Showing papers on "Laplace's equation published in 1974"
TL;DR: In this paper, the interface problem is considered for the Poisson equation in two independent variables and the interface curves along which jump conditions are prescribed are allowed to intersect, and the second derivatives of solutions of the interface problems lie in a twisted Soboiev space.
Abstract: The interface problem is considered for the Poisson equation in two independent variables, The interface curves, along which jump conditions are prescribed, are allowed to intersect, The second derivatives of solutions of the interface problem are shown to lie in a certain "twisted" Soboiev space. The solution operator is shown to be a closed, densely defined, operator in L2 whose domain can be determined exactly
205 citations
TL;DR: An analytical solution for the static temperature distribution on the surface of a heat-sinked integrated chip with one isothermal heat source is derived in this paper, where the mathematical technique applied to solve the Laplace equation is a double Schwarz-Christoffel conformal transformation.
Abstract: An analytical solution for the static temperature distribution on the surface of a heat-sinked integrated chip with one isothermal heat source is derived. The mathematical technique applied to solve the Laplace equation is a double Schwarz-Christoffel conformal transformation. Perturbations in the temperature distribution due to the heat source and the chip finite length are evaluated as well as the temperature losses in the vicinity of the heat source. Explicit expressions of the normalized temperature and thermal input conductance are given and analyzed as a function of the source and chip relative dimensions.
88 citations
TL;DR: In this article, the authors used the direct method of matrix decomposition to formulate an effective numerical algorithm for the discrete biharmonic equation on a rectangular region and the discrete Poisson equation on an irregular region.
Abstract: The discrete biharmonic equation on a rectangular region and the discrete Poisson equation on an irregular region can be treated as modifications to matrix problems with very special structure. We show how to use the direct method of matrix decomposition to formulate an effective numerical algorithm for these problems. For typical applications the operation count is $O(N^3 )$ for an $N \times N$ grid. Numerical comparisons with other techniques are included.
87 citations
TL;DR: In this paper, it was shown that the solution of Nelson's stochastic mechanics equation associated with any stationary solution of the Schrodinger equation is the homogeneous Markov process of the heat equation with Dirichlet boundary condition on the hypersurface ψ = 0.
Abstract: We prove that the solution of Nelson's stochastic mechanics equation associated with any stationary solution ψ of the Schrodinger equation is the homogeneous Markov process of the heat equation with Dirichlet boundary condition on the hypersurface ψ=0
76 citations
TL;DR: In this article, a master equation for the density operator of the light fields alone is derived and analytic solutions are obtained for the diagonal matrix elements of density operator in the Fock representation.
Abstract: The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.
72 citations
TL;DR: In this paper, an exact Fokker-Planck equation for the density function corresponding to the reduced atomic density operator in the diagonal atomic-state representation has been derived in an elementary fashion for arbitrary atomic states which are sufficiently removed from the state of complete inversion.
Abstract: The super-radiant master equation (SME) of Bonifacio et al. is analyzed using the coherent-atomic-state representation. We have succeeded in deriving an exact Fokker-Planck equation for the density function corresponding to the reduced atomic density operator in the diagonal atomic-state representation. A solution to the Fokker-Planck equation has been provided in an elementary fashion for arbitrary atomic states which are sufficiently removed from the state of complete inversion at time zero. The general solution for arbitrary initial conditions (including the initial state of complete inversion) has been obtained using the method of eigen-function expansions and the final result expressed in terms of an integral over the initial density function. The moments of the collective atomic operators are also discussed.
50 citations
TL;DR: 5- and 9-point discretizations of the differential equation of the linear source problem on a rectangular mesh and 3-point approximations to its one-dimensional analogue are studied.
Abstract: A comparative study is undertaken of 5- and 9-point discretizations of the linear source problem on a rectangular mesh, and of 3-point discretizations of its one-dimensional analogue. Traditional difference and (Richardson) extrapolation methods compare very favorably with varia- tional methods. Sample result: Courant's derivation of the standard 5-point formula for the Laplace equation from the Ritz variational method does not generalize to the Poisson or Helmholtz equation. 1. Introduction. We make below a comparative study of 5-point and 9-point discretizations of the differential equation of the linear source problem, (1) -V * (pVu) + qu = f, p(x, y) > O, q _ O, for piecewise smooth p, q and f on a rectangular mesh. We also study 3-point approximations to its one-dimensional analogue:
33 citations
TL;DR: The Boltzmann equation is formulated in a hydrodynamic representation (modified Eulerian) in terms of velocities relative to an accelerated medium as discussed by the authors, where the target particles are treated as fixed in moving hydrodynamics zones, or cells.
Abstract: The Boltzmann equation is formulated in a hydrodynamic representation (modified Eulerian) in terms of velocities relative to an accelerated medium. Treating the target particles as fixed in moving hydrodynamic zones, or cells, and employing a representation in relative velocities gives a well‐defined cross section and permits use of the standard set of velocities and cross sections for computational purposes. Specific one‐dimensional forms for the transport equation are also given in plane and spherical geometries.
30 citations
TL;DR: In this paper, the Fourier method was used to obtain a least square solution of the Poisson's equation on the surface of a sphere, which is economical in both computational time and storage.
30 citations
TL;DR: In this article, a method for analysis of axisymmetric towers partly submerged in water to earthquake ground motion is presented, where the tower is idealized as a finite element system and the hydrodynamic terms are determined by solving the Laplace equation.
Abstract: A method for analysis of response of axisymmetric towers partly submerged in water to earthquake ground motion is presented. The tower is idealized as a finite element system. The hydrodynamic terms are determined by solving the Laplace equation, governing the dynamics of incompressible fluids, subject to appropriate boundary conditions. For cylindrical towers, these solutions are obtained as explicit mathematical solutions of the boundary value problems; whereas they are obtained by the finite element method in case of towers with non-cylindrical outside surface. The response to earthquake ground motion is determined by step-by-step integration of the equations of motion. Analyses of two actual intake towers are presented to illustrate results obtained by this method. The small computation times required for these analyses demonstrate that the method is very efficient. The effectiveness of this formulation lies in avoiding the analysis of a large system by using a substructure approach and in exploiting the important feature that structural response to earthquake ground motion is essentially contained in the first few modes of vibration of the tower with no surrounding water.
01 Jan 1974
TL;DR: In this article, the Laplace integral in the flotation kinetic equation is approximated by the Lobatto quadrature formula and the cumulative distribution is evaluated by optimizing a least square objective function.
TL;DR: In this paper, a simple nonlinear Schrodinger equation for amplitude modulation is derived from the two dimensional Laplace equation for the velocity potential, and it is found from this nonlinear equation that wave trains of finite amplitude are modulationally unstable.
Abstract: Modulational instability of weakly nonlinear capillary waves on a thin liquid sheet with no rigid boundary is theoretically investigated. By using the derivative expansion method, a simple nonlinear Schrodinger equation for the amplitude modulation is derived from the two dimensional Laplace equation for the velocity potential. It is found from this nonlinear equation that wave trains of finite amplitude are modulationally unstable. The phenomenon of the break-up of the sheet may be due to such an instability.
TL;DR: In this article, a numerical method is presented to study the thin skin magnetohydrodynamic stability to kink modes of arbitrary cross section two-dimensional plasmas, where only long wavelength perturbations are considered but no other restrictions are assumed.
Abstract: A numerical method is presented to study the thin skin magnetohydrodynamic stability to kink modes of arbitrary cross section two‐dimensional plasmas. Only long wavelength perturbations are considered but no other restrictions are assumed. Laplace's equation is solved for the perturbed fields by using a variation of Green's third formula.
TL;DR: In this paper, the inverse free boundary problem of electrochemical machining is solved by analytic continuation of the space variables on the plane of the complex potential, thereby obtaining an explicit expression for the geometry of physical boundaries of interest.
Abstract: A Cauchy problem for the Laplace equation is solved by analytic continuation of the space variables on the plane of the complex potential, thereby obtaining an explicit expression for the geometry of physical boundaries of interest. In an illustrative application to the inverse free boundary problem of electrochemical machining, the general solution comprises a closed-form description of a tool family which can be used to machine a prescribed workpiece. The method is extended to include the effects of variable electrolyte conductivity, and a general tool design procedure is suggested in which an analytic series with correct asymptotic behavior is used to represent the given workpiece geometry. Applications in other fields such as heat conduction and hydrodynamics are discussed. The inverted formulation described herein affords considerable advantage and generality in solving Cauchy problems which are encountered in engineering design.
TL;DR: In this article, an inverse method for constructing analytic solutions to the single phase Stefan problem for the heat equation in two space dimensions is presented, where the authors show that it is possible to construct an analytic solution to the Stefan problem in two dimensions.
Abstract: : In this paper the author outlines an inverse method for constructing analytic solutions to the (single phase) Stefan problem for the heat equation in two space dimensions.
TL;DR: In this paper, the Laplace equation for the electrolyte resistance, in conjunction with equations for charge transfer overpotential and electrode resistance phenomena, was used for cylindrical electrolysis cells having a resistive wire electrode along the cell axis.
Abstract: Two‐dimensional current and potential distributions have been calculated for cylindrical electrolysis cells having a resistive wire electrode along the cell axis. Cell behavior has been predicted for both monopolar and bipolar electrode situations. The calculations involve solving the Laplace equation for the electrolyte resistance, in conjunction with equations for charge‐transfer overpotential and electrode resistance phenomena. Over a wide range of parameter space, which includes most practical applications of resistive electrodes, it was found that simple one‐dimensional approximations to Laplace's equation yield reaction rate distributions which are in excellent agreement with more rigorous two‐dimensional calculations. By using the one‐dimensional approximations of monopolar and bipolar electrodes, it may be anticipated that future studies may be conducted with relative ease on mass transport phenomena during high‐rate electrolysis at resistive wire electrodes.
TL;DR: In this article, the Green's function for Laplace's equation in an infinite-length cylinder with a homogeneous mixed boundary condition is considered, and a more general expansion of the green's function is derived.
Abstract: The Green's function for Laplace's equation in an infinite‐length cylinder with a homogeneous mixed boundary condition is considered. Its eigenfunction expansion converges slowly when the axial separation between the source and observation points is small compared to the cylinder radius, and diverges when the axial separation is zero. Applying a modified form of a contour integral method of Watson to an integral representation of the Green's function, a more general expansion of the Green's function is derived. Watson's original method had previously been applied to the case when the source and observation points were both on the axis of the cylinder. The expansion contains a free parameter which may be adjusted to give rapid convergence for any axial separation. It fails, however, when the source and observation points are both near the surface of the cylinder. For two special values of the parameter, the general expansion reduces to the eigenfunction expansion or to the integral representation. The derivation is somewhat obscure, but the resulting formula has a simple interpretation as the superposition of the potential of two related boundary value problems in finite‐length cylinders. Some numerical results are given in the spatial region which previously could not be calculated, for a boundary condition approaching a homogeneous Neumann condition, and for a homogeneous Dirichlet condition.
TL;DR: In this article, a new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed, where the original nonlinear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order.
Abstract: A new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed. The original non-linear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order. The kernel of this new equation has a polynomial form which allows to find analytic expression forg(r) itself without using the Laplace transformation. In addition, the first three moments of the total correlation function can be found.
TL;DR: In this paper, the bound state energies and scattering phase shifts for the Coulomb potential were obtained from both the Schrodinger and Dirac equations by taking a Laplace transform, and the nonrelativistic eigenvalue problem was solved without even obtaining the transforms explicitly.
TL;DR: In this article, Poisson integral formulas are given which solve certain Dirichlet problems for a "quarter-ball" for the equation of generalized bi-axially symmetric potential theory.
Abstract: Poisson integral formulas are given which solve certain Dirichlet problems for a “quarter-ball” for the equation of generalized bi-axially symmetric potential theory. The formulas are useful in studying properties of solutions of this equation and related equations. The equation considered includes the equation of Weinstein's generalized axially symmetric potential theory for which corresponding Poisson integral formulas were given by A. Huber.
TL;DR: In this article, the Laplace equation is applied to the meniscus of a fluid lying in the form of an annulus between two vertical, concentric and coaxial cylindrical tubes.
Abstract: The differential equation which results when the Laplace equation is applied to the meniscus of a fluid lying in the form of an annulus between two vertical, concentric and coaxial cylindrical tubes has been integrated numerically by computer application of Kelvin's graphical method. Results for the capillary rise as a function of the radii of the inner and outer cylinders are presented in tabular form for ranges of the variables likely to be encountered in surface tension studies.
TL;DR: Using the Sommerfeld method, the authors found the Green's function of a mixed boundary-value problem for the Laplace equation in a half-space with circular boundary conditions, and a wide class of stationary problems in heat conduction, electrostatics, and elasticity theory reduce to the solution of this problem.
Abstract: Using the Sommerfeld method we find the Green's function of a mixed boundary-value problem for the Laplace equation in a half-space with circular boundary conditions. A wide class of stationary problems in heat conduction, electrostatics, and elasticity theory reduce to the solution of this problem.
TL;DR: In this paper, it was shown that the BBGKY hierarchy equation can rigorously be transformed into a closed non-markoffian equation for f. The latter equation contains infinite sets of collision and initial correlation terms which are most conveniently represented by connected diagrams and which can be expressed in terms of f and initial (arbitrarily given) correlation functions.
TL;DR: In this paper, Liouville equation and projection operator techniques were used to obtain a compact equation for the rate of change of then-particle momentum distribution function to any order in the density.
Abstract: Starting from the Liouville equation and making use of projection operator techniques we obtain a compact equation for the rate of change of then-particle momentum distribution function to any order in the density. This equation is exact in the thermodynamic limit. The terms up to second order in the density are studied and expressions are given for the errors committed when one makes the usual hypothesis to derive generalized Boltzmann equations. Finally the Choh-Uhlenbeck operator is obtained under additional assumptions.
TL;DR: A Poisson equation for the n th order mean potential has been derived in the case of the primitive model of an electrolyte solution using the Kirkwood integral equations as discussed by the authors, and this analysis has been generalized to an arbitrary short range pair potential between the ions.
05 Feb 1974
TL;DR: In this article, a design method for the steadily loaded, full journal bearing is described as a non-iterative set of algebraic equations, where a dependent bearing parameter, e.g. eccentricity or power-loss, is predicted in terms of known independent parameters which include bearing geometry, running conditions and oil characteristics.
Abstract: A design method is described for the steadily loaded, full journal bearing. This is presented as a non-iterative set of algebraic equations, where a dependent bearing parameter, e.g. eccentricity or power-loss, is predicted in terms of known independent parameters which include bearing geometry, running conditions and oil characteristics. The method is developed from a regression analysis of accurately computed, fully thermohydrodynamic, solutions for the bearing. These solutions are generated by simultaneously solving the Reynolds and energy equations in the oil film, the Laplace equation in the bearing material and the oil-mixing conditions at inlet. A quasi three-dimensional finite-difference technique is used. Both the particular solutions and the predictions of the design method compare favourably with a wide range of experimental data, the latter showing an improvement in accuracy and economy on existing design methods.
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TL;DR: In this article, the smoothness of the potential of a measure on the support of the measure is discussed and some Lemmas regarding smoothness are discussed regarding the potential for a measure.
Abstract: We begin with some Lemmas regarding the smoothness of the potential of a measure on the support of the measure.