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1.
基于拟Shannon小波浅水长波近似方程组的数值解 总被引:1,自引:0,他引:1
本文研究了浅水长波近似方程组初边值问题的数值解.利用小波多尺度分析和区间拟Shannon小波,对浅水长波近似方程组空间导数实施空间离散,用时间步长自适应精细积分法对其变换所的非线性常微分方程组进行求解,得到了浅水长波近似方程组的数值解,并将此方法计算的结果与其解析解进行比较和验证. 相似文献
2.
We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc. 相似文献
3.
Multi-class multi-server queueing problems are a generalisation of the well-known M/M/k queue to arrival processes with clients of N types that require exponentially distributed service with different average service times. In this paper, we give a procedure to construct exact solutions of the stationary state equations using the special structure of these equations. Essential in this procedure is the reduction of a part of the problem to a backward second order difference equation with constant coefficients. It follows that the exact solution can be found by eigenmode decomposition. In general eigenmodes do not have a simple product structure as one might expect intuitively. Further, using the exact solution, all kinds of interesting performance measures can be computed and compared with heuristic approximations (insofar available in the literature). We provide some new approximations based on special multiplicative eigenmodes, including the dominant mode in the heavy traffic limit. We illustrate our methods with numerical results. It turns out that our approximation method is better for higher moments than some other approximations known in the literature. Moreover, we demonstrate that our theory is useful to applications where correlation between items plays a role, such as spare parts management. 相似文献
4.
For a nonlinear functional f, and a function u from the span of a set of tensor product interpolets, it is shown how to compute the interpolant of f (u) from the span of this set of tensor product interpolets in linear complexity, assuming that the index set has a certain
multiple tree structure. Applications are found in the field of (adaptive) tensor product solution methods for semilinear
operator equations by collocation methods, or after transformations between the interpolet and (bi-) orthogonal wavelet bases,
by Galerkin methods. 相似文献
5.
W. G. Litvinov 《Numerical Methods for Partial Differential Equations》2014,30(2):406-450
The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) is widely used for numerical solution of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, there arises in applications a rich variety of problems where approximations of smooth solutions and solutions in the sense of distributions have to be found. This article is devoted to the employment of the Petrov–Galerkin method for solving such problems. The article contains general results on the Petrov–Galerkin approximations of solutions to linear and nonlinear operator equations. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. We apply the general results to two‐dimensional (2D) and 3D problems of the elasticity, to a parabolic problem, and to a nonlinear problem of the plasticity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 406–450, 2014 相似文献
6.
Peter A. Orlin A. Louise Perkins George Heburn 《Numerical Methods for Partial Differential Equations》1997,13(5):549-560
We present a method for designing spatial derivative approximations that achieves a priori accuracy in the spatial frequency domain. We use a general, average value approximation with undetermined coefficients together with a set of constraints that ensure convergence and consistency to formulate a constrained optimal fitting problem. These constraints lead to a linear matrix formulation. We apply the method to the design of spatial approximations for simulating equations with wavelike solutions using both an explicit central difference approximation (which has no phase error) and an upwind design where the level of dissipation can be specified by the designer. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 549–560, 1997 相似文献
7.
XU Yuesheng & ZOU Qingsong Department of Mathematics Syracuse University Syracuse NY USA Institute of Mathematics Academy of Mathematics System Sciences Chinese Academy of Sciences Beijing China Department of Scientific Computing Computer Science Zhongshan University Guangzhou China 《中国科学A辑(英文版)》2005,48(5):680-702
We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence. 相似文献
8.
Nelson Faustino 《PAMM》2006,6(1):735-736
We propose a Wavelet-Galerkin scheme for the stationary Navier-Stokes equations based on the application of interpolating wavelets. To overcome the problems of nonlinearity, we apply the machinery of interpolating wavelets presented in [2] in order to obtain problem-adapted quadrature rules. Finally, we apply Newton's method to approximate the solution in the given ansatz space, using as inner solver a steepest descendent scheme. To obtain approximations of a higher accuracy, we apply our scheme in a multi-scale context. Special emphasize will be given for the convergence of the scheme and wavelet preconditioning. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
N. F. Gasimova 《Mathematical Notes》2012,92(3-4):356-361
We obtain solutions for a class of two-dimensional nonlinear singular integral equations with Hilbert kernel using the contraction mapping method and find the rate of convergence of successive approximations to the exact solution. 相似文献
10.
D. N. Sidorov 《Russian Mathematics (Iz VUZ)》2013,57(1):54-63
We construct an asymptotic approximation for solutions of systems of Volterra integral equations of the first kind with piecewise continuous kernels. We use the asymptotics as an initial approximation in the proposed method of successive approximations to the desired solutions. We prove the existence of a continuous solution depending on free parameters and establish sufficient conditions for the existence of a unique continuous solution. We illustrate the proved existence theorems with examples. 相似文献
11.
We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled
at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh
methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour
of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and
non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous
blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments
that illustrate the behaviour of the adaptive method. 相似文献
12.
We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ?3. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution. 相似文献
13.
In this article we use the C 1 wavelet bases on Powell-Sabin triangulations to approximate the solution of the Neumann problem for partial differential equations. The C 1 wavelet bases are stable and have explicit expressions on a three-direction mesh. Consequently, we can approximate the solution of the Neumann problem accurately and stably. The convergence and error estimates of the numerical solutions are given. The computational results of a numerical example show that our wavelet method is well suitable to the Neumann boundary problem. 相似文献
14.
In this paper, we are concerned with the stochastic differential delay equations with Markovian switching (SDDEwMSs). As stochastic differential equations with Markovian switching (SDEwMSs), most SDDEwMSs cannot be solved explicitly. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEwMSs in the sense of the Lp-norm when the drift and diffusion coefficients are Taylor approximations. 相似文献
15.
Taking linear hyperbolic partial differential equations as an illustration, we attempt to construct weak solutions with higher
integrable gradients, in the sense of Gehring, to hyperbolic diffeential equations with initial and boundary conditions. We
adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic
equations. To establish the scheme, we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic
differential equations. Bibliography: 6 titles.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 30–52. 相似文献
16.
Sanda Micula 《Journal of Fixed Point Theory and Applications》2017,19(3):1815-1824
In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations. 相似文献
17.
This paper deals with finite-difference approximations of Euler equations arising in the variational formulation of image
segmentation problems. We illustrate how they can be defined by the following steps: (a) definition of the minimization problem
for the Mumford–Shah functional (MSf), (b) definition of a sequence of functionals Γ-convergent to the MSf, and (c) definition
and numerical solution of the Euler equations associated to the kth functional of the sequence. We define finite difference approximations of the Euler equations, the related solution algorithms,
and we present applications to segmentation problems by using synthetic images. We discuss application results, and we mainly
analyze computed discontinuity contours and convergence histories of method executions.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
18.
A.Y.T. Leung H.X. Yang Z.J. Guo 《Communications in Nonlinear Science & Numerical Simulation》2012,17(11):4508-4514
We introduce the residue harmonic balance method to generate periodic solutions for nonlinear evolution equations. A PDE is firstly transformed into an associated ODE by a wave transformation. The higher-order approximations to the angular frequency and periodic solution of the ODE are obtained analytically. To improve the accuracy of approximate solutions, the unbalanced residues appearing in harmonic balance procedure are iteratively considered by introducing an order parameter to keep track of the various orders of approximations and by solving linear equations. Finally, the periodic solutions of PDEs result. The proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the sophisticated implicit functions as appearing in most methods. 相似文献
19.
Summary. In this work we address the issue of integrating
symmetric Riccati and Lyapunov matrix differential equations. In
many cases -- typical in applications -- the solutions are positive
definite matrices. Our goal is to study when and how this property
is maintained for a numerically computed solution.
There are two classes of solution methods: direct and
indirect algorithms. The first class consists of the schemes
resulting from direct discretization of the equations. The second
class consists of algorithms which recover the solution by
exploiting some special formulae that these solutions are known to
satisfy.
We show first that using a direct algorithm -- a one-step scheme or
a strictly stable multistep scheme (explicit or implicit) -- limits
the order of the numerical method to one if we want to guarantee
that the computed solution stays positive definite. Then we show two
ways to obtain positive definite higher order approximations by
using indirect algorithms. The first is to apply a symplectic
integrator to an associated Hamiltonian system. The other uses
stepwise linearization.
Received April 21, 1993 相似文献
20.
Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an
adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets
which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations.
The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity
of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are
addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are
appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation
of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets.
The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense
asymptotically optimal. We conclude with a simple numerical example.
Received June 25, 1998 / Revised version received June 5, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献