Showing papers on "Partial differential equation published in 1968"

Reductive Perturbation Method in Nonlinear Wave Propagation. I

Abstract: Presented is a class of nonlinear partial differential equations which admit a reduction to tractable nonlinear equations such as the Burgers and the Kortweg-deVries equation. The method of reduction is based on a singular perturbation expansion. Applications to the hydrodynamics and the plasma physics are discussed.

Inverse problems in partial differential equations.

Abstract: : A procedure for identification in partial differential equations is described and illustrated by the Laplace equation and the unsteady heat conduction equation. The procedure for solution involves the substitution of difference operators for the partial derivatives with respect to all but one of the independent variables. The linear boundary value problem is solved by superposition of particular solutions. For non-linear boundary value problems which arise from the original form of the equation or from the identification procedure, a Newton-Raphson-Kantorovich expansion in function space is used to reduce the solution to an iterative procedure of solving linear boundary value problems. For the problems considered, this procedure has proven to be effective and results in a reasonable approximation to the solution of the boundary value problem in partial differential equations. For the identification problem, it is shown that the constant parameters are identified to the same accuracy as the supplementary data used in the identification procedure. (Author)

Quantum theory of light propagation in a fluctuating laser-active medium

Abstract: The basic equations are derived which describe the propagation of an electromagnetic field in a fluctuating laser-active medium. The well-known methods of Langevinequations and master-equation for a few discrete modes are generalized to meet also the case of a radiation field with continuous spectrum. The medium is described by two-level atoms which are embedded in a merely passive solid matrix and homogeneously distributed over space. They have an inversion which is kept constant by an externally applied pump. The atomic line may be homogeneously or inhomogeneously broadened. We obtain a complete set of partial differential equations for the field operators with damping terms and fluctuating forces homogeneously distributed over the material. The telegraph equation with a fluctuating force occurs as a special case. After the exact elimination of the atomic variables we obtain a nonlinear field equation for the radiation field alone. By means of a pseudo-Hamiltonian and by a simple one-dimensional example we show that in a certain sense there exists a close formal analogy between the present theory and the theory of an interacting Bose gas. The characteristic differences between the two theories are also discussed. We find, that there occurs a phase transition of the radiation field because above a certain threshold of the pump the photons condense into a single mode and establish an “offdiagonal-long-range order”. The amplitude fluctuations and the phase fluctuations, which restore the broken phase symmetry, are calculated in detail. A new condition for the occurrence of undamped spiking (pulse formation) for a continuum of modes is derived.

The Use of Multiple Deflations in the Numerical Solution of Singular Systems of Equations, with Applications to Potential Theory

Abstract: The main emphasis of this paper is on generalizing, to the multi-interface Neumann problem, some of the results established in [14]. In the latter, we considered how deflation techniques could be used both in properly defining and in solving discrete approximations to the integral equation formulation of the single-interface problem. In addition, we observe how deflation techniques can be useful in solving systems obtained from discrete approximations to the partial differential equation formulation of the Neumann problem.

An accurate numerical one-dimensional solution of the p-n junction under arbitrary transient conditions

Abstract: A numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junction devices under arbitrary transient conditions is presented. The method is of a very general character: none of the conventional assumptions and restrictions are introduced and freedom is available in the choice of the doping profile, recombination-generation law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage as a function of time, the solution yields terminal properties and all the quantities of interest in the interior of the device (such as mobile carrier and net electric charge densities, electric field, electrostatic potential, particle and displacement currents) as functions of both position and time. Considerable attention is focused on the numerical analysis of the initial-value-boundary-value problem in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties of the problem, such as stability conditions related to the discretization of partial differential equations of the parabolic type, small differences between nearly equal numbers, and the variation of most quantities over extremely wide ranges within short regions. Results for a particular n + - p single-junction structure under typical external excitations are reported. The iterative scheme of solution for a single device is applicable also to ensembles of active and passive circuit elements. As a simple example, resutls for the combination of an n + - p diode and an external resistor, analyzed under switching conditions, are presented. The inductive behavior of the device for high current pulses, and storage and recovery phenomena under forward-to-reverse bias switching, are also illustrated. ‘Exact’ and conventional approximate analytical results are compared and discrepancies are exposed.

Solitons and Bound States of the Time-Independent Schrödinger Equation

Abstract: A numerical calculation validates the correspondence between the soliton nonlinear asymptotic solutions of the time-dependent Korteweg-deVries equation and the bound states of the one-dimensional time-independent Schr\"odinger equation. Here the attractive potential of the Schr\"odinger equation is equal to the initial condition of the Korteweg-deVries equation. A situation is examined where an oscillatory state remains after solitons have emerged.

Application of finite elements to the solution of Helmholtz's equation

Abstract: A novel method, that of finite elements, for the solution of Helmholtz's equation is suggested. Various 2- and 3-dimensional problems are solved using this method, and the results are compared with more conventional techniques, particularly the finite-difference method, which it may be regarded to supersede. The ease with which various boundary conditions may be handled is discussed and illustrated. Nonhomogeneous configurations present no difficulty, nor do they require any special formulation. There is considerable scope for the further development of the technique, which has, until now, been applied mainly to the solution of Laplace or Poisson equations.

The Coupled Equation Approach to the Numerical Solution of the Biharmonic Equation by Finite Differences. II

Abstract: Coupled equation approach to finite difference solution of biharmonic equation, noting outer iteration scheme for pair of discrete Poisson equations

Ordinary Differential Equations

Abstract: Introduction First-Order Linear Differential Equations Second-Order Differential Equations Power- Series Descriptions The Wronskian Eigenvalue Problems The Second-Order Linear Nonhomogeneous Equation Expansions in Eigenfunctions The Perturbation Expansion Asymptotic Series Special Functions The Laplace Transform Rudiments of the Variational Calculus Separation of Variables and Product Series Solutions of Partial Differential Equations Nonlinear Differential Equations More on Difference Equations Numerical Methods Singular Perturbation Methods.

Accuracy and Convergence of Finite Element Approximations

Abstract: : The paper reports on a theoretical investigation of the convergence properties of several finite element approximations in current use and assesses the magnitude of the principal errors resulting from their use for certain classes of structural problems. The method is based on classical order of error analyses commonly used to evaluate finite difference methods. Through the use of the Taylor series differential or partial differential equations are found which represent the convergence and principal error characteristics of the finite element equations. These resulting equations are then compared with known equations governing the continuum, and the error terms are evaluated for selected problems. Finite elements for bar, beam, plane stress, and plate bending problems are studied as well as the use of Straight or curved elements to approximate curved beams. The results of the study provide basic information on the effect of interelement compatibility, unequal size elements, discrepancies in triangular element approximations, flat element approximations to curved structures, and the number of elements required for a desired degree of accuracy.

Optimal Control Theory Applied to Systems Described by Partial Differential Equations

Abstract: Publisher Summary The study of optimal control problems for distributed parameter systems is intimately connected to the study of partial differential equations and/or integral equations involving two or more independent variables. This chapter discusses optimal control for distributed parameter systems, and reviews algorithms for obtaining numerical solutions. Some of the simplest kinds of equations occurring in initial value problems are the parabolic equations—the heat flow or diffusion equation, the hyperbolic equations (the wave equation), and some of the equations of elasticity (the beam equation or the plate equation). The time-differential operator P may be of order one, such as in the diffusion equation, or of order two as in the wave equation or the beam equation. In cases where P is of order higher than one, the problem can be reduced to an equivalent set of first order equations with respect to time by introducing new variables. The chapter also presents the development of Green's functions from solutions of the eigenvalue-eigenfunction problem.

Computer Simulation of Unsteady Flows in Waterways

Abstract: THE SET OF NONLINEAR, PARTIAL DIFF. EQUATIONS, DESCRIBING ONE-DIMENSIONAL TRANSLATORY WAVE MOTION PROVIDES THE MATHEMATICAL MODEL USED IN THE DEVELOPMENT OF THREE DISTINCTLY DIFFERENT TECHNIQUES FOR THE DIGITAL SIMULATION OF UNSTEADY FLOWS IN RIVERS AND ESTUARIES. THE FIRST TECHNIQUE IS BASED UPON POWER SERIES METHODS AND USES A MACLAURIN SERIES EXPANSION OF THE PARTIAL DIFFERENTIAL EQUATIONS. THE SECOND TECHNIQUE IS PREMISED UPON THE METHOD OF CHARACTERISTICS AND USES A NUMERICAL EVALUATION PROCESS AT SUCCESSIVE SPECIFIED TIME INTERVALS. THE THIRD SIMULATION TECHNIQUE RELIES UPON AN IMPLICIT METHOD OF FLOW SIMULATION WHEREIN THE PARTIAL DIFFERENTIAL EQUATIONS ARE TRANSFORMED TO FINITE DIFFERENCE EQUATIONS. THE EFFECTS OF FLUID FRICTION, VARIABLE CHANNEL GEOMETRY, WIND, LATERAL INFLOW OR OUTFLOW, THE CORIOLIS ACCELERATION, AS WELL AS OVERBANK STORAGE, ARE INCLUDED. FLOWS ARE CONSIDERED TO BE OF HOMOGENOUS DENSITY. EACH OF THE METHODS IS PROGRAMMED FOR HIGH SPEED DIGITAL COMPUTER. COMPARISONS OF THE SIMULATED FLOWS OBTAINED USING EACH OF THE SIMULATION TECHNIQUES WITH THE APPROPRIATE FIELD MEASURED TRANSIENT FLOWS INDICATE GENERALLY GOOD AGREEMENT /ASCE/

Numerical analysis of bod and do profiles

Abstract: The variation of Biochemical Oxygen Demand (BOD) and Dissolved Oxygen (DO) along a stretch of a polluted natural stream can be represented by two second-order partial differential equations. These equations are similar to the equation representing the conduction of heat in solids, with the exception that the stream equations contain additional lower order terms. Analytical solutions have been obtained for many cases of the heat-conduction problem. However, the presence of the lower order terms and the complexities of many of the boundary conditions make it impossible to obtain analytical solutions to the BOD and DO profile equations for most cases of practical interest. Furthermore, the numerical procedures which have been found to work for the solution of the heat conduction equation are not satisfactory for the BOD and DO equations because of the effects introduced by the lower-order terms. Procedures are presented for the numerical solution of the BOD and DO equations under temporally and spatially varying BOD and DO inputs. The procedures described are confined to the condition of steady, uniform stream flow.

WKB methods for difference equations II

Abstract: In a previous paper [1] it was shown how to develop solutions to difference equations analogous to WKB solutions to differential equations. In the work now reported a much more general “comparison equation” theory [2] is developed for difference equations, exploiting the fact that a difference equation can be considered as a differential equation of infinite order. Second order difference equations are considered in the main; by applying the theory to first order difference equations a useful generalization of the Euler-Maclaurin summation formula is found.

Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

Abstract: G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge.

Modes of stable and unstable optical resonators

Abstract: A method is developed for the approximate determination of the normal modes of stable and unstable optical resonators and the associated resonant frequencies and power losses. The method is based on replacing the finite integration limits in the integral equation for the normal modes by infinite limits and, subsequently, finding a differential equation whose solutions coincide with or approximate the solutions of this integral equation. When the end reflectors of the resonator are conical surfaces, a differential equation is found which corresponds exactly to the integral equation with infinite limits. Moreover, the equivalent differential equation is found to be of the same form as the wave equation for a monochromatic transverse electric wave propagating in an inhomogeneous medium of infinite extent with the inhomogeneity being transverse to the direction of propagation, showing the correspondence between the modes of a homogeneously filled conical resonator and the eigenmodes of an infinite inhomogeneous mdium. For the stable, low loss (convergent) region the solutions of the differential equation are readily found. For the unstable, high loss (divergent) region the solutions are found by using the principle of analytic continuation. The specific example of parabolic end reflectors is treated in more detail, and solutions for the eigenvalues and eigenfunctions are given for the case of infinite strip and circular geometries.

Ray theory for an anisotropic inhomogeneous elastic medium

Abstract: A ray theory is developed for an inhomogeneous, anisotropic medium, based on the concept of a wave front as the carrier of a discontinuity in particle velocity. The discontinuity conditions of several field quantities involved are formulated and serve to cast the problem in terms of a partial differential equation. The characteristics of this equation subsequently are identified in terms of rays, represented in parametric form as space curves. The energy transport along the rays is formulated by means of transport equations. The theory is applied in particular to the case of the transversely (horizontally) isotropic, vertically inhomogeneous medium. Equations for rays and travel times are obtained in the form of integrals, which are suited for serving the purpose of numerical computations.

Steady state solutions for axially-symmetric climatic variables

Abstract: Equations governing the axially-symmetric time-average state of the atmosphere and the transient departures from this mean state are set down. As a first step toward a solution of this system for seasonal average conditions, a model is formulated based on the thermodynamical energy equation for the vertical average of the mean state, and on the perturbation solutions of the linearized equations governing the baroclinic growth of transient eddies. All forms of non-adiabatic heating within the atmosphere and at the earth's surface are parameterized. The resulting differential equation governing the axially-symmetric mean potential temperature distribution takes the form of a steadystate diffusion equation in surface spherical coordinates, with a variable Austausch coefficient which is to be determined iteratively as a dependent variable.

Spectral and scattering theory for Maxwell's equations in an exterior domain

Abstract: : The spectral theory and the scattering theory of Maxwell's equations in the exterior of an obstacle are studied using the techniques developed by Phillips and Lax in their book 'Scattering Theory.' (Author)

A new stable computing method for the serial hybrid computer integration of partial differential equations

Abstract: Partial differential equations involving one space dimension and time can be solved by hybrid computers using the serial (or continuous space-discrete time) method. In so doing, the continuous integration capability of the analog computer is used along the space axis while integration along the time axis is performed in a discrete fashion by making use of finite differences.

Exact, Stationary Wave Solutions of the Nonlinear Vlasov Equation

Abstract: A procedure for systematically calculating a wide class of exact, nonlinear wave solutions of the Vlasov equation that are stationary in the wave frame is given. The essential feature of the method is the expansion of the current density in an infinite series of the vector potential or the expansion of the charge density in an infinite series in the scalar potential. The potentials obey a differential equation of the form of the equation of motion for a point particle in a conservative potential. This nonlinear differential equation has numerous analytic solutions depending on the choice of physical parameters.

Solution of the Time-Dependent Navier-Stokes Equations for the Flow around a Circular Cylinder

Abstract: Traditionally, aerodynamicists, in order to simplify the theoretical treatment of determining the flowfield around an object, have developed special methods for treating the various viscous and relatively inviscid zones of the flow, including the shock wave, the shock layer, the boundary layer, and the wake. Thus the total flowfield has generally been obtained by "patchwork." Although the foregoing approach certainly yields useful information, it is desirable to develop a method of solving the complete Navier-Stokes equations in which no such arbitrary assumptions are made. In the present paper, the authors present numerical solutions to the complete time-dependent Navier-Stokes equations for the transient supersonic flow around a right circular cylinder. The nonlinear partial differential equations for the conservation of mass, momentum, and energy were first expressed in cylindrical coordinates and then put into finite-difference form, making use of a new explicit-implicit, finitedifference scheme. These were analyzed for stability and convergence, and specific criteria were established for determining step sizes. Finally, numerical solutions were obtained for several Reynolds numbers for a freestream velocity of 10,000 fps.