*20c Da u(t) + l[ f( t,u(t) ) + q(t) ] = 0, 0 < t < 1, u(0) = 0, u(1) = bu(h), \begin{array}{*{20}{c}} {{{\mathbf{D}}^\alpha }u(t) + {{\lambda }}\left[ {f\left( {t,u(t)} \right) + q(t)} \right] = 0,\quad 0 < t < 1,} \\ {u(0) = 0,\quad u(1) = \beta u(\eta ),} \\ \end{array} 相似文献
9.
Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p( t) in the inverse linear parabolic partial differential equation ut = uxx + p( t) u + φ, in [0,1] × (0, T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫ 01k( x) u( x, t) dx = E( t), 0 ≤ t ≤ T, for known functions E( t), k( x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫ 0s(t) u( x, t) dx = E( t), 0 < t ≤ T, 0 < s( t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
10.
A T-space U of degree k is a ( k + 1)-dimensional vector space over
(the real line) of real-valued functions defined on a linearly ordered set, satisfying the condition: for every nonzero u ε U, Z( u), the number of distinct zeros of u and -(u), the number of alternations in sign of u( t) with increasing t, each do not exceed k. It is demonstrated that given a T-space U of degree k > 0 on an arbitrary linearly ordered set T, there is a subset T′ of the real line and a nonsingular linear map L: U→ C( T′), the set of continuous functions on T′, such that the following hold: L( U) is a T-space of degree k; for u ε U, Z( u) = Z( L( u)), S−(u) = S−(L(u); and for some order-preserving bijection Θ: T → T′, u( t) = O if and only if L( u)(Θ( t) = 0. It is also shown that a T-space on a subset T
can be extended to a T-space on the closure of T in ]inf T, sup T], provided that there are no “interval gaps” in T. Examples show that, in general, a T-space cannot be extended across an “interval gap” in its domain, and cannot be extended to both the infimum and supremum of its domain. Conditions for a T-space to be Markov, and to admit an adjoined function are derived. 相似文献
11.
This paper investigates the boundary value problem for elastic beam equation of the form
u"(t) = q(t)f(t,u(t)u¢(t),u"(t),u"¢(t)),0 < t < 1,u'(t) = q(t)f(t,u(t)u'(t),u'(t),u'(t)),0 < t < 1, 相似文献
12.
The non-characteristic Cauchy problem for the heat equation uxx( x, t) = u1( x, t), 0 ? x ? 1, ? ∞ < t < ∞, u(0, t) = φ( t), ux(0, t) = ψ( t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate. 相似文献
13.
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
14.
The asymptotic conjugation relation is established for all ƒ L2( Rn) under mild assumptions on and g, where
denotes Fourier multiplication. The asymptotic estimate for finite energy solutions u of the wave equation is deduced from (*), along with generalizations to a class of first-order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient, and a weakened version for the Klein-Gordon equation. Also deduced from (*) is the fact that for a free Schrödinger particle the probability of being in the set tA at time t tends to the probability that the velocity is in A as t → ±∞. 相似文献
15.
The problem of determining the pair w:={ F( x, t); T0( t)} of source terms in the parabolic equation ut=( k( x) ux) x+ F( x, t) and Robin boundary condition − k( l) ux( l, t)= v[ u( l, t)− T0( t)] from the measured final data μT( x)= u( x, T) is formulated. It is proved that both components of the Fréchet gradient of the cost functional can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method. 相似文献
16.
For the equation K( t) u
xx
+ u
tt
− b
2
K( t) u = 0 in the rectangular domain D = “( x, t)‖ 0 < x < 1, − α < t < β”, where K( t) = (sgn t)| t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), − α ≤ t ≤ β, u( x, β) = φ( x), u( x,− α) = ψ( x), 0 ≤ x ≤ 1. 相似文献
17.
Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂ t) u=0 in with ( u,∂ tu)| t=0=( u0, u1). Because the L2 estimates and the L∞ estimates of the solution u( t) are well known, in this paper we pay attention to the Lp estimates with 1 p<2 (in particular, p=1) of the solution u( t) for t0. In order to derive Lp estimates we first give the representation formulas of the solution u( t)=∂ tS( t) u0+ S( t)( u0+ u1) and then we directly estimate the exact solution S( t) g and its derivative ∂ tS( t) g of the dissipative wave equation with the initial data ( u0, u1)=(0, g). In particular, when p=1 and n1, we get the L1 estimate: u( t) L1Ce−t/4( u0Wn,1+ u1Wn−1,1)+ C( u0L1+ u1L1) for t0. 相似文献
18.
Let Ω be a plane bounded region. Let U = { Uμ( P):μ( P)ε L∞(Ω), uμ ε H22, 0(Ω) and a( P, μ( P)) uμ,xx + 2 b( P, μ( P)) uμ,xy + c( P, μ( P)) uμ,vv = ƒ( P) for P ε Ω; here we are given a( P, X), b( P, X), c( P, X) ε L∞(Ω × E1), ƒ( P) ε Lp(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ( P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ 0 ε L∞(Ω) such that uμ0 ( P) uμ( P) for all P ε Ω and for each μ ε L∞(Ω). Similar results are obtained for a difference equation and convergence is proved. 相似文献
19.
We consider the nonnegative solutions to the nonlinear degenerate parabolic equation ut = ( D( x, t) um − 1ux) x − b( x, t) up with m > 1, 0 < p < 1, and positive D( x, t), b( x, t). After obtaining the uniqueness and Hölder regularity results, we investigate the dependence of such phenomena as extinction in finite time and instantaneous shrinking of the support on the behaviour of D( x, t) and b( x, t). 相似文献
20.
Let k = (k α) αε, be a positive-real valued multiplicity function related to a root system , and Δ k be the Dunkl-Laplacian operator. For ( x, t) ε N, × , denote by uk( x, t) the solution to the deformed wave equation Δ kuk,( x, t) = δ ttuk( x, t), where the initial data belong to the Schwartz space on N. We prove that for k 0 and N l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if ( N − 3)/2 + Σ αε+ kα ε . Here + is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of uk( x, t) is contained in the conical shell {( x, t), ε N × | | t| − R x | t| + R}. Our approach uses the representation theory of the group SL(2, ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the ( t-independent) energy functional of uk is, for large | t|, partitioned into equal potential and kinetic parts. 相似文献
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