共查询到20条相似文献,搜索用时 406 毫秒
1.
T.A. Suslina 《Functional Analysis and Its Applications》2010,44(4):318-322
Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in Rd is studied. The coefficients are assumed to be periodic in Rd with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(Rd) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the
Sobolev space H
1(Rd) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector
term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H
1(Rd) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained. 相似文献
2.
V. P. Maslov 《Mathematical Notes》1995,58(5):1166-1177
We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space
Γ(L
1(ℝ
n
)) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL
1(ℝ
n
) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L
1(ℝ
n
)) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability
for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with
small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent
characteristic equation.
Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995.
The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible.
This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300. 相似文献
3.
M. S. Ratinsky 《Journal of Mathematical Sciences》2007,141(1):1071-1074
Problems on reciprocal transformation of solutions to equations of Λ2-class (equations related to special coordinate nets on the Lobachevsky plane Λ2) are discussed. A method of construction of solutions to one analytic differential equation of Λ2-class by a given solution of another analytic differential equation of this class is proposed. The reciprocal transformation
of one-soliton solutions of the sine-Gordon equation and one-soliton solutions of the modified Korteweg-de Vries equation
(MKdV) is obtained. This result confirms the possibility of construction of such transition.
__________
Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry,
2005. 相似文献
4.
The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions
is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of
dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L
2(Ω) and H
1(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is
also addressed. 相似文献
5.
On the validity of the Ginzburg-Landau equation 总被引:1,自引:0,他引:1
A. van Harten 《Journal of Nonlinear Science》1991,1(4):397-422
Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when
a control parameterR lies in the unstable regionO(ε
2) away from the critical valueR
c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical
problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation
for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic)
or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation
is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense.
For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem
for G-L's equation on a finite interval of theO(1/ε2)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which
the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution
equation with corresponding initial conditions, has a unique solutionO(ε2) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general
initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited
from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact
solution with a relatively small error. 相似文献
6.
G. I. Shishkin 《Computational Mathematics and Mathematical Physics》2009,49(10):1748-1764
The initial-boundary value problem in a domain on a straight line that is unbounded in x is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation
is multiplied by a parameter ɛ2, where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as x → ∞ at a rate of O(x
2). This causes the unbounded growth of the solution at infinity at a rate of O(Ψ(x)), where Ψ(x) = x
2 + 1. The initialboundary function is piecewise smooth. When ɛ is small, a boundary and interior layers appear, respectively,
in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation
passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid
solution grows unboundedly in the maximum norm as x → ∞ even for smooth solutions when ɛ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem
and its grid approximations is considered in the weighted maximum norm ∥·∥
w
with the weighting function Ψ−1(x); in this norm, the solution of the initial-boundary value problem is ɛ-uniformly bounded. Using the method of special grids
that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite
difference schemes are constructed and studied that converge ɛ-uniformly in the weighted norm. It is shown that the convergence
rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy
problem with the right-hand side and the initial function growing as O(Ψ(x)) that converge ɛ-uniformly in the weighted norm are also considered. 相似文献
7.
M. A. Freedman 《Semigroup Forum》1987,36(1):117-126
In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x0, where A is an accretive operator in a general Banach space X and f ε L1(0,T;X). Their method involves proving convergence in the L∞-norm of a sequence of step function approximations αn(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show
a.e. existence of mild solution by proving convergence of the step functions αn(σ, τ) in the L1-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane. 相似文献
8.
V. G. Gordevskii 《Theoretical and Mathematical Physics》1998,114(1):99-108
An explicit approximate solution to the three-dimensional nonlinear Boltzmann equation for rigid spheres is constructed. It
has the form of a spatially inhomogeneous linear combination of two Maxwellians corresponding to different densities, temperatures,
and mass velocities. It is shown that the integral norm of the discrepancy between the left- and right-hand sides of the equation
can be made arbitrarily small by choosing appropriate values of the parameters entering the distribution.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 1, pp. 126–136, January, 1998. 相似文献
9.
We establish an estimate for the rate of convergence of a solution of an ordinary stochastic differential equation of order
p ≥ 2 with a small parameter in the coefficient of the leading derivative to a solution of a stochastic equation of order p − 1 in the metric ρ(X, Y) = (sup0≤t≤T
M|X(t) − Y(t)|2)1/2
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1587–1601, December, 2006. 相似文献
10.
S. A. Nazarov 《Journal of Mathematical Sciences》2010,167(5):713-725
An asymptotic model is found for the Neumann problem for the second-order differential equation with piecewise constant coefficients
in a composite domain Ω∪ω, which are small, of order ε, in the subdomain ω. Namely, a domain Ω(ε) with a singular perturbed
boundary is constructed, the solution for which provides a two-term asymptotic, that is, of increased accuracy O(ε2| log ε|3/2), approximation to the restriction to Ω of the solution of the original problem. As opposed to other singularly perturbed
problems, in the case of contrasting stiffness, the modeling requires the construction of a contour ∂Ω(ε) with ledges, i.e.,
with boundary fragments of curvature O(ε−1). Bibliography: 33 titles. 相似文献
11.
Kenji Nishihara 《Mathematische Zeitschrift》2003,244(3):631-649
It has been asserted that the damped wave equation has the diffusive structure as t→∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding
parabolic equation, and obtain the L
p
−L
q
estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions
are analyzed for the proof. The second aim is to apply the L
p
−L
q
estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent,
then the time global existence of small weak solution is proved and its optimal decay order is obtained.
Received: 8 June 2001; in final form: 12 August 2002 /
Published online: 1 April 2003
Mathematical Subject Classification (2000): 35L15. 相似文献
12.
For a bounded C
1,α domain in ℝ
d
we show that there exists a strong solution to the multidimensional Skorokhod equation and that weak uniqueness holds for
this equation. These results imply that pathwise uniqueness and strong uniqueness hold for the Skorokhod equation.
Received: 3 February 1999 / Revised version: 2 September 1999 /?Published online: 11 April 2000 相似文献
13.
Criterion of periodicity of solutions of a certain differential equation with a periodic coefficient
F. Neuman 《Annali di Matematica Pura ed Applicata》1967,75(1):385-396
Summary In this paper the differential equation (1) y″=q(t)y is considered where q(t) is a real continuous function with period π.
There is proved a necessary and sufficient condition for the stability of the trivial solution of Equation (1) when the zeros
of the characteristic equation λ2 - Aλ+1=0, coincide. Moreover, there is shown the construction of all Equations (1) admitting only periodic or half-periodic
solutions with period π. 相似文献
14.
LungAnYING 《数学学报(英文版)》2004,20(5):859-868
Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated. 相似文献
15.
G. I. Shishkin L. P. Shishkina 《Computational Mathematics and Mathematical Physics》2011,51(6):1020-1049
In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach
is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order
derivatives converge in the maximum norm uniformly with respect to a perturbation parameter ɛ ∈(0, 1]; the normalized derivatives
are ɛ-uniformly bounded. The key idea of this approach to the construction of ɛ-uniformly convergent finite difference schemes
is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction
technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and
second-order derivatives converge ɛ-uniformly at the rate of O(N
−2ln2
N), where N + 1 is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition
method is constructed such that its solution and its normalized first and second derivatives converge ɛ-uniformly in the maximum
norm at the same rate of O(N
−4ln4
N). 相似文献
16.
We study nonlinear wave and heat equations on ℝ
d
driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ
d
-space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance
kernel of the noise.
Received: 1 April 1998 / Revised version: 23 June 1999 / Published online: 7 February 2000 相似文献
17.
Joanna Janczewska 《Central European Journal of Mathematics》2004,2(4):561-572
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R
k
→Y is a C
2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R
1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R
k
and we describe the solution set of the studied equation in a small neighbourhood of this point. 相似文献
18.
The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space Hs (R) with 1 〈 s ≤ 3/2 is derived. 相似文献
19.
T. Shaposhnikova 《Journal of Mathematical Sciences》1995,75(3):1631-1645
The behavior of the solution of a boundary value problem for a parabolic equation with rapidly oscillating coefficientsɛ
−1
x,ɛ
−2k
t), (k⋝0) in a perforated domain for ε→0 is studied. Some estimates of the deviation of the solution and energy for the original
boundary value problem from the solution and energy of the corresponding homogenized problem are found. In this investigation
methods developed by Oleinik, Zhikov, Kozlov, Bensoussan, Lions, Papanikolaou, Cioranescu, and Paulin are used. Bibliography:
15 titles.
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 27–50, 1994. 相似文献
20.
I. A. Blatov N. V. Dobrobog 《Computational Mathematics and Mathematical Physics》2010,50(9):1476-1493
The boundary value problem for the ordinary differential equation of reaction-diffusion on the interval [−1, 1] is examined.
The highest derivative in this equation appears with a small parameter ɛ2 (ɛ ∈ (0, 1]). As the small parameter approaches zero, boundary layers arise in the neighborhood of the interval endpoints.
An algorithm for the construction of a posteriori adaptive piecewise uniform grids is proposed. In the adaptation process,
the edges of the boundary layers are located more accurately and the grid on the boundary layers is repeatedly refined. To
find an approximate solution, the finite element method is used. The sequence of grids constructed by the algorithm is shown
to converge “conditionally ɛ-uniformly” to some limit partition for which the error estimate O(N
−2ln3
N) is proved. The main results are obtained under the assumption that ɛ ≪ N
−1, where N is number of grid nodes; thus, conditional ɛ-uniform convergence is dealt with. The proofs use the Galerkin projector and
its property to be quasi-optimal. 相似文献