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1.
We consider the periodic boundary-value problem u
tt
− u
xx
= g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u
0(x, t) + ũ(x, t), where u
0(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the
period ω. We show that the relation obtained for a solution includes known results established earlier.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005. 相似文献
2.
Guoxiang Chen Meiying Wang 《分析论及其应用》2007,23(3):266-273
For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family. 相似文献
3.
N. N. Pustovoitov 《Mathematical Notes》1999,65(1):89-98
In the paper order-exact upper bounds for the best approximations of classesH
q
Emphasis>/ω by trigonometric polynomials are obtained. The spectrum of the approximating polynomials lies in sets generated by the level
surfaces of the function ω(t). These sets are a generalization of hyperbolic crosses to the case of an arbitrary function ω(t).
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 107–117, January, 1999. 相似文献
4.
Nils Svanstedt 《Applications of Mathematics》2008,53(2):143-155
Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic
behaviour of a sequence of realizations of the form ∂u
ɛ
ω
/ ∂t+1 / ɛ
3
C(T
3(x/ɛ
3)ω
3) · ∇u
ɛ
ω
− div(α(T
2(x/ɛ
2)ω
2, t) ∇u
ɛ
ω
) = f. It is shown, under certain structure assumptions on the random vector field C(ω
3) and the random map α(ω
1, ω
2, t), that the sequence {u
ɛ
ω
} of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f). 相似文献
5.
The Ulm method is considered to approximate a solution of a nonlinear operator equation F(x) = 0. We study the convergence of this method when F′ is ω-conditioned and prove that the R-order of convergence is at least 1 + p if ω is quasi-homogeneous of type ω(tz)≤ t
p
ω(z), for z > 0, tϵ[0,1] and pϵ[0,1].
Preparation of this paper was partly supported by the Ministry of Education and Science (MTM 2005-03091). 相似文献
6.
M. M. Amangalieva D. M. Akhmanova M. T. Dzhenaliev M. I. Ramazanov 《Differential Equations》2011,47(2):231-243
We continue the study of boundary value problems for spectrally loaded heat equations in unbounded domains for the case in
which the order of the derivative in the loaded term coincides with that of the differential part of the equation and the
motion of the load point with respect to the space variable is given by the law -x(t) = t
ω
, −∞ < ω < 1/2. 相似文献
7.
Roberto Paoletti 《manuscripta mathematica》2002,107(2):145-150
We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic
structure. If X is a compact manifold and the ω
t
are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ
t
of diffeomorphisms of X such that ω
t
=Φ
t
*(ω0). If L⊂X is a Lagrangian submanifold for (X,ω0), L
t
=Φ
t
-1(L) is thus a Lagrangian submanifold for (X,ω
t
). Here we show that if we simply assume that L is compact and ω
t
|
L
is exact for every t, a family L
t
as above still exists, for
sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds,
under perturbation of the ambient Calabi–Yau structure.
Received: 29 May 2001/ Revised version: 17 October 2001 相似文献
8.
If the second order problem u(t) + Bu(t) + Au(t) = f(t), u(0) =u(0) = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ(λ^2 + λB + A)^-1‖ and ‖B(λ^2 + λB + A)^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0. 相似文献
9.
Invariant tori in nonlinear oscillations 总被引:2,自引:0,他引:2
The boundedness of all the solutions for semilinear Duffing equationx″ + ω2
x + φ(x) =p(t), ω ∈ ℝ+ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded. 相似文献
10.
Summary. We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx
t
= ∑
j
=0
m
f
j
(x
t
)∘dW
t
j
and dx
t
=∑
j
=0
m
g
j
(x
t
)∘dW
t
j
in ℝ
d
with smooth coefficients satisfying f
j
(0)=g
j
(0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,
where θ
t
ω(s)=ω(t+s)−ω(t) is the (ergodic) shift on the canonical Wiener space. The normal form problem for SDE consists in finding the “simplest
possible” member in the equivalence class of a given SDE, in particular in giving conditions under which it can be linearized
(g
j
(x)=Df
j
(0)x).
We develop a mathematically rigorous normal form theory for SDE which justifies the engineering and physics literature on
that problem. It is based on the multiplicative ergodic theorem and uses a uniform (with respect to a spatial parameter) Stratonovich
calculus which allows the handling of non-adapted initial values and coefficients in the stochastic version of the cohomological
equation. Our main result (Theorem 3.2) is that an SDE is (formally) equivalent to its linearization if the latter is nonresonant.
As a by-product, we prove a general theorem on the existence of a stationary solution of an anticipative affine SDE.
The study of the Duffing-van der Pol oscillator with small noise concludes the paper.
Received: 19 August 1997 / In revised form: 15 December 1997 相似文献
11.
I. Juhász 《Israel Journal of Mathematics》1993,81(3):369-379
The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT
2 withω(X)≥κ then some λ withκ≤λ≤2<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω
1, and thus solves problem 22 of [M]. Also, it is consistent with 2ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT].
Research supported by OTKA grant no. 1908. 相似文献
12.
Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L0∞ (G,1/ω) can be identified with the strong dual of L1(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L1(G, ω), τ)**, and prove that the topological center of (L1(G, ω), τ)** is (L1(G, ω); this enables us to conclude that (L1(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L0∞ (G, 1/ω)1.
Received: 8 March 2005 相似文献
13.
Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the
nonautonomous ordinary differential equation u
(2m)
=f(t,u,...,u
(m-1)
), where the function f:ℝ×ℝ
m
→ℝ is periodic with respect to the first argument with period ω.
Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001 相似文献
14.
Given a permutation ω of {1, …, n}, let R(ω) be the root degree of ω, i.e. the smallest (prime) integer r such that there is a permutation σ with ω = σ
r
. We show that, for ω chosen uniformly at random, R(ω) = (lnlnn − 3lnlnln n + O
p
(1))−1 lnn, and find the limiting distribution of the remainder term.
Research supported in part by NSF grants CCR-0225610, DMS-0505550 and ARO grant W911NF-06-1-0076.
Research supported by NSF grant DMS-0406024. 相似文献
15.
Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()α≤≤ ()
n
−ω. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour
number μ(G) of G: n− (n−ω)()
n
−ω≤μ(G)≤n−α() α.
Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002 相似文献
16.
E. V. Derets 《Ukrainian Mathematical Journal》2000,52(3):368-378
We establish lower bounds for the Kolmogorov widths d
2n-1(W
r
H
1ω.L
p
) and Gel’fand widths d
2n-1(W
r
H
1ω.L
p
) of the classes of functions W
r
H
1ω with a convex integral modulus of continuity ω(t). 相似文献
17.
We consider the Riemann–Hilbert problem in the following setting: find a function whose boundary values ϕ+(t) satisfy the condition a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and K
p( · )(D;ω) is the set of functions ϕ(z) representable in the form , where ω(z) is a weight function and (K
Γφ
)(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-H?lder condition. The solvability conditions are established and solutions are constructed. These solutions
largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points.
Submitted: May 13, 2007. Revised: August 8, 2007 and August 28, 2007. Accepted: November 8, 2007. 相似文献
18.
We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P
ω(Y
0, λ
0)-solutions (where Y
0 is zero or ±∞ and −∞ ≤ λ0 ≤ +∞) of nonlinear differential equations y
(n) = α
0
p(t)φ(y), where α
0 ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y
0. 相似文献
19.
We obtain lower bounds for solutions of some extremal problems on classes of functions W
rH1ω with integral modulus of continuity ω(t). Some of these bounds are regarded as exact.
Dneprodzerzhinsk Technical University, Dneprodzerzhinsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No.
11. pp. 1499–1503, November, 1997. 相似文献
20.
The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves ei ω tϕω(x) for a nonlinear Schr?dinger equation with an inhomogeneous nonlinearity V (x)|u|p − 1u, where V (x) is proportional to the electron density. Here, ω > 0 and ϕω(x) is a ground state of the stationary problem. When V (x) behaves like |x|−b at infinity, where 0 < b < 2, we show that ei ω tϕω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution
for the case of V (x) = |x|−b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.
Communicated by Bernard Helffer
submitted 14/07/04, accepted 28/02/05 相似文献