Showing papers on "Laplace's equation published in 1969"

Derivation of Kinetic Equations from the Generalized Langevin Equation

Abstract: The projection operator techniques of Zwanzig and Mori are used to obtain a generalized Langevin equation describing the time evolution of the fluctuation of the microscopic phase density 6g(x, p, t) =g(x, p, t) -(g(x, p, t)) for a classical many-particle system. This equation is then used to develop an exact kinetic equation for the time-correlation function (6g(x, p, 0) bg(x', p', t ) ) [which is the generalization of the Van Hove time-dependent pair correlation function G(r, t)]. In the lowest order of approximation, this kinetic description reduces to the Vlasov-like equation which has been used to study neutron scattering from liquids. A less restrictive approximation is obtained by utilizing weak-coupling perturbation theory to yield a generalized Fokker-Planck equation for the time-correlation function. Other possible approximation schemes are also discussed.

Flow Through Spillway Flip Bucket

Abstract: This paper presents two separate solutions for potential flow through circular spillway buckets. The first solution, which considers gravity effects, employs finite difference equations in combination with a successive approximation procedure in order to satisfy the Laplace equation and the boundary conditions. The second solution, which neglects gravity effects, employs a perturbation expansion in combination with conformal mapping and analytical continuation. The two solutions are compared, and recommendations are made for their application.

Group Analysis of Maxwell's Equations

Abstract: In this paper we write down and solve Maxwell's equations without sources when the field variables are considered as functions over the group SU2. A Hilbert space is then constructed out of the field functions. An expansion of the field functions in terms of the matrix elements of the irreducible representation of SU2 is shown to reduce the problem of solving Maxwell's equations to that of solving one partial differential equation with two variables. A Fourier transform reduces this equation into an ordinary differential equation which is identical to the partial‐wave equation obtained from the Schrodinger equation with zero potential. The analogy between the mathematical method used in this paper in relation to the group SU2 and the Fourier transform in relation to the additive group of real numbers is pointed out.

Stream channel concept applied to water flood performance calculations for multi-well, multi-zone three component cases

Abstract: The Higgins-Leighton technique for estimating waterflood performance was expanded to include the calculation of fluid production and injection for a multi-well, multi-zone system with asymmetrical drainage areas. This 2-dimensional, analytical method is based on the computer solution of streamlines, shape factors, waterflood performance by channels, and well-zone-field production combinations. The flow regime for each injector-producer pair in the system is represented by a series of channels whose sides are bounded by streamlines. Each channel may have a unique value for porosity, water saturation, and permeability. The channels are divided into equal volume cells to permit approximation of Buckley-Leverett linear displacement, with radial flow occurring through the cells adjacent to the producer and injector, and linear flow through the remaining cells. Initial conditions may include gas saturation for partial depletion. Streamline positions are generated from the solution of the flow potential in the Laplace equation for steady-state flow of 2 fluids of unit mobility in a uniform bed of constant porosity, water saturation, and absolute permeability.

Calculation of the Modulation Transfer Function of an Image Tube

Abstract: Publisher Summary This chapter presents the calculation of the modulation transfer function of an image tube. Normal practice consists of solving the Laplace equation using relaxation techniques, from which is obtained a value of the electrostatic potential everywhere in the tube. This can then be used to give the field at any point, and by a step-by-step integration process trace the paths of single electrons through the system. In order to choose suitable starting conditions for the electrons, it is necessary to know the distributions of directions and energies from the photocathode used. Using a random number procedure, electrons with cosine and normal distributions of direction and energy respectively are generated. These are traced through the tube and the line-spread functions are built up. The ability to carry out this computation enables the complete electron-optical performance of an electrostatic system to be predicted before a tube is constructed.

Bounds on coupling constants in existence theorems for the Low equation

Abstract: An earlier existence proof for solutions of the one-meson Low equation is extended and improved. It is shown that the Low equation has an iterative solution, which is analytic in the coupling constant. The Low equation is investigated also in a crossing-symmetric inverse amplitude formulation. The inverse equation has an iterative solution, which leads to a ghost-free solution of the Low equation. The proofs are valid if certain upper bounds on the coupling constant are respected. We give numerical values of those bounds (for πN scattering and a particular cut-off function) in each of three formulations of the Low equation: the Low equation itself, the inverse amplitude equation, and theN/D equation. The bounds prove to be disappointingly small, whether the existence proofs are done by Schauder's fixed-point theorem or by the iterative method of Banach's fixed-point theorem. It appears that entirely different methods will be needed to investigate solutions with the coupling constant near its physical value.

Solution of Laplace's equation in a region with dielectric interfaces of arbitrary shape

Abstract: This letter demonstrates an integration method for deriving finite-difference approximations to Laplace's equation for problems which are axially symmetric and involve dielectric interfaces of arbitrary shape. The method generalizes a previous approach, and is easily automated and applicable to other types of boundary conditions as well.

Laplace transform wave functions

Abstract: The wave function defining a quantum-mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen-like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.

An Introduction to Green's Functions

Abstract: : The report is an introduction to Green's functions intended for workers in acoustics but also suitable for general purposes. The first chapter reviews the properties of the various types of second order linear partial differential equations and discusses a simple Green's function as an introductory example. Subsequent chapters discuss the Green's functions associated with the Laplace, diffusion, and wave equations, and the final chapter deals with the Helmholtz equation through some applications from acoustics. Appendix 1 discusses the formal manipulation of delta functions, and Appendix 2 is a brief introduction to Fourier and Laplace transforms. A bibliography, exercises, and solutions are also included.

A weak-separation solution of Helmholtz’ equation with spheroidally symmetric boundaries

Abstract: A solution of Helmholtz’ equation with boundaries having spheroidal symmetry is obtained directly in the form of a power series in eccentricity. The partial differential equation is expressed in a nonorthogonal, « modified spheroidal » co-ordinate system, and a « weak separation » method is used to solve the resulting nonseparable equation. The resulting solution is found to be more convenient than the usual spheroidal-function solution.

Calculation of the distribution of relaxation times from experimental data

Abstract: A complicated dielectric spectrum may consist of a superposition of continuous relaxation contributions, where the physically significant function is the distribution of relaxation times. Methods are described for finding this function by solution of integral equations using inverse Laplace transforms. Realization of the solution for experimental data requires numerical calculation.

On the solution of certain field problems by the schwarz-christoffel method

Abstract: In many practical cases the usefulness of the Schwarz-Christoffel method to solve two-dimensional field problems (Laplace equation with Dirichlet boundary conditions) is limited by the presence of transcendental functions of complex variables. We demonstrate here a new technique whereby, in lieu of qualitative plots of equipotential surfaces and flux lines, field components and potential can be expressed as real power series of the coordinates (x, y). The convergence of these series is only limited by the proximity of singular points corresponding to the physical convex corners. By choosing suitable points on the boundary around which the series of expansion are developed, fringing field components in the regions of interest between the boundaries can be computed directly. In some cases the series converges rapidly and assumes a remarkably simple form.

Determination of the optimum successive overrelaxation parameter for the solution of Laplace's equation

Abstract: A new method for estimating the optimum successive over-relaxation parameter for the solution of Laplace's equation is described. Its effectiveness is compared with that of other methods.

Material coefficients and transfer of polarized radio radiation in a plasma

Abstract: By a perturbation and diagram resummation method, a transport equation for the transverse field polarization matrix is established. This equation is then transformed into an equation for the Stokes parameters of the radiation. The equation takes the usual form of a transfer equation; the absorption and emission coefficients are matrix, the elements of which are given as a function of the dissipative part of the microcurrent correlation tensor and conductivity tensor. Finally this equation is expressed as a system for the intensities of the proper modes. The equations of the system are usually coupled.

Finite Difference Solutions of TEM Mode Structures

Abstract: A finite dilYerence potential solution to a TEM mode trans- mission line cross section may be used to define a continuous potential function, leading to an upper bound for the capacitance. The accuracy of the capacitance calculation is shown to depend on the potential function fitted. A method is developed for interpolating a suitable potential function; in the cases considered, the use of this potential function gave capacitance solutions with an error approximately one-fifth that obtained using the usual methods. N RECENT years a considerable amount of attention has been given to the application of finite difference techniques to the numerical solution of the two-dimen- sional Laplace equation, and the derivation of various TEM mode parameters from this approximate solution (1 )-(6). It is possible to extend these numerical techniques to treat the solution of Laplace's equation in three dimensions (7), but the number of discrete variables becomes very large if an accurate result (say, better than 1 percent) is required. This increased number of variables may be impractical in terms of computer time and storage, so that it is important to dis- cover techniques for improving the accuracy of finite dif- ference solutions without increasing the number of discrete

On the conditions that the net method for the Laplace equation converges with speed h2

Abstract: The paper gives a uniform estimate of order h2 of the error in the net method of solving the Dirichlet problem for the Laplace equation under the assumptions that the modulus of continuity of the second derivatives of the boundary values and the modulus of continuity of the curvature of the region's boundary do not exceed a function satisfying the Dini condition. It is shown that with the removal of the Collatz boundary values the constraints on the boundary values cannot be significantly relaxed in terms of the moduli of continuity of the second derivatives, while the constraints on the moduli of continuity of the curvature of the boundary cannot be completely lifted.