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.     

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.     

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.     

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.     

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.     

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.