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1.
This paper considers a finite-element approximation of a Poissonequation in a region with a curved boundary on which a Neumanncondition is prescribed. Piecewise linear and bilinear elementsare used on unfitted meshes with the region of integration beingreplaced by a polygonal approximation. It is shown, despitethe variational crimes, that the rate of convergence is stillorder (h) in the H1 norm. Numerical examples show that the methodis easy to implement and that the predicted rate of convergenceis obtained. Supported by SERC postdoctoral fellowship RF/5830.  相似文献   

2.
Present address: Department of Mathematics, University of Reading, Reading RG6 2AX. We consider the convergence of solution curves of approximationsto parameter-dependent operator equations of the form G(, x)= 0. Provided Gx(, x) remains non-singular this problem is cateredfor by a simple extension to standard theory. In this paper,however, attention is concentrated on solution curves throughcertain singular points (0, x0), and the main result is thatconvergence depends on consistency and stability results forthe linear eigenvalue problem Gx(0, x0)0 = 0.  相似文献   

3.
N. Karcanias Control Engineering Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square London EC1V OHB, UK Email: n.karcanias{at}city.ac.uk Received on June 14, 2006; Accepted on October 2, 2006 The problem of arbitrary pole placement via dynamic decentralizedoutput feedback is studied for minimal systems described bya proper transfer function matrix P(s) Rm x p(s) (m = mi andp = pi), with McMillan degree n. The family of controllersto be used includes those decentralized controllers with channelswhose ith channel has maximum observability index at most di.The method presented here is based on asymptotic linearizationaround a decentralized degenerate compensator of the pole placementmap related to the problem. It is shown that the method worksgenerically when m+p > n, where m+ = min{di(pi + mi –1) + mi}, i = 1, ..., , and the smallest di of the compensatorof the ith channel is the integral part of (npmi)/p(pi+ mi – 1).  相似文献   

4.
Present address: Department of Mathematics, University of Tabriz, Tabriz, Iran. Accurate end conditions are derived for quintic spline interpolationat equally spaced knots. These conditions are in terms of availablefunction values at the knots and lead to O(h6) convergence uniformlyon the interval of interpolation.  相似文献   

5.
A problem of universal adaptive stabilization in Rn is approachedin a qualitative manner to relate the form of the trajectoriesin the extended state space Rn+1 to the root locus of the associatedfixed-parameter linear system. Relationships are derived betweenthe values of the limit gain and the initial conditions. Numericalstudies are used to illustrate and support these ideas by computationof generic trajectories and attempted computation of certainnon-generic possibilities. The implications of the study formore general dynamic situations are outlined. These authors are also with the Department of Mathematics,University of Exeter, EX4 4QE.  相似文献   

6.
Email: er-wei-bai{at}uiowa.edu Received on July 30, 2005; Accepted on July 23, 2006 In the paper, we discuss identification of a nonlinear systemwithout structural information and propose two methods, thekernel method and the orthonormal basis method. The convergenceresults are established for both methods without a priori structuralinformation. We then apply the results to identification ofHammerstein models with an unknown dynamic nonlinearity. Itis also shown that identification of the linear part in Hammersteinmodels is possible with no knowledge of the dynamic nonlinearity.  相似文献   

7.
Address from 1st April 1985, School of Mathematics, Universityof Bristol, University Walk, Bristol BS8 1TW. The morning finite-element method for evolutionary partial differentialequations leads to a coupled non-linear system of ordinary differentialequations in time, with a coefficien matrix A, say, for thetime derivaties, We show for linear elements in any number ofdimensions, A can be written in the form MTCM, where the matrixC depends solely on the mesh geometry and the matrix M on thegradient of the section, As a simple consequence we show thatA is singular only in the cases (i) element degeneracy () and (ii) collinearity of nodes (M not out of fullrank). We give constructions for the inversion of A in all cases. In one dimension, if A is non-singular, it has a simple explicitinverse. If A is singular we replace it by reduced matrix A*.It can be shown that every case the spectral radius of the Jacobiiteration matrix ia ?and that A or A* can be efficiently invertedby conjugate gradient methods. Finally, we discuss the applicability of these arguments tosystem of equations in any number of dimensions.  相似文献   

8.
It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H2 (D) and (ii) Z–1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary.  相似文献   

9.
Computation and parametrization of periodic and connecting orbits   总被引:1,自引:0,他引:1  
Email: g.moore{at}ma.ic.ac.uk or na.gmoore New algorithms for the computation of periodic and connectingorbits of autonomous differential equations are developed. Previousmethods have parametrized these curves by the independent timevariable, but our procedure is based on the canonical arclengthparametrization.  相似文献   

10.
A recent paper (Delves, 1977) described a variant of the Galerkinmethod for linear Fredholm integral equations of the secondkind with smooth kernels, for which the total solution timeusing N expansion functions is (N2 ln N) compared with the standardGalerkin count of (N3). We describe here a modification of thismethod which retains this operations count and which is applicableto weakly singular Fredholm equations of the form where K0(x, y) is a smooth kernel and Q contains a known singularity.Particular cases treated in detail include Fredholm equationswith Green's function kernels, or with kernels having logarithmicsingularities; and linear Volterra equations with either regularkernels or of Abel type. The case when g(x) and/or f(x) containsa known singularity is also treated. The method described yieldsboth a priori and a posteriori error estimates which are cheapto compute; for smooth kernels (Q = 1) it yields a modifiedform of the algorithm described in Delves (1977) with the advantagethat the iterative scheme required to solve the equations in(N2) operations is rather simpler than that given there.  相似文献   

11.
12.
A linear autonomous system of differential equations =Ax can be transformed to its Jordan normalform, that is, the transformed system is in block diagonal formand the blocks correspond to different eigenvalues. This resultis generalized to arbitrary nonautonomous linear systems =A(t)x with a locally integrable matrix functionA:R RNxN.  相似文献   

13.
An elliptic boundary-value problem on a domain with prescribedDirichlet data on I is approximated using a finite-elementspace of approximation power hK in the L2 norm. It is shownthat the total flux across I can be approximated with an errorof O(hK) when is a curved domain in Rn (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.  相似文献   

14.
Permanent address: Department of Engineering Mathematics, Cairo University, Giza, Egypt. A priori and a posteriori error bounds are given for the computedeigenpair (, ) of the eigenvalue problem Ax = x, which are shownto be more realistic than some of the available ones. A simplemethod is also presented for computing the backward error. Finallya scaling procedure is explained for reducing the residual error.  相似文献   

15.
In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –2u+u=Q(x)|u|q–2u, xRN, uH1(RN) (1.1) where = Ni=12/x2i is the Laplace operator in RN, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on RN satisfying thefollowing conditions. (Q1) Q has a strict local minimum at some point x0RN, that is,for some > 0 Q(x)>Q(x0) for all 0 < |xx0| < . (Q2) There are constants C, > 0 such that |Q(x)–Q(y)|C|xy| for all |xx0| , |yy0| . Our aim here is to show that corresponding to each strict localminimum point x0 of Q(x) in RN, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x0, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x0, while vanishing as 0 everywhere else in RN.  相似文献   

16.
In this paper, it is proved that for n 2, any horizontallyhomothetic submersion : Rn+1 (Nn, h) is a Riemannian submersionup to a homothety. It is also shown that if : Sn+1 (Nn, h)is a horizontally homothetic submersion, then n = 2m, (Nn, h)is isometric to CPm and, up to a homothety, is a standard Hopffibration S2m+1 CPm. 2000 Mathematics Subject Classification53C20, 53C12.  相似文献   

17.
Let f, g: (Rn, 0) (Rp, 0) be two C map-germs. Then f and gare C0-equivalent if there exist homeomorphism-germs h and lof (Rn, 0) and (Rp, 0) respectively such that g = l f h–1.Let k be a positive integer. A germ f is k-C0-determined ifevery germ g with jk g(0) = jk f(0) is C0-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C0-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle Jk(N,P) with fiber Jk(n, p). Let z in Jk(n, p) and let f be suchthat z = jkf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let Wk(N, P) be the subbundle of Jk(N, P) with fiber Wk(n, p).Mather has constructed a finite Whitney (b)-regular stratificationSk(n, p) of Jk(n, p) – Wk(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if Sk(N,P) denotes the corresponding stratification of Jk(N, P) –Wk(N, P) and f C(N, P) is a C map such that jkf is multitransverseto Sk(N, P), jkf(N) Wk(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rn, 0) (Rp, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U Rp such that jkf(U – 0) Wk (Rn, Rp = and suchthat jkf is multitransverse to Sk (Rn, Rp) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate jkf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.  相似文献   

18.
Email: gugat{at}am.uni-erlangen.de Received on April 30, 2006; We consider a finite string that is fixed at one end and subjectto a feedback control at the other end which is allowed to move.We show that the behaviour is similar to the situation whereboth ends are fixed: As long as the movement is not too fast,the energy decays exponentially and for a certain parameterin the feedback law it vanishes in finite time. We considermovements of the boundary that are continuously differentiablewith a derivative whose absolute value is smaller than the wavespeed. We solve a problem of worst-case optimal feedback control,where the parameter in the feedback law is chosen such thatthe worst-case Lp-norm of the space derivative at the fixedend of the string is minimized (p [1, )). We consider the worstcase both with respect to the initial conditions and with respectto the boundary movement. It turns out that the parameter forwhich the energy vanishes in finite time is optimal in thissense for all p.  相似文献   

19.
Consider the bounded linear operator, L: F Z, where Z RN andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, Li:F R. By the Riesz representation theorem, there exist functionsk(xi, ·) F : Lif = f, k(xi, ·)F. The functions,k(xi, ·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel.  相似文献   

20.
A straightforward implementation of the Global Element Method(Delves & Hall, 1979) for two-dimensional partial differentialequations has an operation count: Set up equations: (MN6); solve: (M3N6) where M is the number of elements and N the number of one-dimensionalexpansion functions used in each element. We describe here analternative implementation in which both of these counts arereduced to (MN4). The method used generalizes to p dimensions, with operationcount (MN2p) compared with the "standard" count (MP3p + M3N3p).  相似文献   

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