关于Hardy-Hilbert积分不等式的推广

本文通过引入适当的参数，及如下形式的权系数（ｘ＋β）１－ｔｋｔ（ｒ）－ｌｎ２α＋βｘ＋β１－１／ｒ，ｘ∈［α，∞）（α－β，ｒ＞１，１－１／ｒ＜ｔ１）．而使Ｈａｒｄｙ－Ｈｉｌｂｅｒｔ积分不等式得到有意义的推广．这里ｋｔ（ｒ）＝∫∞０１（１＋ｕ）ｔ１ｕ１／ｒｄｕ，常数ｌｎ２＝０．６９３１４７１８＋．     

关于半线性拟双曲方程的爆破解

本文研究半线性拟双曲方程　的初边值问题和初值问题解的爆破问题，证明了当ｆ（ｔ，ｕ）满足一定条件时，比如ｆ（ｔ，ｕ）≥Ｃｅ－（ｅｔ）ｕ（１＋ｒ）（Ｃ，ｃ，ｒ为正常数，ｔ≥０，ｕ充分大），该方程的初边值和初值问题对大初值其解爆破；而当ｆ（ｔ，ｕ）满足更强的条件时，比如ｆ（ｔ，ｕ）≥Ｃｅ（ｅｔ）ｕ（１＋ｒ）（条件同上），该方程的这类问题对于一些小初值其解也爆破．     

方程u_(tt)=u_(xxt) f(u_x)_x初边值问题的差分法

１．引言方程是在国内外引起广泛关注的一类重要的非线性发展方程．文［１］在函数ｆ（ｓ）满足局部 Ｌｉｐ－ｓｃｈｉｔｚ条件及单调性条件（ｆ（ｓ２）－ｆ（ｓ１））（ｓ２－ｓ１）＞ ０的假设下得到了（１．１）初边值问题整体弱解的存在与唯一性;文［２］用 Ｇａｌｅｒｋｉｎ方法,研究了（１．１）的初边值问题,周期边值问题和初值问题,并在函数ｆ’（ｓ）下方有界的假设下得到了整体强解的存在与唯一性． 本文在有限区域 ＱＴ＝［０,１］×［０,Ｔ］（Ｔ＞ ０）上讨论方程（１．１）带有初值条件和边值条件（ｕ（ｘ,ｔ）为未知…     

一类抛物型偏泛函微分方程解的强迫振动性

本文研究抛物型偏泛函微分方程γ／γｔ［ｕ－ｍΣｔ－１Ｃｔ（ｔ）ｕ（ｘ，ｔ－τｔ）］＝ａ（ｔ）Δｕ－Ｐ（ｘ，ｔ）ｕ－Ｑ（ｘ，ｔ）Ｇ［ｕ（ｘ，ｐ（ｔ）］＋Ｆ（ｘ，ｔ），（ｘ，ｔ）包含Ｄ×［０，＋∞］解的强近振动性，其中Ｄ为Ｒ＾ｎ中具有逐片光滑边办γＤ的有界区域，ｕ＝ｕ（ｘ，ｔ），Δ是Ｒ＾ｎ中的Ｌａｐｌａｃｅ算子。     

平面域上反应扩散方程解的一些性质

本文考虑平面域上反应扩散方程（｜ｕ（ｘ，ｔ）｜＾ｍ－１·ｕ）ｔ－△ｕ＋ｃｕ＝ｆ（ｘ，ｔ，ｕ）的第三非线性初－边值问题。建立了解的积分型极值原理，由此说明非全局解的Ｂｌｏｗｕｐ将在区域边界上发生。     

方程utt-△u-△ut-△utt=f(u)的初边值问题

本文研究一类四阶非线性耗散、色散波动方程的初边值问题｛ｕｔｔ－△ｕ－△ｕｔ－△ｕｔｔ＝ｆ（ｕ）,ｕ│ｔ＝０＝ｕ０（ｘ）,ｕｔ│ｔ＝０＝ｕ１（ｘ）,ｕ│эΩ＝０,得到了问题整体强解的存在性和唯一性,并在一定条件下,研究了解的渐近性质和ｂｌｏｗ ｕｐ现象。     

方程 λ(1 ce~(-τλ) a be~(-τλ)= 0所有根具有负实部的充要条件

考虑方程λ（１＋ｃｅ~（－τλ）＋ａ＋ｂｅ~（－τλ）＝０,（２）其中ａ,ｂ和ｃ为任意常数,τ为正常数,ｃ≠０．方程（１）为中立型方程　　　　　　　　ｘ（ｔ）＋ｃｘ（ｔ－τ）＋ａｘ（ｔ）＋ｂｘ（ｔ－τ）＝０　（２）的特征方程．方程（１）为一常见的拟多项式方程．关于拟多项式函数, Ｐｏｎｔｒｙａｇｉｎ在 １９４２年给出了判断这类函数所有零点位于左半复平面的充要条件．但对中立型方程来说,由于这些条件往往难以验证,使得人们长期以来无法用Ｐｏｎｔｒｙａｐｉｎ定理找出方程（１）所有根具有负实部的充要条件．本文在克服了上述困难后,用Ｐｏｎｔｒｇｉｎ定理找出方程（１）所有根具有负实部的充要条件．     

交替方向隐格式稳定性和收敛性的改进

交替方向隐格式是数值求解高维抛物型方程的主要方法之一,考虑二维变系数抛物型方程ｕｔ－ｘａ（ｘ,ｙ,ｔ）ｕｘ－ｙｂ（ｘ,ｙ,ｔ）ｕｙ＝ｆ本文研究两个著名的交替方向隐式差分格式———Ｐ＿Ｒ格式和Ｄｏｕｇｌａｓ格式的稳定性和收敛性,对常系数情形（即函数ａ和ｂ均为常数）,文献已证明了按离散Ｌ２范数的绝对稳定性和二阶收敛性,结论是完善的,但所用Ｆｏｕｒｉｅｒ分析方法不能推及一般变系数问题·文献采用了能量方法研究Ｐ＿Ｒ格式的稳定性和收敛性,但由于目的是Ｌ２估计以及使用了“Ｌ２范数与Ｈ１半范数等价”,所得到的Ｌ２稳定性和收敛性结论是很不完善的·本文采用Ｈ１能量估计方法,证明了格式按离散Ｈ１范数是稳定的,并且收敛阶为Ｏ（Δｔ２＋ｈ２）,改进了已有结果     

一类奇异抛物方程的超过程解法

对于如下一类奇异非线性抛物型方程：ｔｕ（ｔ,ｘ）＝Δｕ（ｔ,ｘ）－ｕβ（ｔ,ｘ）,ｘ∈Ｂ（０,ａ）,ｕ（０,ｘ）＝０,ｕ（ｔ,ｘ）｜ｘ｜→ａ－→∞,ｔ＞０,这里,１≤β≤２,Ｂ（ｏ,ａ）表示以ｏ为中心,以ａ为半径的Ｒｄ中的闭球,利用超过程的理论,给出了它的概率解法,从而推广了文［１］相应的结论     

非强制性Kirchhoff方程

本文讨论下述非齐次Ｋｉｒｃｈｈｏｆ方程的Ｃａｕｃｈｙ问题ｕｔｔ－ｍ∫Ｒｎ｜Δｕ｜２ｄｘΔｕ＝ｆ（ｔ，ｘ），ｕ（０，ｘ）＝ｕ０（ｘ），ｕｔ（０，ｘ）＝ｕ１（ｘ），其中ｍ（ｒ）∈Ｃ１［０，＋∞）且ｍ（ｒ）＞０．在初值和右端项“小”的条件下，我们获得了此问题整体解的存在唯一性     

Valuing Asian options using the finite element method and duality techniques

The main objective of this paper is to develop an adaptive finite element method for computation of the values, and different sensitivity measures, of the Asian option with both fixed and floating strike. The pricing is based on Black–Scholes PDE-model and a method developed by Ve?e? where the resulting PDEs are of parabolic type in one spatial dimension and can be applied to both continuous and discrete Asian options. We propose using an adaptive finite element method which is based on a posteriori estimates of the error in desired quantities, which we derive using duality techniques. The a posteriori error estimates are tested and verified, and are used to calculate optimal meshes for each type of option. The use of adapted meshes gives superior accuracy and performance with less degrees of freedom than using uniform meshes. The suggested adaptive finite element method is stable, gives fast and accurate results, and can be applied to other types of options as well.     

On the Galerkin finite element approximations to multi-dimensional differential and integro-differential parabolic equations

The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev inequality and some operator representations of the finite element solutions. The results of the present paper lead to the error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements.     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

Superconvergence of the continuous Galerkin finite element method for delay-differential equation with several terms

The continuous Galerkin finite element method for linear delay-differential equation with several terms is studied. Adding some lower terms in the remainder of orthogonal expansion in an element so that the remainder satisfies more orthogonal condition in the element, and obtain a desired superclose function to finite element solution, thus the superconvergence of p  -degree finite element approximate solution on (p+1)$(p+1)$-order Lobatto points is derived.     

A characteristic finite element method for optimal control problems governed by convection-diffusion equations

In this paper we analyze a characteristic finite element approximation of convex optimal control problems governed by linear convection-dominated diffusion equations with pointwise inequality constraints on the control variable, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by either piecewise constant functions or piecewise linear discontinuous functions. A priori error estimates are derived for the state, co-state and the control. Numerical examples are given to show the efficiency of the characteristic finite element method.     

High-order finite element methods for time-fractional partial differential equations

The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for space and finite difference method for time. Optimal convergence rate O((Δt)^2−α+N^−r) is proved for the (r−1)th-order finite element method (r≥2).     

A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart-Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L² norm as well as for the approximate pressures in the L² norm.     

Weighted quadrature rules for finite element methods

We discuss the numerical integration of polynomials times non-polynomial weighting functions in two dimensions arising from multiscale finite element computations. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic integration. We validate this approach by introducing the new quadrature formulas into a multiscale finite element method for the two-dimensional reaction–diffusion equation.     

A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H¹-norm is proved to be O(h+H³|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.     

Expanded mixed finite element methods for the problem of purely longitudinal motion of a homogeneous bar

In this paper, expanded mixed finite element methods for the initial-boundary-value problem of purely longitudinal motion equation of a homogeneous bar are proposed and analyzed. Optimal error estimates for the approximations of displacement in L² norm and stress in H¹ norm are obtained.