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1.
众所周知,不等式a≤c≤a中蕴涵着等量关系c=a,不等式g(x)≤f(x+k)-f(x)≤g(x)(x∈R)中蕴涵着等量关系f(x+k)-f(x)-g(x).若函数g(x)已知,再给出f(x0)的值以及n(n∈R且n≥2),就可以求出f(x0+nk)=f(x0)+∑i=0^n-1g(x0+ik)这一函数值.  相似文献   

2.
This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut+(u^2/2)x+px=εuxx, t〉0,x∈R, -αPxx+P=f(u)+α/2ux^2-1/2u^2, t〉0,x∈R, (E) with the initial data u(0,x)=u0(x)→u±, as x→±∞ (I) Here, u_ 〈 u+ are two constants and f(u) is a sufficiently smooth function satisfying f" (u) 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u+, the above Riemann problem admits a unique global entropy solution u^R(x/t) u^R(x/t)={u_,(f′)^-1(x/t),u+, x≤f′(u_)t, f′(u_)t≤x≤f′(u+)t, x≥f′(u+)t. Let U(t, x) be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0(x) - U(0,x) ∈ H^1(R) and u_ 〈 u+, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends to the rarefaction wave u^R(x/t) as → +∞ in the maximum norm. The proof is given by an elementary energy method.  相似文献   

3.
构造二次函数巧用判别式解一类题   总被引:1,自引:1,他引:0  
田辉 《数学通讯》2009,(12):17-19
判别式△=b^2-4ac是二次函数f(x)=ax^2+bx+c(a≠0)的一个重要的特征数字,其一条性质:若f(x)=ax^2+bx+c且a〉0,则f(x)≥0对x∈R恒成立 △≤0,为我们利用二次函数解决一些数学问题提供了突破IZl.本文将利用这一性质,构造适当二次函数,灵活解决一类问题.  相似文献   

4.
解法辩析     
百友 《中学生数学》2010,(2):48-48,45
1.问题 题1已知二次函数f(x)满足条件:(1)f(-1)=0;(2)对一切x∈R,都有x≤f(x)≤1+x^2/2,求f(x)的解析式. 杨忠老师在《求解不等式命题的另类方法》(见本刊2009,1上期)中给出解答如下:  相似文献   

5.
Let Ω IR^N, (N ≥ 2) be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- := inf pΩ(x) ≤ p+ = supp(x) Ω〈∞,
and f:Ω^-× IR be a C^1 function with f(x,s) ≥ 0, V (x,s) ∈Ω × R^+ and sup ∈Ωf(x,s) ≤ C(1+s)^q(x), Vs∈IR^+,Vx∈Ω for some 0〈q(x) ∈C(Ω^-) satisfying 1 〈p(x) 〈q(x) ≤p^* (x) -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^+ ≤ q- ≤ q+. As usual, p* (x) = Np(x)/N-p(x) if p(x) 〈 N and p^* (x) = ∞- if p(x) if p(x) 〉 N. Consider the functional I: W0^1,p(x) (Ω) →IR defined as
I(u) def= ∫Ω1/p(x)|△|^p(x)dx-∫ΩF(x,u^+)dx,Vu∈W0^1,p(x)(Ω),
where F (x, u) = ∫0^s f (x,s) ds. Theorem 1.1 proves that if u0 ∈ C^1 (Ω^-) is a local minimum of I in the C1 (Ω^-) ∩C0 (Ω^-)) topology, then it is also a local minimum in W0^1,p(x) (Ω)) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below.  相似文献   

6.
邓聚成 《数学季刊》1992,7(3):77-87
本文讨论含有溶质的流体在两层多孔介质中的渗流问题,即(θ(x,U)t=(K(x,U)Ux-K(x,U))x,(x,t)∈GT,(θ(x,U)V(x,t)t=(DθVx)x-(V(KUx-K))x,(x,t)∈GT,U(x,0)=U0(x),V(x,0)=V0(x),0≤x≤2,U(0,t)-h0(t),U(2,t)=h2(t),0≤t≤T,V(0,t)=g0(t),V(2,t)=g2(t),0≤t≤T。其中θ(x,U)=θ1(x,U),当(x,t)∈D1={0≤x≤1,0≤t≤T};θ(x,U)=θ2(x,U)当(x,t)∈D2+1{1<x≤2,0≤t≤T}。K(x,U)=K1(x,U)当(x,t)∈D1;K(x,U)=K2(x,U),当(x,t)∈D2。θi,Ki分别是Di上的介质含水率及水力传导率,V是溶质的浓度,此外还要求U,V,K(x,U)(Ux-1)及DθVx V(KUx-K)在x=1连续。  相似文献   

7.
On Approximation by Reciprocals of Spherical Harmonics in L p Norm   总被引:1,自引:0,他引:1  
Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f(x)∈L^p(S^q-1),f(x)≥0,f absohutely unegual to 0,1≤p≤+∞,Then,it is proved in the present paper that there is a spherical harmonics PN(x) of order≤N and a constant C〉0 such that where ω(f,δ)L^p=sup 0〈t≤δ‖St(f)-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω(f,N^-1)L^p,St(f,μ)=1/|S^q-2|Sin^2λt ∫-μμ’=t f(μ')dμ' is a translation operator.  相似文献   

8.
金山  鲁世平 《数学研究》2008,41(3):280-286
考虑共振情形下三阶微分方程m-点边值问题x'''(t)=f(t,x(t),x'(t),x"(t))+p(t),t∈(0,1), x(0)=0,x"(0)=0,x'(0)=0,x'(1)=∑i=1^m-2 aix'(ξi),其中ai≥0,0〈ξ1〈ξ2〈…〈ξm-2〈1且∑i=1^m-2 ai=1.利用Mawhin重合度拓展定理,得到该问题解存在性的新的结果.  相似文献   

9.
This paper is concerned with a nonlocal hyperbolic system as follows utt = △u + (∫Ωvdx )^p for x∈R^N,t〉0 ,utt = △u + (∫Ωvdx )^q for x∈R^N,t〉0 ,u(x,0)=u0(x),ut(x,0)=u01(x) for x∈R^N,u(x,0)=u0(x),ut(x,0)=u01(x) for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.  相似文献   

10.
In this paper, we obtain some nonoscillatory theories of the functional differential equation (r(t)ψ(x(t))x (t)) + f(t, x(t), x(σ(t))) = 0, t ≥ t 0 , where r ∈ C 1 ([t 0 , ∞); (0, ∞)), ψ∈ C 1 (R, R) and f ∈ C([t 0 , ∞) × R × R, R).  相似文献   

11.
用消元法程序在计算机上求含m个方程n个未知量的线性代数方程组的通解的问题现在仍然没有得到解决.本文首先给出一个称为“通解矩阵”的新定义,然后证明两个有关定理.以此为基础,使解决上述问题成为一件容易的事.  相似文献   

12.
本文从分析含时矢量的Stokes-Helmholtz分解着手,给出了均匀各向同性介质中弹性动力学Lamé方程的通解.  相似文献   

13.
本文研究退化时滞差分系统Ex(k+ 1)= Ax(k)+ ∑li= 1Bix(k- i)+ f(k) (k= 0,1,2,…),x(k)= φ(k) (k= 0,- 1,- 2,…,- l),其中E、A、Bi∈Rm ×n,x(k)∈Rn,f(k)∈Rm ,rank(E)< n.给出了上述系统解的存在性条件及通解表达式.  相似文献   

14.
对于3阶非齐次线性微分方程y''+py'+qy'+ry=f,由它对应齐次方程的2个线性无关特解y1,y2与其Wronski行列式W,应用降阶法推导出一个求解公式为y=y2(C3+∫w/y21(C2+∫y1/w2 e-∫pdx(c1+∫w2/y21 fe∫pdx dx)dx)dx).  相似文献   

15.
求矩阵广义逆的另一种初等变换方法   总被引:1,自引:0,他引:1  
讨论了当矩阵A为满秩矩阵时求其广义逆的一种方法,并将此方法推广,给出当A为非满秩矩阵时求其广义逆的一般方法,同时给出算例.本文推广了文献[1]的结果.  相似文献   

16.
17.
U型波纹管整体弯曲问题的一般解   总被引:6,自引:4,他引:2  
利用朱卫平等提出的柔性细环壳整体弯曲的一般解和相应的环板方程的解给出了U型波纹管在子午面内受纯弯矩作用整体弯曲的应力分布和转角公式。把计算的结果和按EJMA(Stan-dardsofthe Expansion Joint Manufacturers Association)计算的结果以及黎廷新等的实验结果作了分析比较。  相似文献   

18.
The general measurable solution of (A) was found by Stamate [8]. Aczél [3] and Lajkô [6] proved that the general solution of (A) for unknown functions ψ, g, h: ? → ? are (1), (2) and (3), respectively. Filipescu [5] found the general measurable solution of (B). We establish an elementary prof for the general solution of equation (A) (Theorem 1.). Our method is suitable for finding the general solution of (B) (Theorem 2.).  相似文献   

19.
线性规划问题通解表示的注记   总被引:5,自引:2,他引:3  
本讨论了线性规划(LP)的多解问题,且给出LP问题多解时通解的表示,以及如何探求。  相似文献   

20.
关于圆柱壳定解方程的通解问题   总被引:2,自引:0,他引:2  
薛毅  薛大为 《应用数学和力学》1996,17(12):1045-1050
本文证明了Vlasov[5]所提出的以应变一应力函数F(ξ,η)所表达的解,为圆柱壳的定解偏微分方程组的通解。  相似文献   

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