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Regular solutions to second-order elliptic systems on the plane are representable in terms of A-analytic functions satisfying an operator equation of the Beltrami type. We prove Carleman-type formulas for reconstruction of solutions from data on a part of the boundary of the domain. We use these formulas for solving the Cauchy problems for the system of Lame equations, the Navier–Stokes system, and the system of equations of elasticity with resilience.  相似文献   

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We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.  相似文献   

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We consider the problem of analytic continuation of a solution to the system of Maxwell equations in a bounded spatial domain from data on part of the boundary of the domain. We construct an approximate solution to the problem using the Carleman matrix method.  相似文献   

5.
In this work, some classes of initial boundary-value problems are investigated in the Sobolev space for the third-order linear pseudoparabolic equation having, in general, nonsmooth coefficients. A new type of Riemann function concept is given for these problems, which is more natural than the classical Riemann-type function concept, and an integral form of the solutions of nonhomogeneous problems can be found more naturally using this concept.  相似文献   

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Based on the Tikhonov regularization method, we explicitly construct a Carleman function in an ill-posed mixed problem for the Laplace equation.  相似文献   

7.
郭基风  李红 《数学季刊》2007,22(3):415-425
This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation.By the priori estimates and the method in [9],It proves that the Cauchy problem admits a unique global classical solution.And by the concave method,we give sufficient conditions on the blowup of the global solution for the Cauchy problem.  相似文献   

8.
Helmholtz方程Cauchy问题是严重不适定问题,本文我们在一个带形区域上考虑了一类Helmholtz方程Cauchy问题:已知Cauchy数据u(0,y)=g(y),在区间0<x<1上求解.我们用半离散的中心差分方法得到了这一问题的正则化解,给出了正则化参数的选取规则,得到了误差估计.  相似文献   

9.
陈国旺 《数学学报》2012,(5):797-810
证明下列非线性波动方程的Cauchy问题v_(tt)-α△v_(tt)-Δv=g(v)-αΔg(v),x∈R~N,t>0,(1)v(x,0)=v_0(x),v_t(x,0)=v_1(x),x∈R~N(2)在空间C~2([0,∞);H~s(R~N))(s>N/2)中存在唯一整体广义解v和在空间C~2([0,∞);H~s(R~N))(s>2+N/2N)中存在唯一整体古典解v,即u∈C~2([0,∞);C_B~2(R~N)).还证明Cauchy问题(1),(2)在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m≥0,1≤p≤∞)中有唯一整体广义解v和在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m>2+N/P)中有唯一整体古典解v,即v∈C~3([0,∞);C~2(R~N)∩L~∞(R~N)).  相似文献   

10.
A Carleman estimate for the stationary anisotropic Maxwell system is established. Its proof adopts a technique pioneered by Calderón to an overdetermined systems with rough coefficients. As an application, the conditional stability of the Cauchy problem is discussed.  相似文献   

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In this paper, we consider the reconstruction of the wave field in abounded domain. By choosing a special family of functions, the Cauchy problemcan be transformed into a Fourier moment problem. This problem is ill-posed. Wepropose a regularization method for obtaining an approximate solution to the wavefield on the unspecified boundary. We also give the convergence analysis and errorestimate of the numerical algorithm. Finally, we present some numerical examples toshow the effectiveness of this method.  相似文献   

13.
The authors prove Carleman estimates for the Schr(o)dinger equation in Sobolev spaces of negative orders,and use these estimates to prove the uniqueness in the inverse problem of determining Lp-potentials.An L2-1evel observability inequality and unique continuation results for the Schr(o)dinger equation are also obtained.  相似文献   

14.
A global existence theorem for the discrete Carleman system in the Sobolev class W 1,2 is proved by the Leray-Schauder topological degree method, which was not previously applied to discrete kinetic equations. The instability of the nonequilibrium steady flow on a bounded interval is established in the linear approximation.  相似文献   

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主要研究时标上三阶动力学方程u~(△△△)(t)=f(t,u(t),u~(△△)(t))在三点边值条件下正解的存在性.主要方法应用格林函数和锥上凹凸函数的性质给出其正解存在的充分条件.  相似文献   

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通过研究非线性分数阶微分方程边值问题D_(0+)~αu(t)+y(t,u(t))=0,0相似文献   

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The Cauchy problem for the generalized Korteweg-de Vries-Burgers equation is considered and the local existence and uniqueness of solutions in L^q(0, T;L^p) ∩ L^∞(0, T; \dot{H}^{-s})(0 ≤ s < 1) are obtained for initial data in \dot{H}^{-s}. Moreover, the local solutions are global if the initial data are sufficiently small in critical case. Particularly, for s = 0, the generalized Korteweg-de Vries-Burgers equation satisfies the energy equality, so the initial data can be arbitrarily large to obtain the global solution.  相似文献   

19.
Laplace方程边值问题的边界积分方程法   总被引:1,自引:1,他引:1  
孙建设  叶留青 《数学季刊》2003,18(4):429-434
§ 1. Introduction  Inengineeringandtechnology ,theproblemofstaticelectricfieldscanbeattributedtotheboundaryproblemofLaplaceequationofstaticeletricpotentialfunction .Themethodsofclassi calmathematicalphysicscanbeonlyusedtosolveboundaryproblemofverysimpledomainandspecialboundarycondition .Althoughthemethodsoflimitedelementscanbeusedtosolvetheproblemsonarbitrarydomain ,butitneedstopartitionthewholedomainandtocalculateverycomplex .Theapproachofboundaryintegralequationistosolverelatedproblemsb…  相似文献   

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