Showing papers on "Laplace's equation published in 2003"
TL;DR: It is found that nanotubes imbibe very rapidly (
Abstract: Our understanding of the static properties of capillaries filled by a wetting liquid is based on the 19th century work of Laplace, Poisson, and Young [1]. The dynamics of capillary flow were elucidated much later by Washburn [2] using the Poiseuille equation with the driving force for flow described by the Laplace equation for the pressure difference across the invading liquid meniscus. More recently Kalliadasis and Chang [3,4] have reanalyzed the classical problem using a scaling law for the dynamic contact angle of the liquid phase and find agreement with the classic experimental results of Blake et al. [5]. There is extensive literature on the wetting dynamics of homogeneous and structured surfaces [6]. In this Letter we extend the research to times and lengths characteristic of nanomaterials by examining, using molecular dynamics, the imbibition of oil into a nanoscale capillary formed by a single wall carbon nanotube (SWNT). The simulation [7] geometry is shown in Fig. 1. A liquid-vapor interface of 507 decane molecules was brought to equilibrium at 298 K. The decane model potential is a short range united atom chain model with intramolecular bond bending and torsion terms, but bond lengths are fixed at 0.153 nm. The bond bending is described by a harmonic Van der Ploeg and Berendsen [8] potential
222 citations
TL;DR: In this paper, a time fractional advection-dispersion equation was obtained from the standard advective dispersion equations by replacing the first-order derivative in time by a fractional derivative in order α(0 < α<-1).
Abstract: A time fractional advection-dispersion equation is obtained from the standard advection-dispersion equation by replacing the firstorder derivative in time by a fractional derivative in time of order α(0<α<-1). Using variable transformation, Mellin and Laplace transforms, and properties of H-functions, we derive the complete solution of this time fractional advection-dispersion equation.
210 citations
TL;DR: In this paper, the authors applied particle image velocimetry (PIV) to landslide impact and wave generation in an extremely unsteady three-phase flow consisting of granular matter, air, and water.
Abstract: Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. The challenges posed to the measurement system in an extremely unsteady three-phase flow consisting of granular matter, air, and water were considered. The complex flow phenomena in the first stage of impulse wave initiation are: high-speed granular slide impact, slide deformation and penetration into the fluid, flow separation, hydrodynamic impact crater formation, and wave generation. During this first stage the three phases are separated along sharp interfaces changing significantly within time and space. Digital masking techniques are applied to distinguish between phases thereafter allowing phase separated image processing. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, elongational, and shear strain were extracted from the velocity vector fields. The fundamental assumption of irrotational flow in the Laplace equation was confirmed experimentally for these non-linear waves. Applicability of PIV at large scale as well as to flows with large velocity gradients is highlighted.
169 citations
147 citations
TL;DR: In this paper, the authors prove that spherically symmetric solutions of the Cauchy problem for the linear wave equation with the inverse-square potential satisfy a modified dispersive inequality that bounds the $L^\infty$ norm of the solution in terms of certain Besov norms of the data.
Abstract: We prove that spherically symmetric solutions of the Cauchy problem for the
linear wave equation
with the inverse-square potential satisfy a modified dispersive inequality
that bounds the $L^\infty$ norm
of the solution in terms of certain Besov norms of the data, with a factor
that decays in $t$ for positive potentials. When the potential is negative
we show that the decay is split between $t$ and $r$, and the estimate blows up
at $r=0$. We also provide a counterexample
showing that the use of Besov norms in dispersive inequalities for the wave
equation are in general unavoidable.
69 citations
TL;DR: In this article, it was shown that for simple polygons and for a large class of boundary conditions, the above Riemann-Hilbert problem can either be reduced to a triangular RH problem which can be solved in closed form or bypassed, and the ρ j can be obtained using only algebraic manipulations.
Abstract: Let q(x, y) satisfy a boundary value problem for the Laplace equation in an arbitrary convex polygon with n sides An integral representation in the complex k-plane is given for q(x, y) in terms of n functions ρ j (k), j = 1,,n The function ρ j consists of an integral over the jth side involving both q x and q y , thus each ρ j involves one unknown boundary value The functions ρ j are not independent but they satisfy the important global relation that their sum vanishes The solution of a given boundary value problem reduces to the analysis of this single relation for the n unknown ρ j For a general polygon with general Poincare boundary conditions, this gives rise to a matrix Riemann-Hilbert problem In this paper it is shown that for simple polygons and for a large class of boundary conditions, the above Riemann-Hilbert problem (a) can either be reduced to a triangular RH problem which can be solved in closed form or (b) can be bypassed, and the ρ j can be obtained using only algebraic manipulations As an illustration of these 'triangular' and 'algebraic' cases we solve the Laplace equation in the quarter-plane, the semi-infinite strip and the right isosceles triangle with certain Poincare boundary conditions These boundary value problems, which include the Dirichlet and the Neumann problems as particular cases, cannot be solved by conformal mappings
62 citations
TL;DR: In this article, the authors studied the properties of the Brownian self-transport operator for planar and irregular membranes and showed that the net effect of a decrease of the surface permeability is to induce a broadening of the region where a particle, first hitting the surface on one point, is finally absorbed.
Abstract: The problem of the Laplacian transfer across an irregular resistive interface (a membrane or an electrode) is investigated with use of the Brownian self-transport operator. This operator describes the transfer probability between two points of a surface, through Brownian motion in the medium neighbouring the surface. This operator governs the flux across a semi-permeable membrane as diffusing particles repetitively hit the surface until they are finally absorbed. In this paper, we first give a theoretical study of the properties of this operator for a planar membrane. It is found that the net effect of a decrease of the surface permeability is to induce a broadening of the region where a particle, first hitting the surface on one point, is finally absorbed. This result constitutes the first analytical justification of the Land Surveyor Approximation, a formerly developed method used to compute the overall impedance of a semi-permeable membrane. In a second step, we study numerically the properties of the Brownian self-transport operator for selected irregular shapes.
52 citations
TL;DR: In this article, the Laplace transform of the unknown function φ with respect to time t was used to solve the one-dimensional heat equation, and the physical solution was recovered with the help of a numerical technique for inverting the Laplacian transform.
Abstract: The problem of solving the one-dimensional heat equation ∂φ/∂t - ∂2φ/∂x2 = f(x, t) subject to given initial and nonlocal conditions is considered. It is solved in the Laplace transform domain by taking the Laplace transform of the unknown function φ with respect to time t. The physical solution is recovered with the help of a numerical technique for inverting the Laplace transform.
51 citations
TL;DR: In this article, it is proved that any solution u1 and u2 of the generalized Cauchy problem for the Laplace equation satisfies the stability estimate where α (0, 1) is arbitrary, y [0, 0] and c is a countable positive constant independent of the data of the problem.
Abstract: Let p [1, ∞], Lp () and e, M be given constants such that 0 < e < M < ∞. It is proved that any solutions u1 and u2 of the 'generalized' Cauchy problem for the Laplace equation satisfy the stability estimate where α (0, 1) is arbitrary, y [0, 1] and c is a countable positive constant independent of the data of the problem. Similar local estimates for derivatives of the solutions of any order with respect to both x and y are established. The proofs of the results are based on the construction of stable solutions to the problem by the mollification method of Dinh Nho Hao (1994 Numer. Math. 68 469–506).
46 citations
TL;DR: In this paper, the effect of irregularities on the rate of heat conduction from a two-dimensional isothermal surface into a semi-infinite medium is considered, and the effect is quantified in terms of the displacement of the linear temperature profile prevailing far from the surface.
46 citations
TL;DR: In this paper, the simultaneous flow of two phases through a three-dimensional porous medium is calculated by means of a Lattice-Boltzmann algorithm, and the time-dependent phase configurations can be derived and also macroscopic quantities such as the relative permeabilities.
Abstract: The simultaneous flow of two phases through a three-dimensional porous medium is calculated by means of a Lattice-Boltzmann algorithm. The time-dependent phase configurations can be derived and also macroscopic quantities such as the relative permeabilities. When one phase only is supposed to be conductive, the Laplace equation which governs electrical conduction can be solved in each phase configuration; an instantaneous value of the macroscopic conductivity is obtained and it is averaged over many configurations. The influence of saturation on the resistivity index is studied for six different samples and two viscosity ratios. The saturation exponent is systematically determined. The numerical results are also compared to other possible models and also to experimental results; finally, they are discussed and criticized.
TL;DR: This paper estimates the dimensionless Eotvos number limits above which the deviation of the drop profile from the spherical is significant enough to result in contact angle deviation of 1 degrees and checks if the spherical section assumption can be used for contact angle estimation and other shape-related analysis for a system.
TL;DR: In this article, the authors consider the Laplace equation in a smooth bounded domain and prove logarithmic estimates of solutions on a part of the boundary or of the domain without known boundary conditions.
Abstract: We consider the Laplace equation in a smooth bounded domain. We prove logarithmic estimates, in the sense of John [5] of solutions on a part of the boundary or of the domain without known boundary conditions. These results are established by employing Carleman estimates and techniques that we borrow from the works of Robbiano [8,11]. Also, we establish an estimate on the cost of an approximate control for an elliptic model equation.
TL;DR: In this article, the convergence and accuracy of various velocity formulations for water waves in the framework of Boussinesq theory are discussed and analyzed in comparison with the target cosh-and sinh-functions from linear wave theory.
Abstract: The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the still-water datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh- and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary -level. This is shown to have a remarkable influence on the convergence and to improve accuracy considerably. Fifth, we derive and analyse a new formulation which doubles the power of the vertical coordinate without increasing the order of the horizontal derivatives. Finally, we involve the kinematic and dynamic boundary conditions at the free surface and discuss the linear dispersion relation and a spectral solution for steady nonlinear waves.
TL;DR: In this paper, the Laplace transform is used to eliminate the time dependence of the variables and the meshless approximation based on the moving least square method is employed for the implementation.
TL;DR: In this article, a system of functions 0-normalized with respect to the operator ∆ in some domain Ω ⊂ R n is constructed, and application of this system to boundary value problems for the polyharmonic equation is considered.
TL;DR: In this article, an approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized.
Abstract: An approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized. The result of the study is explicit analytical solutions of three-dimensional problems. Bodies are examined whose boundary surface is the coordinate surfaces in coordinate systems that permit the separation of the variables in the three-dimensional Laplace equation. The stress concentration in bodies near necks, cavities, inclusions, and cracks is investigated. The stress intensity factors of the force field and electric induction near elliptic and parabolic cracks are determined. The contact interaction of a piezoceramic half-space with elliptic and parabolic dies is studied. The bodies are under various mechanical, thermal, and electric loads
TL;DR: In this article, the authors considered the inverse problem of identifying a Robin coefficient on some part of the boundary of a smooth 2D domain from overdetermined data available on the other part of boundary, for Laplace equation in the domain.
Abstract: We consider the inverse problem of identifying a Robin coefficient on some part of the boundary of a smooth 2D domain from overdetermined data available on the other part of the boundary, for Laplace equation in the domain. Using tools from complex analysis and analytic functions theory, we provide a constructive and convergent identification scheme for this inverse problem, together with numerical experiments.
TL;DR: In this article, the authors explore the usefulness of the Schrodinger approach by applying it to a number of simple examples of self-gravitating systems in the weakly nonlinear regime.
Abstract: The dynamical equations describing the evolution of a self-gravitating fluid can be rewritten in the form of a Schrodinger equation coupled to a Poisson equation determining the gravitational potential. This wave-mechanical representation allows an approach to cosmological gravitational instability that has numerous advantages over standard fluid-based methods. We explore the usefulness of the Schrodinger approach by applying it to a number of simple examples of self-gravitating systems in the weakly non-linear regime. We show that consistent description of a cold self-gravitating fluid requires an extra ‘quantum pressure’ term to be added to the usual Schrodinger equation and we give examples of the effect of this term on the development of gravitational instability. We also show how the simple wave equation can be modified by the addition of a non-linear term to incorporate the effects of gas pressure described by a polytropic equation of state.
TL;DR: The Cauchy problem is transformed to a moment problem so that numerical computations using a regularization technique can be achieved and a stability estimation and a suitable choice of regularization parameter are given.
Abstract: This paper devises a computational method for solving a Cauchy problem of Laplace's equation in multidimensional space. By using the Green formula, the Cauchy problem is transformed to a moment problem so that numerical computations using a regularization technique can be achieved. A stability estimation and a suitable choice of regularization parameter for the proposed method are also given. For numerical verification, a numerical example in the three-dimensional case is presented.
TL;DR: This study presents a Feynman-Kac path-integral implementation for solving the Dirichlet problem for Poisson's equation that is a modified "walk on spheres" (WOS) that includes the Feynmans' path-Integral contribution for the source term.
TL;DR: In this paper, Buffa et al. studied the Dirichlet problem for the Laplace operator and the electric boundary problem of the Maxwell operator and showed that regularity results in two families of weighted Sobolev spaces: a classical isotropic family and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account.
TL;DR: The first part of a program aimed at studying the regularity of sub-elliptic free boundaries in the setting of Carnot groups is presented in this article, where the optimal interior regularity is established for the solution to the obstacle problem in terms of the Folland-Stein non-isotropic class Γ 1,1.
Abstract: This paper is the first part of a program aimed at studying the regularity of sub-elliptic free boundaries. In the setting of Carnot groups we establish the optimal interior regularity of the solution to the obstacle problem in terms of the Folland-Stein non-isotropic class Γ 1,1 . This result constitutes the sub-elliptic counterpart of the classical C 1,1 regularity for Laplace equation. We also prove non-degeneracy properties of the solution and of its free boundary.
TL;DR: In this paper, a regularization method for the inversion of real-valued Laplace transforms is presented and error analysis based on an analytical link between the regularized and exact solutions is given.
Abstract: The paper presents a regularization method for the inversion of real-valued Laplace transforms. A regularizing inverse Laplace operator is obtained for a subspace of Laplace transforms directly from the Laplace transformation definition. The fact that a regularized solution converges to the exact one when input data inaccuracy tends to zero is proven. Error analysis based on an analytical link between the regularized and exact solutions is given. This error analysis reflects some general limitations for any method of inversion of real-valued Laplace transforms. The numerical results are briefly discussed.
TL;DR: In this paper, the fundamental equations of generalized thermoelasticity with one relaxation parameter including heat sources in infinite rotating media have been written in the form of a vector-matrix differential equation in the Laplace transform domain for a one-dimensional problem.
Abstract: The fundamental equations of the problems of generalized thermoelasticity with one relaxation parameter including heat sources in infinite rotating media have been written in the form of a vector-matrix differential equation in the Laplace transform domain for a one-dimensional problem. These equations have been solved by the eigenvalue approach. The results have been compared to those available in the existing literature. The graphs have been drawn to show the effect of rotation.
TL;DR: In this paper, a new abstract function space formulation of the subsonic small disturbance potential field equations of aeroelasticity and an operator theoretic treatment of the Possio integral equation in the generality of the Laplace transform variable λ were presented.
Abstract: This paper presents a new abstract function space formulation of the subsonic small disturbance potential field equations of aeroelasticity and an operator theoretic treatment of the Possio integral equation in the generality of the Laplace transform variable λ. A key result is the new form of the kernel—which is shown to be analytic in the whole plane, excepting the negative real axis—using an existence and uniqueness theorem is proved valid for small |λ|. The main new feature is the use of spatial Lp-Lq Fourier transforms for 1
TL;DR: In this paper, the Laplace equation with inhomogeneous Dirichlet and Neumann boundary conditions in the half-space of ℝN is studied and the behaviour of solutions at infinity is described by means of a family of weighted Sobolev spaces.
Abstract: The purpose of the present paper is twofold. The first object is to study the Laplace equation with inhomogeneous Dirichlet and Neumann boundary conditions in the half-space of ℝN. The behaviour of solutions at infinity is described by means of a family of weighted Sobolev spaces. A class of existence, uniqueness and regularity results are obtained. The second purpose is to investigate some properties of grad, div and curl operators in order to treat curl–div systems of the form
curl w = u, div w = 0
and problems related to vector potentials and Helmholtz decomposition.Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: In this article, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution, and the unknown parameters included in the particular solutions are determined by the iterative process.
Abstract: Trefftz method is the boundary-type solution procedure using regular T-complete functions satisfying the governing equation. Until now, it has been mainly applied to numerical analyses of the problems governed with the homogeneous differential equations such as the two- and three-dimensional Laplace problems and the two-dimensional elastic problem without body forces. On the other hand, this paper describes the application of the indirect Trefftz method to the solution of the boundary value problems of the two-dimensional Poisson equation. Since the Poisson equation has an inhomogeneous term, it is generally difficult to determine the T-complete function satisfying the governing equation. In this paper, the inhomogeneous term containing an unknown function is approximated by a polynomial in the Cartesian coordinates to determine the particular solutions related to the inhomogeneous term. Then, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution. Once the boundary value problem of the Poisson equation is solved according to the ordinary Trefftz formulation, the solution of the boundary value problem of the Poisson equation is estimated from the solution of the Laplace equation and the particular solution. The unknown parameters included in the particular solution are determined by the iterative process. The present scheme is applied to some examples in order to examine the numerical properties.
TL;DR: In this paper, an iterative solution of the Berger equation by the local boundary integral equation method with meshless approximation of physical quantities is proposed, where the governing equation is decomposed into two coupled partial differential equations of the second order, one of them is Poisson's equation whereas the other one is Helmholtz's equation.
Abstract: The nonlinear integro-differential Berger equation is used for description of large deflections of thin plates. An iterative solution of Berger equation by the local boundary integral equation method with meshless approximation of physical quantities is proposed. In each iterative step the Berger equation can be considered as a partial differential equation of the fourth order. The governing equation is decomposed into two coupled partial differential equations of the second order. One of them is Poisson's equation whereas the other one is Helmholtz's equation. The local boundary integral equation method is applied to both these equations. Numerical results for a square plate with simply supported and/or clamped edges as well as a circular clamped plate are presented to prove the efficiency of the proposed formulation.
TL;DR: In this paper, the grain-void interface is considered to be one-dimensional, and the physical formulation of an electromigration and diffusion model results in two coupled, fourth-order, time-dependent PDEs, with the boundary conditions imposed at the electrode points and at the triple point which belongs to two neighboring grains and the void.