二维连续信号的近似采样定理

利用加细方程的面具,给出该方程的一个近似解,并根据这个近似解构造出二维连续信号的近似采样定理.其近似采样函数是所求加细方程的近似解,它是由加细方程的面具唯一确定的逐段线性函数,且有显示的计算公式.因此可以根据需要选择加细方程的面具,从而达到控制近似采样函数的衰减速度.     

广义Benjamin-Bona-Mahony方程Fourier谱方法的大时间误差估计

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｅｎｊａｍｉｎ Ｂｏｎａ Ｍａｈｏｎｙｅｑｕａｔｉｏｎｕｔ+ｕｘ+ｕｕｘ -ｕｘｘ-ｕｘｘｔ =0 ( 1 .1 )ｉｎｃｏｒｐｏｒａｔｅｓｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅａｎｄｄｉｓｓｉｐａｔｉｖｅｅｆｆｅｃｔｓ ,ａｎｄｈａｓｂｅｅｎｐｒｏｐｏｓｅｄａｓａｍｏｄｅｌｆｏｒｂｏｔｈｔｈｅｂｏｒｅｐｒｏｐａｇａｔｉｏｎａｎｄｔｈｅｗａｔｅｒｗａｖｅｓ[1,2 ] .Ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｉｓｅｑｕａｔｉｏｎｈａｖｅｂｅｅ…     

THE LARGE TIME BEHAVIOR OF SPECTRAL APPROXIMATION FOR A CLASS OF PSEUDOPARABOLIC VISCOUS DIFFUSION EQUATION

The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estiation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d(AN,A)→0 are proved.     

Solution of the Cauchy problem for a system of integrodifferential equations of viscoelasticity

Existence and uniqueness of the solution of the Cauchy problem is proved for a system of integrodifferential equations of the hereditary theory of viscoelasticity. A method of constructing an approximate solution is proposed. An estimate of the error of the approximate solution is presented.     

Algorithm of Barrier Objective Penalty Function

In this paper, an algorithm of barrier objective penalty function for inequality constrained optimization is studied and a conception–the stability of barrier objective penalty function is presented. It is proved that an approximate optimal solution may be obtained by solving a barrier objective penalty function for inequality constrained optimization problem when the barrier objective penalty function is stable. Under some conditions, the stability of barrier objective penalty function is proved for convex programming. Specially, the logarithmic barrier function of convex programming is stable. Based on the barrier objective penalty function, an algorithm is developed for finding an approximate optimal solution to an inequality constrained optimization problem and its convergence is also proved under some conditions. Finally, numerical experiments show that the barrier objective penalty function algorithm has better convergence than the classical barrier function algorithm.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

间断系数边值问题有限元分析

在数学物理问题中,诸如不同介质的热传导,电场分布等问题,都将产生求解具有间断系数的椭圆型微分方程边值问题,即所谓内表面问题。对具有光滑内表面的问题,早在60年代初,就研究得相当深入了,例如参看[5]和[8],以后,人们又开始注意研究非光滑     

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.     

Approximate Controllability of Fractional Order Semilinear Delay Systems

In this paper, the approximate controllability for a class of semilinear delay control systems of fractional order is proved under the natural assumption that the linear system is approximately controllable. The existence and uniqueness of the mild solution is also proved under suitable assumptions. An example is given to illustrate our main results.     

半空间上Boussinesq方程组弱解的L2衰减估计

主要研究外力函数f 满足一定条件时,半空间上Boussinesq 方程组当n≥3时弱解的L2衰减。先假设解光滑,给出了光滑解的一致L2衰减估计,再通过构造逼近解,对逼近解序列作一致衰减估计,取极限得到弱解的一致L2衰减估计。     

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

The present work considers a nonlinear abstract hyperbolic equation with a self-adjoint positive definite operator, which represents a generalization of the Kirchhoff string equation. A symmetric three-layer semi-discrete scheme is constructed for an approximate solution of a Cauchy problem for this equation. Value of the gradient in the nonlinear term of the scheme is taken at the middle point. It makes possible to find an approximate solution at each time step by inverting the linear operator. Local convergence of the constructed scheme is proved. Numerical calculations for different model problems are carried out using this scheme.     

Two-dimensional integral-algebraic systems: Analysis and computational methods

In this article we formulate sufficient conditions for the existence and uniqueness of solution to systems of two-dimensional Volterra integral equations, in which the coefficient of the main term is a singular matrix. A numerical method is introduced which can be applied to approximate the solution when the given conditions are satisfied. The convergence of this method is proved and illustrated by numerical examples.     

奇摄动分数阶微分方程的渐近解

本文研究了一类奇摄动非线性分数阶微分方程初值问题.利用伸长变量构造出解的形式展开式,并利用微分不等式理论,证明了解的一致有效的渐近式.所得的结果具有较好精度的近似解.     

The reconstruction of a solely time-dependent load in a simply supported non-homogeneous Euler–Bernoulli beam

In this paper, the theoretical and numerical determination of a solely time-dependent load distribution is investigated for a simply supported non-homogeneous Euler–Bernoulli beam. The missing source is recovered from an additional “local” integral measurement. The existence and uniqueness of a solution to the corresponding variational problem is proved by employing Rothe’s method. This method also reveals a time-discrete numerical scheme based on the backward Euler method to approximate the solution. Corresponding error estimates are proved and assessed by two numerical experiments.     

Solving a nonlinear system of second order boundary value problems

In this paper, a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems. The analytical solution is represented in the form of series in the reproducing kernel space. In the mean time, the approximate solution un(x) is obtained by the n-term intercept of the analytical solution and is proved to converge to the analytical solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Two-dimensional integral–algebraic systems: Analysis and computational methods

In this article we formulate sufficient conditions for the existence and uniqueness of solution to systems of two-dimensional Volterra integral equations, in which the coefficient of the main term is a singular matrix. A numerical method is introduced which can be applied to approximate the solution when the given conditions are satisfied. The convergence of this method is proved and illustrated by numerical examples.     

State-constrained optimal control of nonlinear elliptic variational inequalities

An optimization control problem for systems described by abstract variational inequalities with state constraints is considered. The solvability of this problem is proved. The problem is approximated by the penalty method. The convergence of this method is proved. Necessary conditions of optimality for the approximation problem are obtained. Its solution is an approximate optimal control of the initial problem.     

The Spectral Method for the Generalized Kuramoto-Sivashinsky Equation

A spectral method is proposed, the existence and uniqueness of the global and smooth solution are proved for the periodic initial value problem of the generalized K-S equation. The error estimates are established and the convergence is proved for the approximate solution of the spectral method.     

关于非均匀变截面梁阶梯折算法的极限

本文讨论叶开沅教授提出的计算非均匀变截面梁的阶梯折算法.并证明了如下结论:用这个方法得出的近似解,当阶梯数趋于无穷时趋于真解.不过极限解和真解之间差别的度量不是纯数学意义的而是在力学意义下的.     

Iterative solution of matrix games using grid variational inequalities

A new approach to the approximate solution of matrix games is proposed. It is based on the reduction of the original problem to a variational inequality of a special form. In particular, this makes it possible to design preconditioned iterative methods, which proved to be effective as a tool for the numerical solution of large and ill-conditioned systems of linear algebraic equations.