首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献,搜索用时 31 毫秒
1.
For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q t and q xxx have the same sign (KdVI) or two boundary conditions if q t and q xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q x (0,t),q xx (0,t)} and {q xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ (t)(t,k), where Φ (t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ (t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ (t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ (t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.  相似文献   

2.
We consider the asymptotic nonlinear filtering problem dx=f(x)dt + ?1/2 dw,dy=h(x) dt + ? dv and obtain lim?→0 ? log q 2(x,t) = -W(x,t) for unnormalized conditional densities q 2(x,t) using PDE methods. HereW(x,t) is the value function for a deterministic optimal control problem arising in Mortensen's deterministic estimation, and is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. ijab has also studied this filtering problem, and we extend his large deviation result for certain unnormalized conditional measures. The resulting variational problem corresponds to the above control problem  相似文献   

3.
We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D q x(t)=Ax(t)+t n f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ+, x(0)+g(x)=x 0, where 0<q<1, A is the infinitesimal generator of a C 0-semigroup of bounded linear operators on a Banach space X.  相似文献   

4.
An Application of a Mountain Pass Theorem   总被引:3,自引:0,他引:3  
We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, uH 1 0(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L -function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ s 0 f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.  相似文献   

5.
6.
We develop the Krasnoselskii–Krein type of uniqueness theorem for an initial value problem of the Riemann–Liouville type fractional differential equation which involves a function of the form f?(t,?x(t),?D q?1 x(t)), for 1<q<2 and establish the convergence of successive approximations. We prove a few other uniqueness theorems.  相似文献   

7.
When m = qt, g(xt+1, x2t+1,…, x(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x1, x2,…, xm) = x1 + g(xt+1, x2t+1,…, x(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).  相似文献   

8.
Sufficient conditions for continuability, boundedness, and convergence to zero of solutions of (a(t)x′)′ + h(t, x, x′) + q(t) f(x) g(x′) = e(t, x, x′) are given.  相似文献   

9.
We study the initial-boundary value problem for ?t2u(t,x)+A(t)u(t,x)+B(t)?tu(t,x)=f(t,x) on [0,T]×Ω(Ω??n) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given.  相似文献   

10.
Summary Using the integral average method, we give some new oscillation criteria for the second order differential equation with damped term (a(t)Ψ(x(t))K(x'(t)))'+p(t)K(x'(t))+q(t)f(x(t))=0, t<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>≧t0. These results improve and generalize the oscillation criteria in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1], because they eliminate both the differentiability of p(t) and the sign of p(t), q(t). As a consequence, improvements of Sobol's type oscillation criteria are obtained.  相似文献   

11.
A detailed analysis is given to the solution of the cubic Schrödinger equation iqt + qxx + 2|q|2q = 0 under the boundary conditions as |x|→∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|→∞] is discussed.  相似文献   

12.
This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u t −Δu+b(x)·∇(u|u| q −1)=f(x, t) in ℝ n ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u 0 is supposed to be in an appropriate L p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant  相似文献   

13.
We consider the existence and uniqueness of singular solutions for equations of the formu 1=div(|Du|p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon  相似文献   

14.
Let E be a pre-ordered real Banach space and f:[0,TEE a quasimontone increasing function. We prove one-sided estimates of the form +q[yx,f(t,y)–f(t,x)](t,q(yx)) with respect to seminorms q generated by a single positive linear functional. Such estimates lead to growth conditions, for example for the total variation of the solution of u=f(t,u) in function spaces.Mathematics Subject Classification (2000): 34C11, 34C12, 34G20  相似文献   

15.
We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(xt) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).  相似文献   

16.
The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q t +q xxx +6q 2 q x =0, 0x<, –<t<,q t +q xxx –6q 2 q x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L 1(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q t +q xxx –6|q|2 q x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.  相似文献   

17.
Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x¢(t))¢+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ¥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.  相似文献   

18.
Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q 1(t)q 2(t)2 ... q m (t)m, where each q i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q j (t). We give an algorithm to construct the polynomials q i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λi| of G, where λ1, λ2, ..., λn are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.  相似文献   

19.
We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:
tu−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)  相似文献   

20.
Necessary and sufficient conditions for an arbitrary q-variate stationary sequence xt, tZ, to be deterministic are presented. A characterization of the rank r(x) of xt, tZ, and a method to construct the Wold-Cramér decomposition for xt, tZ, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号