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The optimal free knot spline approximation of stochastic differential equations with additive noise  [PDF]
Mehdi Slassi
Mathematics , 2013,
Abstract: In this paper we analyse the pathwise approximation of stochastic differential equations by polynomial splines with free knots. The pathwise distance between the solution and its approximation is measured globally on the unit interval in the $L_{\infty}$-norm, and we study the expectation of this distance. For equations with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary spline approximation methods, which use $k$ free knots. The optimal order is achieved by an approximation method $\hat{X}_{k}^{\dagger}$, which combines an Euler scheme on a coarse grid with an optimal spline approximation of the Brownian motion $W$ with $k$ free knots.
Characterization theorem for best polynomial spline approximation with free knots  [PDF]
Nadezda Sukhorukova,Julien Ugon
Mathematics , 2014,
Abstract: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a special property of the knots. Then, using this property, we construct a characterization theorem for best free knots polynomial spline approximation, which is stronger than the existing characterisation results when only continuity is required.
Free-Knot Spline Approximation of Stochastic Processes  [PDF]
J. Creutzig,T. Mueller-Gronbach,K. Ritter
Mathematics , 2006,
Abstract: We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the $s$-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average $L_p$-distance to the splines spaces, as the (expected) number $k$ of free knots tends to infinity.
Free Vibration of Layered Circular Cylindrical Shells of Variable Thickness Using Spline Function Approximation  [PDF]
K. K. Viswanathan,Kyung Su Kim,Kyung Ho Lee,Jang Hyun Lee
Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/547956
Abstract: Free vibration of layered circular cylindrical shells of variable thickness is studied using spline function approximation by applying a point collocation method. The shell is made up of uniform layers of isotropic or specially orthotropic materials. The equations of motions in longitudinal, circumferential and transverse displacement components, are derived using extension of Love's first approximation theory. The coupled differential equations are solved using Bickley-type splines of suitable order, which are cubic and quintic, by applying the point collocation method. This results in the generalized eigenvalue problem by combining the suitable boundary conditions. The effect of frequency parameters and the corresponding mode shapes of vibration are studied with different thickness variation coefficients, and other parameters. The thickness variations are assumed to be linear, exponential, and sinusoidal along the axial direction. The results are given graphically and comparisons are made with those results obtained using finite element method. 1. Introduction Circular cylindrical shells are used in various fields like aviation, missiles, ship buildings, and chemical industries. Shells made of composite materials with variable thickness are used increasingly, since composite structures are having high specific stiffness, better damping, and shock absorbing characters over the homogeneous ones. The study of vibrational behavior of such shells is very important. The effect of variation of thickness on frequency parameter of the shell, which is made up of different layered materials, has been studied by very few researchers. Baker and Herrmann [1] analysed three layered (Sandwich) shells, including the effects of shear deformation, rotary inertia, and initial stress. Sivadas and Ganesan [2] studied the vibration of circular cylindrical shells having the thickness variations of linear and quadratic along the axial direction. A series of studies has been made on vibration of cylindrical shells by Tonin and Bies [3], Takahashi et al. [4], Suzuki et al. [5] and Sivadas and Ganesan [6]. Hinton et al. [7] presented free vibration analysis of variable thickness of plates and curved shells using a finite strip formulation. The fundamental frequencies of laminated anisotropic circular cylindrical shells are presented by Sun et al. [8] using finite element method (FEM). Zhang [9] used a propagation approach to analyse the cross-ply laminated composite cylindrical shells. Hufenbach et al. [10] presented a study on vibration and damping behaviour of multilayered
Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation  [PDF]
Vincent Duchene
Mathematics , 2010, DOI: 10.1051/m2an/2011037
Abstract: We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one dimensional waves, and consider the case of a flat bottom. Starting from the classical Boussinesq/Boussinesq system, we introduce a new family of equivalent symmetric hyperbolic systems. We study the well-posedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, and the precise behavior of the KdV approximations depending on the depth and density ratios is discussed for both rigid lid and free surface configurations. The fact that we obtain {\it simultaneously} the four KdV equations allows us to study extensively the influence of the rigid lid assumption on the evolution of the interface, and therefore its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are rigorously compared and numerically computed.
Composite Scattering of Ship on Sea Surface with Breaking Waves
Min Zhang;Wei Luo;Gen Luo;Chao Wang;Hong-Cheng Yin
PIER , 2012, DOI: 10.2528/PIER11100811
Abstract: The composite backscattering of the ship model on sea surface is investigated with the spilling breaking waves and ship bow waves. The spilling breakers are approximately modeled with the wedge-like waves, and the ship bow waves are simulated based on the Kelvin model. With the modified four-path model, each scattering component is evaluated with the high frequency approximation methods for the total composite scattering. Due to the volume scattering, the composite scattering at large incidence angles is strongly enhanced by the non-Bragg scattering. The relationship of the composite scattering and the ship motion is analyzed. The numerical results of sea surface scattering agree with the measured data well, and the complex physical mechanism of the low-grazing-angle composite scattering is explicitly evaluated in this paper.
Fundamental solution of the problem describing ship motion in waves
Jan Jankowski
Opuscula Mathematica , 2006,
Abstract: The problem describing a ship motion in waves comprises the Laplace equation, boundary condition on wetted surface of the ship, condition on the free surface of the sea in the form of a differential equation, the radiation condition, and a condition at infinity. This problem can be transformed to a Fredholm equation of second kind, and then numerically solved using the boundary element method, if the fundamental solution of the problem is known. This paper presents the derivation of the fundamental solution. In physical interpretation, the fundamental solution represents the moving and pulsating source under free surface of the sea. The free surface elevation, generated by the source for different forward speed and frequency of pulsation, is presented in this paper.
Ship waves in the presence of uniform vorticity  [PDF]
Simen ?. Ellingsen
Physics , 2014, DOI: 10.1017/jfm.2014.28
Abstract: Lord Kelvin's result that waves behind a ship lie within a half-angle 19 deg 28' is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity S, and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the "shear Froude number" Frs=VS/g (based on length g/S^2 and the ship's speed V), and on the angle between current and the ship's line of motion. In all directions except exactly along the shear flow there exists a critical value of Frs beyond which no transverse waves are produced, and where the full wake angle reaches 180 deg. Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed 90 deg. On the other hand, the angle of maximum wave amplitude scales as 1/Fr (Fr based on size of ship) when Fr >> 1, a scaling virtually unaffected by the shear flow.
An asymptotic approximation to the free-surface elevation around a cylinder in long waves  [PDF]
Michael T. Morris-Thomas
Physics , 2013,
Abstract: Strong nonlinear effects are known to contribute to the wave run-up caused when a progressive wave impinges on a vertical surface piercing cylinder. The magnitude of the wave run-up is largely dependent on the coupling of the cylinder slenderness, $ka$, and wave steepness, $kA$, parameters. This present work proposes an analytical solution to the free-surface elevation around a circular cylinder in plane progressive waves. It is assumed throughout that the horizontal extent of the cylinder is much smaller than the incident wavelength and of the same order of magnitude as the incident wave amplitude. A perturbation expansion of the velocity potential and free-surface boundary condition is invoked and solved to third-order in terms of $ka$ and $kA$. The validity of this approach is investigated through a comparison with canonical second-order diffraction theory and existing experimental results. We find that for small $ka$, the long wavelength theory is valid up to $kA \approx 0.16-0.2$ on the up wave side of the cylinder. However, this domain is significantly reduced to $kA<0.06$ when an arbitrary position around the cylinder is considered. An important feature of this work is an improved account of the first-harmonic of the free-surface elevation over linear diffraction theory.
A Low-cost Four-degree-of-freedom Motion-platform for Ship-borne Free Space Laser Communication  [cached]
Tingyu Wang,Sha Lu
Journal of Communications , 2010, DOI: 10.4304/jcm.5.9.647-653
Abstract: Free space laser communication (FSOC) has been an emerging technology that is deployed in the marine ship communication field for enabling high-quality communications among moving ships due to its high anti-EM interference capability and good security. However, the system typically requires six dimensional supports of Acquisition, tracking and pointing in the moving environment, which is rather expensive. In this paper, we propose a novel four-degree-of-freedom solution based on the law of motion of the ship in wind waves derived from linear system theory; the solution uses a horizontal, a vertical and two rotating motions to simulate the laser movement caused by ship movement in the wind waves. To the best of our knowledge this is the first effort in this field; our approach significantly reduces the complexity and cost of the six-dimension based solution. Our experimental results demonstrate a realization of laser movement caused by ship roll movement in the wind waves, and verify the convenience and accuracy of the simulated ship-borne free space optical communication system.
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