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121.
Frank Koster 《Numerical Functional Analysis & Optimization》2013,34(7-8):802-819
Because of the implementation of numerical solution algorithms for the nonstationary Navier–Stokes equations of an incompressible fluid on massively parallel computers iterative methods are of special interest. A red–black pressure–velocity iteration that allows an efficient parallelization based on a domain decomposition [3] will be analyzed in this paper. We prove the equivalence of the pressure–velocity iteration (PUI) by Chorin/Hirt/Cook [1, 2] with a SOR iteration to solve a Poisson equation for the pressure. We show this on a 2D rectangle with some special outflow boundary conditions and Dirichlet data for the velocity elsewhere. This equivalence allows us to prove the convergence of that iteration scheme. We also discuss the stability of the occurring discrete Laplacian in discrete Sobolev spaces. 相似文献
122.
A. L. Dontchev W. W. Hager K. Malanowski 《Numerical Functional Analysis & Optimization》2013,34(5-6):653-682
We examine convergence of the Euler approximation to a nonlinear optimal control problem subject to mixed state-control and pure state constraints. We prove that under smoothness, independence, controllability and coercivity conditions at a reference solution of the continuous problem, there exists a locally unique solution to the Euler approximation, for sufficiently fine discretization, which converges to the reference solution with rate proportional to the mesh size. 相似文献
123.
We extend the well-known Peano Kernel Theorem to a class of linear operators L : Cn+1([a,b];X}→ X, X being a Branch space, which vanish on abstract polynomials of degree ≤ n. We then recover, in the abstract setting, classical estimates of remainders in polynomials interpolation and quadrature formulas. Finally, we present an application to the error analysis of the trapezoidal time discretization scheme for parabolic evolution equations. 相似文献
124.
Projection methods are efficient operator-splitting schemes to approximate solutions of the incompressible Navier-Stokes equations. As a major drawback, they introduce spurious layers, both in space and time. In this work, we survey convergence results for higher order projection methods, in the presence of only strong solutions of the limiting problem; in particular, we highlight concomitant difficulties in the construction process of accurate higher order schemes, such as limited regularities of the limiting solution, and a lack of accurate initial data for the pressure. Computational experiments are included to compare the presented schemes, and illustrate the difficulties mentioned. 相似文献
125.
126.
** Email: marian.slodicka{at}ugent.be This paper is devoted to the study of a non-linear degeneratetransient eddy current model of the type [graphic: see PDF] for some 0 < < 1 subject to homogeneous Dirichlet boundarycondition H x = 0. Here, the magnetic properties of a softferromagnet are linked by a power material law. The well-posednessof the problem is proved and a non-linear time-discrete numericalscheme for the approximation in suitable function spaces isdesigned. Convergence of the approximation to a weak solutionis proved and error estimates are derived. 相似文献
127.
Devika Shylaja 《Numerical Methods for Partial Differential Equations》2020,36(5):972-997
The Hessian discretization method (HDM) for fourth-order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit-conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators. A generic error estimate has been established in L2, H1, and H2-like norms in literature. In this paper, we establish improved L2 and H1 error estimates in the framework of HDM and illustrate it on various schemes. Since an improved L2 estimate is not expected in general for FVM, a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini ncFEM, in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented. 相似文献
128.
In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations. 相似文献
129.
Tamás Szabó 《Journal of Computational and Applied Mathematics》2010,235(2):478-490
In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments. 相似文献
130.
This article reports a numerical discretization scheme, based on two‐dimensional integrated radial‐basis‐function networks (2D‐IRBFNs) and rectangular grids, for solving second‐order elliptic partial differential equations defined on 2D nonrectangular domains. Unlike finite‐difference and 1D‐IRBFN Cartesian‐grid techniques, the present discretization method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian‐grid‐based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high‐order integration schemes to evaluate flux integrals arising from a control‐volume discretization on irregular domains. A new preconditioning scheme is suggested to improve the 2D‐IRBFN matrix condition number. Good accuracy and high‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 相似文献