress intensity factor. Numerical experiments show that the results coincide with the exact solution of the infinite plate. In particular, this case of the double external tension applied on the outer boundary is seldom studied by boundary collocation method.     

A New Collocation Algorithm for Solving Even-Order Boundary Value Problems via a Novel Matrix Method

This paper is dedicated to presenting and analyzing a numerical algorithm for the solution of even-order boundary value problems. The proposed solutions are spectral and they depend on introducing a new matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the introduced matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm.     

Analytical solution for the fully coupled static response of variable stiffness composite beams

An analytical solution is presented for the 3D static response of variable stiffness non-uniform composite beams. Based on Euler-Bernoulli theory, a set of governing differential equations are obtained, in which four degrees of freedom are fully coupled. For the variable stiffness beam, the governing field equations have variable coefficients reflecting the stiffness variation along the beam. Using the direct integration technique, the general analytical solution is derived in the integral form and the closed-form expressions of the obtained solutions are presented employing a series expansion approximation. The series expansion representation enables the proposed approach to be applicable for variable stiffness composite beams with arbitrary span-wise variation of properties. As an alternative solution, the Chebyshev collocation method is applied to the proposed formulation to verify the results obtained from the analytical solution. A number of variable stiffness composite beams made by fibre steering with various boundary conditions and stacking sequences are considered as the test cases. The static response are presented based on the analytical solution and Chebyshev collocation method and excellent agreement is observed for all test cases. The proposed model presents a reliable and efficient approach for capturing the complicated behaviour of variable stiffness non-uniform composite beams.     

Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.     

The Linear Rational Pseudospectral Method for Boundary Value Problems

We consider the version of the pseudospectral method for solving boundary value problems which replaces the differential operator with a matrix constructed from the elementary differentiation matrices whose elements are the derivatives of the Lagrange fundamental polynomials at the collocation points. The iterative solution of the resulting system of equations then requires the recurrent application of that differentiation matrix. Since global polynomial interpolation on the interval only gives useful approximants for points which accumulate in the vicinity of the extremities, the matrix is ill-conditioned. To reduce this drawback, we use Kosloff and Tal-Ezer's suggestion to shift the collocation points closer to equidistant by a conformal map. However, instead of applying their change of variable setting, we extend to stationary equations the linear rational collocation method introduced in former work on partial differential equations. Numerically about as efficient, this does not require any new coding if one starts from an efficient program for the polynomial differentiation matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Boundary value problem for second-order differential operators with integral conditions

In this article, we study a second-order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called regular and nonregular cases, we prove that the resolvent decreases with respect to the spectral parameter in L ^p ?(0,?1), but there is no maximal decreasing at infinity for p?>?1. Furthermore, the studied operator generates in L ^p ?(0,?1) an analytic semigroup for p?=?1 in regular case, and an analytic semigroup with singularities for p?>?1 in both cases, and for p?=?1 in the nonregular case only. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with nonregular boundary conditions.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

Least squares collocation solution of differential equations on irregularly shaped domains using orthogonal meshes

The least squares collocation (LESCO) method has been formulated to solve differential equations defined over irregular domains using a more convenient orthogonal computational mesh. The LESCO method is described in detail for second-order boundary value problems and applied to the time-dependent diffusion and advection-diffusion equations defined over two-dimensional irregular domains. Particular attention is given to the proper procedure for applying boundary conditions. Accuracy, convergence, and consistency are examined. For cubic elements with arbitrary location of collocation points, the convergence rate is between 3rd and 4th order. The major advantages of this method are reduced input data requirements, a more robust procedure for forming the equations, positive definite matrices, and flexibility in distrbuting errors.     

Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs

In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.