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## Dirichlet Problem (dirichlet + problem)
## Selected Abstracts## Linear augmented Slater-type orbital method for free standing clusters JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 8 2009K. S. KangAbstract We have developed a Scalable Linear Augmented Slater-Type Orbital (LASTO) method for electronic-structure calculations on free-standing atomic clusters. As with other linear methods we solve the Schrödinger equation using a mixed basis set consisting of numerical functions inside atom-centered spheres and matched onto tail functions outside. The tail functions are Slater-type orbitals, which are localized, exponentially decaying functions. To solve the Poisson equation between spheres, we use a finite difference method replacing the rapidly varying charge density inside the spheres with a smoothed density with the same multipole moments. We use multigrid techniques on the mesh, which yields the Coulomb potential on the spheres and in turn defines the potential inside via a Dirichlet problem. To solve the linear eigen-problem, we use ScaLAPACK, a well-developed package to solve large eigensystems with dense matrices. We have tested the method on small clusters of palladium. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009 [source] ## Stopping-time resampling for sequential Monte Carlo methods JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 2 2005Yuguo ChenSummary., Motivated by the statistical inference problem in population genetics, we present a new sequential importance sampling with resampling strategy. The idea of resampling is key to the recent surge of popularity of sequential Monte Carlo methods in the statistics and engin-eering communities, but existing resampling techniques do not work well for coalescent-based inference problems in population genetics. We develop a new method called ,stopping-time resampling', which allows us to compare partially simulated samples at different stages to terminate unpromising partial samples and to multiply promising samples early on. To illustrate the idea, we first apply the new method to approximate the solution of a Dirichlet problem and the likelihood function of a non-Markovian process. Then we focus on its application in population genetics. All our examples show that the new resampling method can significantly improve the computational efficiency of existing sequential importance sampling methods. [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] ## On a Penrose,Fife type system with Dirichlet boundary conditions for temperature MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2003Gianni GilardiWe deal with the Dirichlet problem for a class of Penrose,Fife phase field models for phase transitions. An existence result is obtained by approximating the non-homogeneous Dirichlet condition with classical third type conditions on the heat flux at the boundary of the domain where the model is considered. Moreover, we prove a regularity and uniqueness result under stronger assumptions on the regularity of the data. Suitable assumptions on the behaviour of the heat flux at zero and +,are considered. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Integral equation methods for scattering by infinite rough surfaces MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2003Bo ZhangAbstract In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of time-harmonic acoustic scattering of an incident field by a sound-soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double- and single-layer potential and a Dirichlet half-plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half-plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single-layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface MATHEMATISCHE NACHRICHTEN, Issue 9 2010Giuseppe CardoneAbstract It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the essential spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The Dirichlet problem for second order parabolic operators in non-cylindrical domains MATHEMATISCHE NACHRICHTEN, Issue 4 2010Roberto ArgiolasAbstract In this paper we develope a perturbation theory for second order parabolic operators in non-divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with Lp -data on the parabolic boundary (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients MATHEMATISCHE NACHRICHTEN, Issue 9 2009Vladimir KozlovAbstract We consider the Dirichlet problem for non-divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi-linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,,, , , [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the decay of the solutions of second order parabolic equations with Dirichlet conditions MATHEMATISCHE NACHRICHTEN, Issue 8 2007Brice FrankeArticle first published online: 8 MAY 200Abstract We use rearrangement techniques to investigate the decay of the parabolic Dirichlet problem in a bounded domain. The coefficients of the second order term are used to introduce an isoperimetric problem. The resulting isoperimetric function together with the divergence of the first order coefficients and the value distribution of the zero order part are then used to construct a symmetric comparison equation having a slower heat-flow than the original equation. (© 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] ## Comparison results for nonlinear elliptic equations with lower,order terms MATHEMATISCHE NACHRICHTEN, Issue 1 2003Vincenzo FeroneAbstract We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) + g(x, u) = f, where the principal term is a Leray,Lions operator defined on and g(x, u) is a term having the same sign as u and satisfying suitable growth assumptions. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric. [source] ## Analysis of parameterized quadratic eigenvalue problems in computational acoustics with homotopic deviation theory NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2006F. Chaitin-ChatelinAbstract This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partly impedance, with a complex impedance parameter ,. The admittance t = 1/, is the classical homotopy parameter. In particular, we study the spectrum when t , ,. We show that in the limit part of the eigenvalues remain bounded and converge to the so-called kernel points. We also show that there exist the so-called critical points that correspond to frequencies for which no finite value of the admittance can cause a resonance. Finally, the physical interpretation that the impedance condition is transformed into a pressure release condition when |t| , , enables us to give the kernel points in closed form as eigenvalues of the discrete Dirichlet problem. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Dirichlet duality and the nonlinear Dirichlet problem COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2009F. Reese HarveyWe study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain , , ,n. In our approach the equation is replaced by a subset F , Sym2(,n) of the symmetric n × n matrices with ,F , {F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric "F -convexity" assumption on the boundary ,,. We also study the topological structure of F -convex domains and prove a theorem of Andreotti-Frankel type. Two key ingredients in the analysis are the use of "subaffine functions" and "Dirichlet duality." Associated to F is a Dirichlet dual set F, that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F, is F, and in the analysis the roles of F and F, are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge-Ampère equation over ,, ,, and ,; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p -convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc. [source] ## The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2004Mu-Tao WangLet , be a bounded C2 domain in ,n and , ,, , ,m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : , , ,m with f|,, = , and with the graph of f a minimal submanifold in ,n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if , : ¯, , ,m satisfies 8n, sup, |D2,| + ,2 sup,, |D,| < 1, then the Dirichlet problem for ,|,, is solvable in smooth maps. Here , is the diameter of ,. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with , as initial data. © 2003 Wiley Periodicals, Inc. [source] ## Numerical analysis of a non-singular boundary integral method: Part II: The general case MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2002P. DreyfussIn order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non-singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non-singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single-layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any ,,, domains. We prove that the properties of stability and convergence remain valid. Copyright © 2002 John Wiley & Sons, Ltd. [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] |