BOUNDEDNESS AND CONVERGENCE FOR THE NON-LINARD TYPE DIFFERENTIAL EQUATION

In this article, the author studies the boundedness and convergence for the non-Lienard type differential equation (x|·)=a(y)-f(x) (y|·)=b(y)β(x)-g(x) e(t) where a(y),b(y),f(x),g(x),β(x) are real continuous functions in y∈R or x∈R,β(x)≥0 for all x and e(t) is a real continuous function on R = {t: t≥0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

一类带弱奇异核非线性偏积分微分方程的全离散有限元

1引言我们将研究下面一类带弱奇异核非线性偏积分微分方程的数值解:u_t-▽·(a(u)▽u)-integral from n=0 to tβ(t-s)△u(s)ds=f(u),x∈Ω,t∈(?),(1.1) u(·,t)=0,x∈(?)Ω,t∈J,(1.2) u(·,0)=v(x),x∈Ω,(1.3)其中Ω为平面上的凸角域,J=(0,T],α和f为R上的光滑函数,满足0相似文献   

Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRDINGER EQUATION

In this paper,the boundary control problem of a distributed parameter system described by the Schrdinger equation posed on finite intervalα≤x≤β: iyt yxx |y|~2y=0, y(α,t)=h_`1(t),y(β,t)=h_2(t)for t>0(S) is considered.It is shown that by choosing appropriate control inputs(h_j),(j=1,2)one can always guide the system(S)from a given initial stateφ∈H~s(α,β),(s∈R)to a terminal stateΨ∈H~s(α,β),in the time period[0,T].The exact boundary controllability is obtained by considering a related initial value control problem of Schrdinger equation posed on the whole line R.The discovered smoothing properties of Schrdinger equation have played important roles in our approach;this may be the first step to prove the results on boundary controllability of(semi-linear)nonlinear Schrdinger equation.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)>0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N>0,k(s)>0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

两参数分数Wiener过程增量的一般形式

§1Introductionandresults A2-parameterGaussianprocess{Z(t,s);t,s≥0}iacalleda2-parameterfractional Wienerprocesswithorderα(0<α<1),ifZ(0,0)=0a.s.EZ(s,t)=0anditscovariance EZ(t1,s1)Z(t2,s2)={|t1|2α+|t2|2α-|t2-t1|2α}{|s1|2α+|s2|2α-|s2-s1|2α}/4.LetR=[x1,x2]×[y1,y2],DT={(x,y)∶0≤x,y≤bT,xy≤T}.Let0相似文献   

Well-posedness of equations with fractional derivative via the method of sum

We study the well-posedness of the equations with fractional derivative Dαu(t)=Au(t)+f(t)(0 ≤t≤2π),where A is a closed operator in a Banach space X,0α1 and Dα is the fractional derivative in the sense of Weyl.Although this problem is not always well-posed in Lp(0,2π;X) or periodic continuous function spaces Cper([0,2π];X),we show by using the method of sum that it is well-posed in some subspaces of L p(0,2π;X) or C per([0,2π];X).     

一类四阶抛物型方程的解的渐近行为

本文考虑一维空间中四阶抛物型方程Cauchy问题{ut-(e)2xu+(e)4xu=(e)xf(u), x∈R,t＞0,u(x,0)=u0(x), x∈R,的整体解u=u(x,t)的大时间渐近行为和时间衰减速率,其中f(u)∈C1(R), |f(u)|≤C|u|q, q＞5/2.     

某类沿曲线的振荡积分

对R2上沿曲线(t,γ(t))的振荡积分算子τα,βf(x, y) = ∫R f(x - t, y - γ(t))e-i|t|-βt-1|t|-αdt进行了研究,其中γ(t)=|t|κ或γ(t)=sgn(t)|t|κ,α,β,κ为使得算子τα,β有定义的任意实数.假设αβ＞0,|β|＞3|α|以及β≠1,得到τα,β在Lp(R2)上有界,当且仅当κ≠β,其中p ∈ (2β/(2β-3α),2β/3α).     

Boundedness and convergence for the non-Liénard type differential equation

In this article, the author studies the boundedness and convergence for the {(x) = a(y) - f(x),(y) = b(y)β(x) - g(x) e(t),where a(y), b(y), f(x),g(x),β(x) are real continuous functions in y ∈ R or x ∈ R,β(x) ≥ 0 for all x and e(t) is a real continuous function on R = {t: t ≥ 0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

综合题新编选登

题158已知函数f(x)的导数f′(x)满足0α时,总有f(x)α,在α与β之间存在一点c,αα,所以f′(c)=1,与已知0相似文献   

2006年高考数学全国卷Ⅰ

一、选择题:共12小题,每小题5分,共60分.1.设集合M={x|x2-x<0},N={x||x|<2},则A.M∩N=B.M∩N=MC.M∪N=MD.M∪N=R2.已知函数y=ex的图像与函数y=f(x)的图像关于直线y=x对称,则A.f(2x)=e2x(x∈R)B.f(2x)=ln2·lnx(x>0)C.f(2x)=2ex(x∈R)D.f(2x)=lnx+ln2(x>0)3.双曲线mx2+y2=1的虚轴长是实轴长的2倍,则m=A.-41B.-4C.4D.414.如果复数(m2+i)(1+mi)是实数,则实数=A.1B.-1C.2D.-25.函数f(x)=tanx+4π的单调增区间为A.kπ-2π,kπ+2π,k∈ZB.(kπ,(k+1)π),k∈ZC.kπ-34π,kπ+4π,k∈ZD.kπ-4π,kπ+34π,k∈Z6.△ABC的内角A、B、…     

具有无穷时滞的中立型积分微分系统的平稳振荡

利用重合度理论中的延拓定理和微分积分不等式讨论具有无穷时滞的中立型积分微分系统其中x(t)=(x1(t),…,xn(t))T,G∈C2(Rn,R),f∈C(R×R×Rn×Rn,Rn),e∈C(R,Rn),e(t ω)≡e(t),f(t ω,u ω,x,y)≡f(t,u,x,y),f(t,u,0,0)≡0,t,u∈R,x,y∈Rn,ω>0为常数,获得了该系统平稳振荡的易于检验的判别条件.     

一类双曲型方程反问题的存在性与唯一性

针对双曲型方程定解问题{utt=a2uxx+f(t),0xπ,a∈R且a≠0,u(0,t)=v1(t),u(π,t)=v2(t),t0,u(x,0)=g(x),ut(x,0)=h(x),0≤x≤π研究了可以唯一决定未知函数组{v1(t),v2(t),f(t)}的基本条件,提出了该定解问题的反问题,并且讨论了此反问题的存在性与唯一性.     

非线性两点边值问题的存在唯一性

对于非线性两点边值问题x″=f(t,x,x′),　x(0)=A,　x(1)=B,我们在f,fx,fx′,β(t),α′(t)都连续,且fx≥-β(t), -α(t)≤fx′≤M(1+|x′|),max β(t)-α2(t)+2α′(t)4t∈[0,1] ≤λ2<π24,α(1)≤2λcotλ,这些条件之下证明解之存在且唯一.     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

Lienard方程周期解、概周期解的存在性

本文考虑 Lienard方程 x″+f (x) x′+g(x) =e(t) ,我们得到 :当 -∞ 0且 0 相似文献   

陕西省第六次大学生高等数学竞赛本科组初赛试题

一、填空题:(6×4′=24′)1·设[x]表示不超过x的最大整数,则limx→0sinx|x|-2[x]=1.2·d4dx42 x1-x2x=0=48.3·设函数f(x,y)可微,f(0,0)=0,fx(0,0)=m,fy(0,0)=n,φ(t)=f[t,f(t,t)],则φ′(0)=m mn n2.4·设ddx∫2xf(2t)dt=x(x>0),则∫f(x)dx=-61x3 c.5·设f(x)在区间[-π,π]上连续,且满足f(x-π)=-f(x),则f(x)的傅立叶系数a2n=0.6·设质点在变力F=(3x 4y)i (7x-y)j的作用下,沿椭圆ax2 y2=4的逆时针方向运动一周所作的功等于6π,则a=4.二、选择题(8×4′=32′)7·当x→0时,下列无穷小量中最高阶的无穷小量是(D)A·∫0x1n(1 t3/2)dt;B·ta…     

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

A Note on First Order Differential Equation

In this note, a theorem and its three corollaries on solution of the first order ordinary differential equation are given. Theorem Suppose that b, F∈C,a∈C~1,b(y)≠0. If a(t) and b(t) satisfy the equality a′(t)b(t)=1, (1) then the first order differential equation y′=b(y)F(x,a(y)) (2) has a solution y=f(u) (3) where u=u(x) is a solution of the equatien