共查询到20条相似文献,搜索用时 484 毫秒
1.
P. V. Vinogradova 《Differential Equations》2010,46(7):962-972
In the present paper, we consider the Galerkin method for a quasilinear differentialoperator equation with a leading self-adjoint
operator A(t) and a subordinate monotone operator K. For the projection subspaces we take linear spans of eigenelements of an operator similar to the leading operator A(t). We obtain new estimates for the Galerkin method and consider applications to an initial-boundary value problem for a parabolic
equation of higher order. 相似文献
2.
Polina Vinogradova 《Journal of Computational and Applied Mathematics》2009,231(1):1-10
This article investigates the projection-difference method for a Cauchy problem for a linear operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in Hilbert space. This method leads to the solution of a system of linear algebraic equations on each time level; moreover, the projection subspaces are linear spans of eigenvectors of an operator similar to A(t). The convergence estimates are obtained. The application of the developed method for solving the initial boundary value problem is given. 相似文献
3.
P. V. Vinogradova 《Russian Mathematics (Iz VUZ)》2010,54(7):1-11
We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations
with a leading self-adjoint operator A(t) and a subordinate linear operator K(t), whose definition domain is independent of t. Operators A(t) and K(t) are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact
solution as well as those for fractional degrees of an operator similar to A(0). 相似文献
4.
We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem. 相似文献
5.
P. V. Vinogradova 《Differential Equations》2008,44(7):970-979
We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution. 相似文献
6.
Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations
Polina Vinogradova 《Numerical Functional Analysis & Optimization》2013,34(3):339-365
In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type. 相似文献
7.
A. G. Ramm 《Mathematical Methods in the Applied Sciences》2013,36(4):422-426
Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ0(t), λ0(t) > 0, but assume that limt → ∞λ0(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u0 = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ0(t) with λ0(t) > 0 and limt → ∞λ0(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
8.
《Optimization》2012,61(1):79-87
Let K be a closed convex order-bounded set of an order-complete vector lattice and let A be a him continuous linear operator. Then the equality AK = A(ext K) is proved, where ext K is the set of all extremal points of K. It is shown that various generalizations of the Ljapunov-theorem on the range of vector-measures are special cases of this general statement. 相似文献
9.
Peter D. Miletta 《Mathematical Methods in the Applied Sciences》1994,17(10):753-763
The convergence of the Galerkin approximations to solutions of abstract evolution equations of the form u′(t)= ? Au(t) + M(u(t)) is shown. Here A is a closed, positive definite, self-adjoint linear operator with domain D(A) dense in a Hilbert space H and M is a non-linear map defined on D(A½) which satisfies a Lipschitz condition on balls in D(A½). 相似文献
10.
In this note, we present a Massera type theorem for the existence of almost automorphic solutions of periodic linear evolution equations of the form x′(t)=A(t)x(t)+f(t), where A(t) is unbounded linear operator depending on t periodically and generates a τ-periodic evolutionary process, f is almost automorphic. The main results are stated in terms of the almost automorphy of solutions and their Carleman spectra. 相似文献
11.
Lech Zarȩba 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(4):445-467
In this paper we consider the mixed problem for the equation u
tt
+ A
1
u + A
2(u
t
) + g(u
t
) = f(x, t) in unbounded domain, where A
1 is a linear elliptic operator of the fourth order and A
2 is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially,
in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution
at infinity.
相似文献
12.
We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described. 相似文献
13.
Jitsuro Sugie 《Monatshefte für Mathematik》2009,157(2):163-176
Sufficient conditions are given for asymptotic stability of the linear differential system x′ = B(t)x with B(t) being a 2 × 2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′ = B(t)x.
相似文献
14.
Jitsuro Sugie 《Monatshefte für Mathematik》2009,110(1):163-176
Sufficient conditions are given for asymptotic stability of the linear differential system x′ = B(t)x with B(t) being a 2 × 2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′ = B(t)x. 相似文献
15.
Let K be a compact subgroup of a locally compact group G. Completely complemented ideals in A(G/K) are characterised. Biprojectivity and biflatness for the Fourier algebra A(G/K) are studied. A(G/K) is operator biprojective precisely when K is open and if this happens, then G does not contain the free group on two generators as a closed subgroup. 相似文献
16.
This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. LetA be a closed linear operator with domainD(A) dense in a Banach spaceX. We first characterize the exponential wellposedness of ACP onD(A
k+1),k teN. Next let {C(t);t teR} be a family of generalized solution operators, on [D(A
k)] toX, associated with an exponentially wellposed ACP onD(A
k+1). Then we define a new family {T(t); Ret>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H
k) onX.
Research of the second-named author was partially supported by Grant-in-Aid for Scientific Research (No. 63540139), Ministry
of Education, Science and Culture. 相似文献
17.
Uwe Franken 《Mathematische Nachrichten》1993,164(1):119-139
For a weight function ω and a closed set A ? ?N let ?(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?N there exists an ω-extension operator EA: ?(ω)(A) → ?(ω)(?N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?N there exists an ω-extension operator EK: ?(ω)(K) → ?(ω)(?N) if and only if the Fréchet space ?(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.). 相似文献
18.
Yu. M. Semenov 《Differential Equations》2011,47(11):1668-1674
We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ
n
, u ∈ U ⊆ ℝ
m
, with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives
of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria. 相似文献
19.
Larbi Berrahmoune 《Rendiconti del Circolo Matematico di Palermo》2009,58(2):275-282
We consider bilinear control systems of the form y′(t) = Ay(t) + u(t)By(t) where A generates a strongly continuous semigroup of contraction (e
t A
)
t⩾0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. The function u denotes the scalar control. We suppose that B is a linear bounded operator from the state Y into itself. Tacking into account the control saturation, we study the problem of stabilization by feedback of the form u(t)=−f(〈By(t), y(t)〉). Application to the heat equation is considered.
相似文献
20.
Peter Otte 《Journal of Mathematical Analysis and Applications》2004,289(1):167-179
Given a semi-group U(t) of bounded linear operators with bounded self-adjoint generator A we estimate the logarithm of the section determinants of U(t) in terms of A. When A is subject to an additional condition, which is related to so-called Følner sequences of orthogonal projections, this estimate implies a Szeg? type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when A is a Toeplitz operator or a compact operator. 相似文献