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## Boundary Data (boundary + data)
## Selected Abstracts## A hypersingular time-domain BEM for 2D dynamic crack analysis in anisotropic solids INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2009M. WünscheAbstract A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## STORMFLOW SIMULATION USING A GEOGRAPHICAL INFORMATION SYSTEM WITH A DISTRIBUTED APPROACH, JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION, Issue 4 2001Zhongbo YuABSTRACT: With the increasing availability of digital and remotely sensed data such as land use, soil texture, and digital elevation models (DEMs), geographic information systems (GIS) have become an indispensable tool in preprocessing data sets for watershed hydrologic modeling and post processing simulation results. However, model inputs and outputs must be transferred between the model and the GIS. These transfers can be greatly simplified by incorporating the model itself into the GIS environment. To this end, a simple hydrologic model, which incorporates the curve number method of rainfall-runoff partitioning, the ground-water base-flow routine, and the Muskingum flow routing procedure, was implemented on the GIS. The model interfaces directly with stream network, flow direction, and watershed boundary data generated using standard GIS terrain analysis tools; and while the model is running, various data layers may be viewed at each time step using the full display capabilities. The terrain analysis tools were first used to delineate the drainage basins and stream networks for the Susquehanna River. Then the model was used to simulate the hydrologic response of the Upper West Branch of the Susquehanna to two different storms. The simulated streamflow hydrographs compare well with the observed hydrographs at the basin outlet. [source] ## Finite energy solutions of self-adjoint elliptic mixed boundary value problems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2010Giles AuchmutyAbstract This paper describes existence, uniqueness and special eigenfunction representations of H1 -solutions of second order, self-adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H1(,). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H1(,). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Multi-periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n -torus MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009Denis ConstalesAbstract In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n -torus. Using the Clifford algebra calculus, the solutions can be expressed as multi-periodic eigensolutions to the Dirac operator associated with a complex parameter ,,,. Physically, these can be interpreted as the solutions to the time-harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green-type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n -torus. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009Yi ZhouAbstract In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263,1317; Nonlinear Anal. 1997; 28:1299,1322; Chin. Ann. Math. 2004; 25B:37,56). We give a new, very simple proof of this result and also give a sharp point-wise decay estimate of the solution. Then, we consider the mixed initial-boundary-value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59,78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point-wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2008Clair PoignardAbstract We study the behaviour of steady-state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, ,) surrounded by a thin membrane of thickness h and of complex permittivity , (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady-state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter , is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady-state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and , with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Weak solutions for time-dependent boundary integral equations associated with the bending of elastic plates under combined boundary data MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2004Igor ChudinovichAbstract The existence, uniqueness, stability, and integral representation of distributional solutions are investigated for the equations of motion of a thin elastic plate with a combination of displacement and moment-stress components prescribed on the boundary. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Approximation by herglotz wave functions MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2004Norbert WeckBy a general argument, it is shown that Herglotz wave functions are dense (with respect to the C,(,)-topology) in the space of all solutions to the reduced wave equation in ,. This is used to provide corresponding approximation results in global spaces (eg. in L2-Sobolev-spaces Hm(,)) and for boundary data. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Complex-distance potential theory, wave equations, and physical wavelets MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002Gerald KaiserPotential theory in ,n is extended to ,n by analytically continuing the Euclidean distance function. The extended Newtonian potential ,(z) is generated by a (non-holomorphic) source distribution ,,(z) extending the usual point source ,(x). With Minkowski space ,n, 1 embedded in ,n+1, the Laplacian ,n+1 restricts to the wave operator ,n,1 in ,n, 1. We show that ,,(z) acts as a propagator generating solutions of the wave equation from their initial values, where the Cauchy data need not be assumed analytic. This generalizes an old result by Garabedian, who established a connection between solutions of the boundary-value problem for ,n+1 and the initial-value problem for ,n,1 provided the boundary data extends holomorphically to the initial data. We relate these results to the physical avelets introduced previously. In the context of Clifford analysis, our methods can be used to extend the Borel,Pompeiu formula from ,n+1 to ,n+1, where its riction to Minkowski space ,n, 1 provides solutions for time-dependent Maxwell and Dirac equations. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2002Krzysztof CheThis article introduces the notion of weak-type solutions for systems of equations from the theory of inelastic deformations, assuming that the considered model is of monotone type (for the definition see [Lecture Notes in Mathematics, 1998, vol. 1682]). For the boundary data associated with the initial-boundary value problem and satisfying the safe-load condition the existence of global in time weak-type solutions is proved assuming that the monotone model is rate-independent or of gradient type. Moreover, for models possessing an additional regularity property (see Section 5) the existence of global solutions in the sense of measures, defined by Temam in Archives for Rational Mechanics and Analysis, 95: 137, is obtained, too. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Explicit polynomial preserving trace liftings on a triangle MATHEMATISCHE NACHRICHTEN, Issue 5 2009Mark AinsworthAbstract We give an explicit formula for a right inverse of the trace operator from the Sobolev space H1(T) on a triangle T to the trace space H1/2(,T) on the boundary. The lifting preserves polynomials in the sense that if the boundary data are piecewise polynomial of degree N, then the lifting is a polynomial of total degree at most N and the lifting is shown to be uniformly stable independently of the polynomial order. Moreover, the same operator is shown to provide a uniformly stable lifting from L2(,T) to H1/2(T). Finally, the lifting is used to construct a uniformly bounded right inverse for the normal trace operator from the space H(div; T) to H,1/2(,T) which also preserves polynomials. Applications to the analysis of high order numerical methods for partial differential equations are indicated (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Determining the temperature from incomplete boundary data MATHEMATISCHE NACHRICHTEN, Issue 16 2007B. Tomas JohanssonAbstract An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L2 -space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Equivalence of weak Dirichlet's principle, the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functions MATHEMATISCHE NACHRICHTEN, Issue 4 2006Christian G. SimaderAbstract For boundary data , , W1,2(G ) (where G , ,N is a bounded domain) it is an easy exercise to prove the existence of weak L2 -solutions to the Dirichlet problem ",u = 0 in G, u |,G = , |,G". By means of Weyl's Lemma it is readily seen that there is , , C,(G ), ,, = 0 and , = u a.e. in G . On the contrary it seems to be a complicated task even for this simple equation to prove continuity of , up to the boundary in a suitable domain if , , W1,2(G ) , C0(). The purpose of this paper is to present an elementary proof of that fact in (classical) Dirichlet domains. Here the method of weak solutions (resp. Dirichlet's principle) is equivalent to the classical approaches (Poincaré's "sweeping-out method", Perron's method). (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005Mehdi DehghanAbstract Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] ## Semilinear parabolic problem with nonstandard boundary conditions: Error estimates NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Marián SlodiAbstract We study a semilinear parabolic partial differential equation of second order in a bounded domain , , ,N, with nonstandard boundary conditions (BCs) on a part ,non of the boundary ,,. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ,non is given, and the solution along ,non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ,(t). We prove the well-posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167,191, 2003 [source] ## About smoothness of solutions of the heat equations in closed, smooth space-time domains COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2005Hongjie DongWe consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ,2 = {(x1, x2)} of class C2k. We give sufficient conditions for u to have kth -order continuous derivatives with respect to (x1, x2) in D, for integers k , 2. The equation is supplemented with C2k boundary data, and we assume that f , C2(k,1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc. [source] ## An inverse problem for the dynamical Lamé system with two sets of boundary data COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2003Oleg ImanuvilovWe prove uniqueness and a Hölder-type stability of reconstruction of all three time-independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements. In proofs we use Carleman-type estimates in Sobolev spaces of negative order. © 2003 Wiley Periodicals, Inc. [source] |