An adaptive method of lines solution of the Korteweg-de Vries equation

Following a method of lines formulation, the Korteweg-de Vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle, which allows the number of nodes to be significantly reduced as compared to a fixed-grid solution. Several finite difference schemes, including direct and stagewise procedures, are compared and the results of a large number of computational experiments are presented, which demonstrate that the selection of a spatial approximation scheme for the third-order derivative term is the primary determinant of solution accuracy.     

Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.     

Double ultraspherical solution of eigenvalue problems for partial differential equations

Particular spectral method based on an expansion in double ultraspherical polynomials is proposed to solve a number of eigenvalue problems defined by partial differential equations with constant and variable coefficients.We obtain comparable results with those computed by Liu and Oritz [14] by using Tau-Lines method, El-Hawary [8] by using El-Gendi-Lines method, El-Hawary [7] by using double Chebyshev approximation and Brandt et al.[1] by using multigrid method.     

全局正定径向基函数在图像分割中的应用

将全局正定径向基函数和图像分割中基于偏微分方程水平集方法的发展方程相结合,提出了一种基于全局正定径向基函数的图像分割算法。用全局正定径向基函数插值发展方程中的水平集函数,得到的插值函数具有较高的精度和光滑性,克服了传统水平集方法中复杂费时的重新初始化过程和水平集对初始轮廓位置敏感等缺点,非线性发展方程最终被转化成常微分方程组并用Euler法求解。实验结果表明该算法不需要重新初始化过程,并且在没有初始轮廓时也能够快速正确地分割图像。     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

Prediction of magnetic field near power lines by normalized radial basis function network

Over the past several decades, concerns have been raised over the possibility that the exposure to extremely low frequency electromagnetic fields from power lines may have harmful effects on human and living organisms. This work involved the computation of the magnetic field generated by 110 kV overhead power lines using a normalized radial basis function (NRBF) network. Training of the evolving NRBF network is achieved by using the data generated from the numerical simulation based on Charge Simulation method (CSM). Then, NRBF has been used to determine the magnetic field distribution in a new geometry differing from the geometries used for training. These test results show that proposed NRBF network can be used as useful tool to calculate the magnetic fields from power lines, alternative to the conventional methods.     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.     

结合权余量和粒子群方法求解变分优化问题

鉴于对变分问题的研究,发现了可以将微分方程与变分问题进行转化的原理,将求解微分方程的权余量方法与标准粒子群算法进行结合,旨在找到一种求解变分优化问题的较好算法。在权余量方法的计算中,把方程组转化为求近似目标函数的极小值的变分问题,接着用标准粒子群算法对其求解。对于因变量的幂为高次时的变分问题,可以直接用改进的算法对其求解。对于因变量的幂为一次时的变分问题,需要对等式左右分别平方后,再转化为求极小值的变分问题并求解。用一个变分问题的实例进行检验,发现改进后的算法容易,计算过程简单,结果较好。对近似目标函数的基的个数取三四个进行比较,对于具体问题,应该选取较好的适应实际问题的次数,得到较好的结果。从理论上分析,该算法具有可行性与一定的实际意义。     

非线性l-1模极小化问题的极大熵差分进化算法

针对一类非线性l-1模极小化问题目标函数非光滑的特点给求解带来的困难,利用差分进化算法并结合极大熵函数法给出了解决此类问题的一种有效算法。利用极大熵函数将l-1模极小化问题转化为一个光滑函数的无约束最优化问题,利用差分进化算法对其进行求解。实验结果表明,该方法是有效的。     

Robust adaptive boundary control of a flexible marine riser with vessel dynamics

In this paper, robust adaptive boundary control for a flexible marine riser with vessel dynamics is developed to suppress the riser’s vibration. To provide an accurate and concise representation of the riser’s dynamic behavior, the flexible marine riser with vessel dynamics is described by a distributed parameter system with a partial differential equation (PDE) and four ordinary differential equations (ODEs). Boundary control is proposed at the top boundary of the riser based on Lyapunov’s direct method to regulate the riser’s vibration. Adaptive control is designed when the system parametric uncertainty exists. With the proposed robust adaptive boundary control, uniform boundedness under the ocean current disturbance can be achieved. The proposed control is implementable with actual instrumentation since all the required signals in the control can be measured by sensors or calculated by a backward difference algorithm. The state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Simulations are provided to illustrate the applicability and effectiveness of the proposed control.     

A general approach to hyperbolic partial differential equations by homotopy perturbation method

The hyperbolic partial differential equations (PDEs) have a wide range of applications in science and engineering. In this article, the exact solutions of some hyperbolic PDEs are presented by means of He's homotopy perturbation method (HPM). The results reveal that the HPM is very effective and convenient in solving nonlinear problems.     

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.     

Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation

We introduce semi-implicit complementary volume numerical scheme for solving the level setformulation of Riemannian mean curvature flow problem arising in image segmentation, edge detection, missing boundary completion and subjective contour extraction. The scheme is robust and efficient since it is linear, and it is stable in L_∞ and weighted W ^1,1 sense without any restriction on a time step. The computational results related to medical image segmentation with partly missing boundaries and subjective contours extraction are presented.     

Analysis of inhomogeneous anisotropic viscoelastic bodies described by multi-parameter fractional differential constitutive models

The response to static loads of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. Multi-parameter differential viscoelastic constitutive equations are employed, which are generalized using fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacement components. Using the Analog Equation Method (AEM) in conjunction with the Boundary Element Method (BEM) these equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using a numerical method for FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the response of plane bodies described by different viscoelastic models.     

Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics

In this paper we propose a two-grid quasilinearization method for solving high order nonlinear differential equations. In the first step, the nonlinear boundary value problem is discretized on a coarse grid of size H. In the second step, the nonlinear problem is linearized around an interpolant of the computed solution (which serves as an initial guess of the quasilinearization process) at the first step. Thus, the linear problem is solved on a fine mesh of size h, h?H. On this base we develop two-grid iteration algorithms, that achieve optimal accuracy as long as the mesh size satisfies h=O(Hr²), r=1,2,… , where r is the rth Newton's iteration for the linearized differential problem. Numerical experiments show that a large class of NODEs, including the Fisher-Kolmogorov, Blasius and Emden-Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations.     

On control and optimization of elastic multilink mechanisms

This paper presents a simple method for control of nonlinear elastic multilink mechanisms. The associated control law consists of open and closed loop components. The open loop component produces the desired overall rigid body motion, while the closed loop component suppresses the elastic motion relative to rigid body motion. Both the control law and the structural dynamics are referred to an inertial coordinate system. As a consequence, the control forces and control moments are easily simulated in a dynamic finite element analysis program. A series of numerical simulations illustrate the generality of this new method.     

Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors. This work was supported by DARPA/AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425,the Singapore-MIT Alliance,the Pappalardo MIT Mechanical Engineering Graduate Monograph Fund,and the Progetto Roberto Rocca Politecnico di Milano-MIT.We acknowledge many helpful discussions with Professor Yvon Maday of University Paris6.