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At least 1,531 records · Page 85
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NASA NTRS1972

Stationary conditions for stochastic differential equations

This is a preliminary study of possible necessary and sufficient conditions to insure stationarity in the solution process for a stochastic differential equation. It indirectly sheds some light on ergodicity properties and shows that the spectral density is generally inadequate as a statistical measure of the solution. Further work is proceeding on a more general theory which gives necessary and sufficient conditions in a form useful for applications.

Adomian, G.↗
NASA NTRS1973

A stability analysis for a semilinear parabolic partial differential equation

The parabolic partial differential equation considered is u sub t = u sub xx + f(u), where minus infinity x plus infinity and o t plus infinity. Under suitable hypotheses pertaining to f, a class of initial data is exhibited: phi(x), minus infinity x plus infinity, for which the corresponding solutions u(x,t) appraoch zero as t approaches the limit of plus infinity. This convergence is uniform with respect to x on any compact subinterval of the real axis.

Chafee, N.↗
NASA NTRS1973

A qualitative study of the complete set of solutions of the differential equation of motion of a test particle in the equatorial plane of the Kerr gravitational field

A study is made of the mathematical solution of the differential equation of motion of a test particle in the equatorial plane of the Kerr gravitational field, using S (Schwarzschild-like) coordinates. A qualitative solution of this equation leads to the conclusion that there can only be 25 different types of orbits. For each value of a, the results are presented in a master diagram for which h and e are the parameters. A master diagram divides the h, e parameter space into regions such that at each point within one of these regions the types of admissible orbits are qualitatively the same. A pictorial representation of the physical orbits in the r, phi plane is also given.

Montgomery, H. E.↗
NASA NTRS1973

Classical eighth- and lower-order Runge-Kutta-Nystroem formulas with a new stepsize control procedure for special second-order differential equations

New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.

Fehlberg, E.↗
NASA NTRS1972

Differential maneuvering simulator data reduction and analysis software

A multielement data reduction and analysis software package has been developed for use with the Langley differential maneuvering simulator (DMS). This package, which has several independent elements, was developed to support all phases of DMS aircraft simulation studies with a variety of both graphical and tabular information. The overall software package is considered unique because of the number, diversity, and sophistication of the element programs available for use in a single study. The purpose of this paper is to discuss the overall DMS data reduction and analysis package by reviewing the development of the various elements of the software, showing typical results that can be obtained, and discussing how each element can be used.

Beasley, G. P.↗
NASA NTRS1973

Pioneer-10 measurements of the differential and integral cosmic-ray gradient between 1 and 3 AU

The cosmic-ray experiment on Pioneer has provided measurements of the differential radial gradient of cosmic-ray helium nuclei between 1 and 3 AU. Upper limits are quoted between 10 and 500 MeV/nuc which are everywhere 25%/AU and in some cases substantially smaller. The integral proton gradient ( 56 MeV) was also measured and found to have the following values: 2.4 + or - 0.3%/AU (1-2 AU), 7.6 + or - 0.7%/AU (2-3 AU).

Teegarden, B. J.↗
NASA NTRS1973

A comparison of digital computer programs for the numerical solution of ordinary differential equations

Recently the determination of the best technique for numerically solving systems of ordinary differential equations on a digital computer has received much attention. The use of these formulas in conjunction with a stepsize control developed is explained, and one of the formulas is chosen for comparison with other integration techniques. This comparison of one of the best of Fehlberg's formulas with the different numerical techniques described in previous studies on a variety of test problems clearly shows the superiority of Fehlberg's formula. That is, on each of the test problems, the chosen Fehlberg formula is able to achieve a given accuracy in less computer time than any of the other techniques tested. Also, the computer program for the chosen Fehlberg formula is less complex and easier to use than the computer programs for most of the other techniques. To illustrate the use of the chosen Fehlberg formula, a computer listing of its application to several example problems is included along with a sample of the computer output from these applications.

Ingram, H. L.↗
NASA NTRS1972

Differential Pricing Policy

Differential pricing is a valid means of improving profits, keeping the total cost of air transportation down, and making it possible for more people to use air transportation.

Gebhardt, J. B.↗
NASA NTRS1972

Control of functional differential equations of retarded and neutral type to target sets in function space.

Optimal control of systems governed by functional differential equations of retarded and neutral type is considered. Problems with function space initial and terminal manifolds are investigated. Existence of optimal controls, regularity, and bang-bang properties are discussed. Necessary and sufficient conditions are derived and several solved examples which illustrate the theory are presented.

Banks, H. T.↗
NASA NTRS1973

A lunar differentiation model in light of new chemical data on Luna 20 and Apollo 16 soils.

Fines from a Luna 20 soil sample and from three Apollo 16 deep drill core samples have been analyzed for major-minor element abundances by a combined, semi-micro atomic absorption spectrophotometric and colorimetric method. Both the major element and large ion lithophile trace element abundances in these soils, the first from interior highland sites, are greatly influenced by the very high normative plagioclase content, being distinctly richer in Al and Ca, and poorer in K, P, Cr, Mn, Fe, and Ti, than most bulk soil samples from previous lunar missions. The relatively large compositional variations in the Apollo 16 core can be ascribed almost entirely to decreasing plagioclase with increasing depth. The chemical composition of the Luna 20 soil indicates less plagioclase and less KREEP than in the Apollo 16 soils. A lunar differentiation model is presented in which is made the suggestion that KREEP is the result of a second fusion event in a lunar crust consisting of early feldspathic cumulates and primary aluminous 'liquid.'

Nava, D. F.↗
NASA NTRS1972

Control of functional differential equations with function space boundary conditions.

The results of various authors dealing with problems involving functional differential equations with terminal conditions in function space are reviewed. The review includes not only very recent results, but also some little known results of Soviet mathematicians prior to 1970. Particular attention is given to results concerning controllability, existence of optimal controls, and necessary and sufficient conditions for optimality.

Banks, H. T.↗
NASA NTRS1972

Fast differential thermal analysis.

A study is conducted of the limits of time-dependent differential thermal analysis (DTA). Applications of DTA to the hysteresis phenomena of glasses are also reported. The computation of DTA sample temperatures is discussed, giving attention to feasible heating rates, transient effects to be measured, and the simulation of DTA responses to changes in thermal diffusivity similar to hysteresis phenomena in glasses. The reproducibility of temperature recording is checked with zone-refined organic chemicals.

Wunderlich, B.↗
NASA NTRS1973

Precision selenodesy via differential interferometry.

Review of the first results obtained from the development and use of a new type of differential receiver designed to determine the relative locations on the lunar surface of two Apollo-deposited transmitters. This receiver has made it possible to reduce random and systematic instrumental errors to nearly negligible levels (displacement uncertainties equivalent to a few centimeters).

Counselman, C. C., III↗
NASA NTRS1973

Advanced methods for the solution of differential equations

This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.

Goldstein, M. E.↗
NASA NTRS1973

Differential energy spectra of low energy (less than 8.5 MeV per nucleon) heavy cosmic rays during solar quiet times

Explorer 47 satellite observations of carbon, oxygen, and heavier nuclei differential energy spectra below 8.5 MeV/nucleon are presented for solar quiet time periods. A dE/dx vs E method for particle identification and energy determination was used. The instrumentation telescope included an isobutane proportional counter, a surface barrier Si detector, and a cylindrical plastic scintillator anticoincidence shield. The observations were performed outside the bow-shock and in the ecliptic plane. Results show an anisotropy of about 25% at 22 degrees west of the sun with a C/O ratio of 0.5 supporting a solar origin. The low energy portions of the C and O spectra have steep negative slopes, and the corresponding power law is given. Peculiarities in the O spectrum are discussed.

Hovestadt, D.↗
NASA NTRS1973

Investigation into the propagation of Omega very low frequency signals and techniques for improvement of navigation accuracy including differential and composite omega

An analysis of Very Low Frequency propagation in the atmosphere in the 10-14 kHz range leads to a discussion of some of the more significant causes of phase perturbation. The method of generating sky-wave corrections to predict the Omega phase is discussed. Composite Omega is considered as a means of lane identification and of reducing Omega navigation error. A simple technique for generating trapezoidal model (T-model) phase prediction is presented and compared with the Navy predictions and actual phase measurements. The T-model prediction analysis illustrates the ability to account for the major phase shift created by the diurnal effects on the lower ionosphere. An analysis of the Navy sky-wave correction table is used to provide information about spatial and temporal correlation of phase correction relative to the differential mode of operation.

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