Difference equations and iterative processes
Divergence equations and iterative processes
Cavanagh, R. C.↗
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Results for “DIFFERENCE EQUATION”
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Some MACSYMA program for solving difference equations
A set of MACSYMA programs are described for finding closed form solutions to linear recurrence relations in equations having either constant or variable coefficients. In the homogenous case, a polymonial equation is obtained and the solution to the recurrence relation can be written as a linear combination of the roots of the polynomial. Exponential generating functions are used to solve variable coefficient relations. Taking successive derivatives and using the recurrence relation, an ordinary differential equation is obtained. Expanding the solution to the differential equation in a Taylor series, shows that the nth term of the series is the solution to the recurrence relation. For second order recurrences, a check is made for those that can be solved in terms of Bessel functions.
Ivie, J.↗
Difference-Equation/Flow-Graph Circuit Analysis
Numerical technique enables rapid, approximate analyses of electronic circuits containing linear and nonlinear elements. Practiced in variety of computer languages on large and small computers; for circuits simple enough, programmable hand calculators used. Although some combinations of circuit elements make numerical solutions diverge, enables quick identification of divergence and correction of circuit models to make solutions converge.
Mcvey, I. M.↗
Discrete potential theory for two-dimensional Laplace and Poisson difference equations
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Saltzer, Charles↗
Pole-equator difference and the variability of the brightness of the chromospheric CaII-K-network elements in quiet regions over the solar cycle
The dependence of the brightness of chromospheric network elements on latitude was investigated for quiet solar regions. Calibrated photographic CaII K-spectroheliograms were used to compare the variation in brightness at the center of the disk with higher latitude of chromospheric network elements in a quiet region as a function of solar activity. It was found that there was no significant difference in brightness between the center of the solar disk and higher latitude. It is concluded that the brightness of the chromospheric network elements in a quiet region does not depend on the latitude, but that the variation in the intensity enhancement is related to the level of solar activity.
Kariyappa, R.↗
Bioelectrical impedance analysis prediction equations differ between African Americans and Caucasians, but it is not clear why
No abstract available
Non-NASA Center↗
The coupled equation approach to the numerical solution of the biharmonic equation by finite differences. I.
Coupled equation approach to finite difference solution of biharmonic equation, noting outer iteration scheme for pair of discrete Poisson equations
Smith, J.↗
On the stability of processes defined by stochastic difference-differential equations
Stability of processes defined by stochastic difference and differential equations
DIFFERENCE EQUATION↗
On the application of difference methods for the asymptotic estimates of errors at numerical integration of systems of ordinary differential equations
Difference methods for asymptotic estimates of errors at numerical integration of systems of ordinary differential equations
NUMERICAL INTEGRATION↗
Digital filter synthesis computer program
Digital filter synthesis computer program expresses any continuous function of a complex variable in approximate form as a computational algorithm or difference equation. Once the difference equation has been developed, digital filtering can be performed by the program on any input data list.
Moyer, R. A.↗
On the stability of processes defined by stochastic difference-differential equations.
Stability of stochastic processes defined by difference differential equations, noting properties of solution, weak infinitesimal operator, etc
Kushner, H. J.↗
Stochastic Mixing Model with Power Law Decay of Variance
Here we present a simple stochastic mixing model based on the law of large numbers (LLN). The reason why the LLN is involved in our formulation of the mixing problem is that the random conserved scalar c = c(t,x(t)) appears to behave as a sample mean. It converges to the mean value mu, while the variance sigma(sup 2)(sub c) (t) decays approximately as t(exp -1). Since the variance of the scalar decays faster than a sample mean (typically is greater than unity), we will introduce some non-linear modifications into the corresponding pdf-equation. The main idea is to develop a robust model which is independent from restrictive assumptions about the shape of the pdf. The remainder of this paper is organized as follows. In Section 2 we derive the integral equation from a stochastic difference equation describing the evolution of the pdf of a passive scalar in time. The stochastic difference equation introduces an exchange rate gamma(sub n) which we model in a first step as a deterministic function. In a second step, we generalize gamma(sub n) as a stochastic variable taking fluctuations in the inhomogeneous environment into account. In Section 3 we solve the non-linear integral equation numerically and analyze the influence of the different parameters on the decay rate. The paper finishes with a conclusion.
Fedotov, S.↗
Local behavior of autonomous neutral functional differential equations.
Basic problems for a special class of neutral functional differential equations (NFDE) are formulated, and some contributions to a general qualitative theory in the neighborhood of an equilibrium point are indicated. The properties of a NFDE (G,f) are examined to determine in what sense these properties are insensitive to small changes in (G,f) in the topology G x F. The special class of equations that is introduced includes retarded functional differential equations and difference equations.
Hale, J. K.↗
Frequency response fo multiple-sampling rate systems
Analytical procedure simplifies prediction of frequency response of multirate digital control systems. Although developed for Space Shuttle flightcontrol system, procedure is applicable to any multirate system describable by linear, constant-coefficient differential equations of difference equations.
Scharmack, D. K.↗
Finite difference methods for transient signal propagation in stratified dispersive media
Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.
Lam, D. H.↗
Linear functional-differential equations with constant coefficients
Linear differential-difference equations with constant coefficients
LINEAR EQUATION↗
Wave propagation in a random medium - A complete set of the moment equations with different wavenumbers
The propagation of waves in a random medium is studied in the 'quasi-optics' and the 'Markov random process' approximations. Under these assumptions, a Fokker-Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of moment equations with different transverse coordinates and different wave numbers is then obtained from the Fokker-Planck equation of the characteristic functional. The application of those results to the pulse smearing of the pulsar signal and the frequency correlation function of the wave intensity in interstellar scintillation is briefly discussed.
Lee, L. C.↗
Discrete approximation methods for parameter identification in delay systems
Approximation schemes for parameter identification problems in which the governing state equation is a linear functional differential equation of retarded type are constructed. The basis of the schemes is the replacement of the parameter identification problem having an infinite dimensional state equation by a sequence of approximating parameter identification problems in which the states are given by finite dimensional discrete difference equations. The difference equations are constructed using linear semigroup theory and rational function approximations to the exponential. Sufficient conditions are given for the convergence of solutions to the approximating problems, which can be obtained using conventional methods, to solutions to the original parameter identification problem. Finite difference and spline based schemes using Paderational function approximations to the exponential are constructed, and shown to satisfy the sufficient conditions for convergence. A discussion and analysis of numerical results obtained through the application of the schemes to several examples is included.
Rosen, I. G.↗