A method for the numerical integration of coupled first-order differential equations with greatly different time constants.
Numerical integration of coupled first order ODE OF greatly differing time constant
DIFFERENTIAL EQUATION↗
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Optimal control with Chebyshev minimax performance index.
Optimal control of dynamic systems with minimax type performance index, discussing application of proposed method of solution
CHEBYSHEV APPROXIMATION↗
Numerical integration of nonlinear differential equations by use of rational approximation Final report, 15 Jan. 1965 - 14 Jul. 1967
Numerical integration of nonlinear differential equations by use of rational approximation
NUMERICAL INTEGRATION↗
Multichannel photo-ionization of atomic systems.
Multichannel photoionization of atomic systems obtained with dipole approximation assuming LS COUPLING, using wave functions
PHOTOIONIZATION↗
Sample quadratic variation of sample continuous second order martingales.
Sample continuous second order martingale process, determining existence of limit of stochastic processes using approximation theorems
RANDOM PROCESS↗
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.↗
Two-way coherent tracking systems.
Review of some results pertinent to the development of nonlinear theory on two-way coherent tracking systems. In particular, the model of cascaded systems is presented, and approximations to steady state probability density functions of the two-way system phase and Doppler error are developed. From these, certain numerical results required in the design and planning of these systems are derived.
Lindsey, W. C.↗
Power series evaluation of transition and covariance matrices.
Reexamination power series solutions to the matrix covariance differential equation and the transition differential equation. Truncation error bounds are derived which are computationally attractive and which extend previous results. Polynomial approximations are obtained by exploiting the functional equations satisfied by the transition and covariance matrices. The series-functional equation propagation technique represents a fast and accurate alternative to the numerical integration of the time-invariant transition and covariance equations.
Bierman, G. J.↗
Computer enhancement of radiographs
Examination of three relevant noise processes and the image degradation associated with Marshall Space Flight Center's (MSFC) X-ray/scanning system was conducted for application to computer enhancement of radiographs using MSFC's digital filtering techniques. Graininess of type M, R single coat and R double coat X-ray films was quantified as a function of density level using root-mean-square (RMS) granularity. Quantum mottle (including film grain) was quantified as a function of the above film types, exposure level, specimen material and thickness, and film density using RMS granularity and power spectral density (PSD). For various neutral-density levels the scanning device used in digital conversion of radiographs was examined for noise characteristics which were quantified by RMS granularity and PSD. Image degradation of the entire pre-enhancement system (MG-150 X-ray device; film; and optronics scanner) was measured using edge targets to generate modulation transfer functions (MTF). The four parameters were examined as a function of scanning aperture sizes of approximately 12.5 25 and 50 microns.
Dekaney, A.↗
Emissivity of half-space random media
Scattering of electromagnetic waves by a half-space random medium with three-dimensional correlation functions is studied with the Born approximation. The emissivity is calculated from a simple integral and is illustrated for various cases. The results are valid over a wavelength range smaller or larger than the correlation lengths.
Tsang, L.↗
The effect of suprathermal protons on the physical conditions in Seyfert galaxy nuclei. II
Radiative transfer of Ly-alpha, Ly-beta, and H-alpha in a hydrogen gas containing dust is considered with application to the emitting gas in Seyfert galaxies. The line radiation originates from the kinetic energy of suprathermal particles lost initially to a free-electron gas, which then delivers it to the radiation through excitation of neutral hydrogen atoms. By neglecting the direct escape of line radiation and by averaging over the gas, transfer in space is suppressed, and only transfer in frequency is considered. The dust degrades the line radiation via frequency-independent absorption, converting the energy to infrared luminosity. Solutions are obtained for the source functions in the lines, using appropriate approximations, in order to determine under what conditions the narrow component of the Balmer-line radiation from the gas can be self-absorbed and degraded without similar degradation of the broad component, which originates from the suprathermals themselves. The results of the transfer calculation are employed to find self-consistent values for the temperature and ionization of the gas for various amounts of dust and various concentrations of suprathermal particles. The luminosity in the Balmer continuum is examined which would result under these conditions.
Stoner, R.↗
A triangular element based on generalized potential energy concepts
Stiffness equations are formulated for a doubly-curved triangular thin shell finite element. The strain energy component of the potential energy is first expressed in terms of displacements and displacement gradients with the aid of consistent deep shell strain-displacement equations. The element in-plane and normal displacement fields are approximated by complete cubic polynomials. These functions do not satisfy the interelement displacement admissibility conditions. Satisfaction is forced by the imposition of constraint conditions on the interelement boundaries; the constraints represent the modification of the potential energy. Some numerical results for a pinched cylinder, a cylindrical sphere, and a pinched sphere are examined.
Thomas, G. R.↗
M2, S2, K1 models of the global ocean tide
Ocean tidal signals appear in many geophysical measurements. Geophysicists need realistic tidal models to aid in interpretation of their data. Because of the closeness to resonance of dissipationless ocean tides, it is difficult for numerical models to correctly represent the actual open ocean tide. As an approximate solution to this problem, test functions derived by solving Laplace's Tidal Equations with ocean loading and self gravitation are used as a basis for least squares dynamic interpolation of coastal and island tidal data for the constituents M2, S2, and Kl. The resulting representations of the global tide are stable over at least a ?5% variation in the mean depth of the model basin, and they conserve mass. Maps of the geocentric tide, the induced free space potential, the induced vertical component of the solid earth tide, and the induced vertical component of the gravitational field for each contituent are presented.
Parke, M. E.↗
Atomic electron excitation probabilities during orbital electron capture by the nucleus
Approximate probabilities of electron excitation (shakeup/shakeoff) from various atomic states during nuclear ns electron capture have been calculated in the sudden approximation, using Hartree-Fock wave functions. Total excitation probabilities are much lower than during inner-shell ionization by photons or electrons, and ns states are more likely to be excited than np states. This latter result is borne out by K-alpha X-ray satellite spectra.
Crasemann, B.↗
Preliminary design of composite wing-box structures for global damage tolerance
A procedure is presented that incorporates the influence of potential global damage conditions into the design process for minimum-mass wing-box structures. The procedure is based on mathematical-programming optimization techniques. Material-strength, minimum-gage, and panel-buckling constraints are introduced by penalty functions, and Newton's method with approximate second derivatives of the penalty terms is used as the search algorithm to obtain minimum-mass designs. A potential global damage condition is represented by a structural model with the damaged components removed. Example minimum-mass designs are obtained that simultaneously satisfy the constraints of the damaged and undamaged configurations of both graphite-epoxy and aluminum wing-box structural models. These examples are designed with and without the influence of potential damage conditions, and results indicate that for equal mass cases the residual strength of a damaged structure is higher when the influence of potential damage is properly included in the design from the outset. Results of these examples also identify the minimum structural mass increase required to increase residual strength levels.
Starnes, J. H., Jr.↗
Theory of inhomogeneous fluids
A theory of nonuniform liquids is presented which is based on the Yvon-Born-Green equation for the one-particle density and the Ornstein-Zernike equation of an inhomogeneous system. The necessary closure is effected by exploiting the solution of a modified hypernetted-chain equation and making a local approximation on the (highly universal) bridge function. The theory is successfully applied to model problems that have also been studied by direct-simulation methods. A brief generalization to quantum liquids is also given.
Nieminen, R. M.↗
The galactic cosmic-ray radial intensity gradient and large-scale modulation in the heliosphere
Measurements of fluxes from 1 AU on the IMP 8 satellite to 11 AU on the Pioneer 11 and to greater than 24 AU on the Pioneer 10 spacecraft show that the nucleon integral gradient is a function of the level of the approximately 11-year solar modulation cycle. The gradient decreases from approximately 4.5% to 1.5% throughout the solar minimum of 1975-1977, and increases to approximately 2.5-3.0% per AU near maximum modulation. Pioneer 10 at greater than 24 AU is still deep in the modulation region, leading to estimates of 50-70 AU for the radial distance to the heliopause at both solar minimum and solar maximum modulation.
Mckibben, R. B.↗
A numerical study of the effects of ambipolar diffusion on the collapse of magnetic gas clouds
The gravitational collapse of isothermal, nonrotating magnetic gas clouds have been calculated numerically, including the effects of ambipolar diffusion. The fractional ionization in the clouds is approximated by a power-law function of the gas density, f = K/n to the q-power, where K and q are adjustable parameters. Eleven numerical experiments were run, and the results indicate that the asymptotic character of collapse is determined mainly by the value of q and is largely independent of the other parameters characterizing a cloud (e.g., K, cloud mass). In particular, there is nearly a one-to-one correspondence between q and the slope, x, of the central magnetic field strength-gas density relationship. If q is no more than 0.8, a cloud collapses asymptotically, as though the magnetic field were 'frozen' to the neutral matter. The magnetic field strength at the center of a collapsing cloud is strongly amplified during collapse even for values of q of about 1, despite extremely low values of fractional ionization. A discussion of the theoretical basis for this unexpected behavior is given. Possible implications of our results for the problems of magnetic braking of rotating protostars and star formation in general are also presented.
Black, D. C.↗