摘要:

AbstractAlthough the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2and given Neumann data on the remaining part Γ2of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solutionuon Γ2and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity ofubut, on the other hand, gradupossesses singularities at the collision points Γ1∩ Γ2even forC∞data,uis decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2providing an improvement of regularity for the smooth part ofu. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1and Γ2augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm ofuand also superconvergence as well as high orders in smoother norms if the given data are smooth (and not t

ISSN:0170-4214    

DOI:10.1002/mma.1670010302

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

摘要:

AbstractIn this paper boundary controllability of one‐dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off.This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null‐controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported b

ISSN:0170-4214    

DOI:10.1002/mma.1670010303

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

摘要:

AbstractThis part is concerned with the applications of the theorems on error estimates for asymptotic expansions in the central limit theorem for general linear stochastic processes of Part I to two important typical models. These are the general random noise process and a class of pulse train processes that occur in stochastic signal theory and in a variety of digital communication schemes. The assumptions on the inherent structure of the general theorems turn out to be simple here.

ISSN:0170-4214    

DOI:10.1002/mma.1670010304

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

摘要:

AbstractIn plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh‐Clough‐Tocher elements on regular triangulations as conformC1‐finite elements a new error analysis is performedwithouthigher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2ndorder and its Laplacian are approximated simultaneously byC0‐finite elements being piecewise poly

ISSN:0170-4214    

DOI:10.1002/mma.1670010305

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

摘要:

AbstractAs, in general, the projections of characteristics into thex‐space intersect for finite values oft, the global solution of a conservation lawcannot be determined from the characteristic system of the equation,is considered. Only in the linear case, this equation coincides with the equation of the projections of characteristics. For convexhand allx0this equation has a solution almost everywhere, and the properties of this solution permit to construct a global solution of the conservation law using strips, in the same way as this is done for linear problems by the method of characteristic

ISSN:0170-4214    

DOI:10.1002/mma.1670010306

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

摘要:

AbstractGiven a semilinear reaction‐diffusion equation. If the corresponding ordinary differential equation admits a convex compact positively invariant set and the boundary data assume their values in this set then the first and third boundary value problem have stationary solutions. The proofs are based on Weinberger's strong invariance principle, some related tools and the Leray‐Schauder degree. The theorem is applied to several equations from theoretical biology, also in the case of distinct diffusion ra

ISSN:0170-4214    

DOI:10.1002/mma.1670010307

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

摘要:

AbstractWith the maximum principle for differential equations asymptotic estimates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to the stationary solution of the Kolmogorov backward equation in population genetics.

ISSN:0170-4214    

DOI:10.1002/mma.1670010308

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY

ISSN:0170-4214    

DOI:10.1002/mma.1670010301

出版商:John Wiley&Sons, Ltd    

年代:1979

数据来源: WILEY