PERIODIC SOLUTIONS OF THE RESTRICTED THREE BODY PROBLEM REPRESENTING ANALYTIC CONTINUATIONS OF KEPLERIAN ELLIPTIC MOTIONS
Periodic solution of restricted three body problem as analytic continuation of keplerian elliptical motions
KEPLER LAW↗
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Simultaneous integration of n-bodies by analytic continuation with recursively formed derivatives.
Motion equations of sun, moon, earth and lunar probe integrated by power series solution and analytic continuation with recursively formed derivatives
Hartwell, J. G.↗
Analytic continuation of Appell's hypergeometric series F sub 2 to the vicinity of the singular point x equals 1, y equals 1.
Analytic continuation of Appell hypergeometric series into linear combination of four infinite series
Hahne, G. E.↗
Computer extension and analytic continuation of Stokes' expansion for gravity waves
Stokes' infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes' expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Pade approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth.
Schwartz, L. W.↗
Analytical continuation for calculating multigroup-multizone eigenvalue problems using Lie series.
Critical volume of cylindrical reactors calculated using n-group diffusion theory
REACTOR CORE↗
Simultaneous integration of n-bodies by analytic continuation with recursively formed derivatives.
Recursion formulae to generate Taylor series expansion for motion of n-point masses subject to gravitational attractions
Hartwell, J. G.↗
Global solutions by analytic parameter continuation for a class of non-linear singular integral equations
Solution of non-linear singular integral equations from hydroelasticity and electroelasticity by analytic parameter continuation
Sloss, J. M.↗
Series solution for the planar asymmetric blunt-body problem
Accurate semianalytic solutions to the inverse blunt-body problem have been obtained using a method of series expansion. Rational fractions are employed for series summation and analytic continuation. Angles of incidence up to 30 deg and Mach numbers as low as 2 have been considered. The maximum-entropy streamline will not wet the body surface in asymmetric flow. It may pass either above or below the stagnation streamline. Limit lines appear in the supersonic portion of the flow field, both in the shock layer and in its upstream analytic continuation.
Schwartz, L. W.↗
NASA’s System-Wide Safety Project: System-Wide Predictive Analytics and Continuous Learning for In-time Aviation Safety Management Systems (IASMS)
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Michael James Vincent↗
Analytic theory for the selection of 2-D needle crystal at arbitrary Peclet number
An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Peclet number for small values of the surface tension parameter. The velocity selection is caused by the effect of transcendentally small terms which are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small Peclet number analytical results of other investigators, though there are some discrepancies in details. It also addresses questions raised on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.
Tanveer, Saleh↗
Analytic theory for the selection of a two-dimensional needle crystal at arbitrary Peclet number
An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Peclet number for small values of the surface tension parameter. The velocity selection is caused by the effect of transcendentally small terms which are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small Peclet number analytical results of other investigators, though there are some discrepancies in details. It also addresses questions raised on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.
Tanveer, S.↗
Reflecting Solutions of High Order Elliptic Differential Equations in Two Independent Variables Across Analytic Arcs
Consideration is given specifically to sixth order elliptic partial differential equations in two independent real variables x, y such that the coefficients of the highest order terms are real constants. It is assumed that the differential operator has distinct characteristics and that it can be factored as a product of second order operators. By analytically continuing into the complex domain and using the complex characteristic coordinates of the differential equation, it is shown that its solutions, u, may be reflected across analytic arcs on which u satisfies certain analytic boundary conditions. Moreover, a method is given whereby one can determine a region into which the solution is extensible. It is seen that this region of reflection is dependent on the original domain of difinition of the solution, the arc and the coefficients of the highest order terms of the equation and not on any sufficiently small quantities; i.e., the reflection is global in nature. The method employed may be applied to similar differential equations of order 2n.
Carleton, O.↗
Use of analyticity in the calculation of nonrelativistic scattering amplitudes.
Two body and two channel nonrelativistic scattering amplitudes calculation method using numerical analytic continuation by rational fractions
Schlessinger, L.↗
A theory of supercritical wing sections, with computer programs and examples.
Mathematical methods for the design of supercritical wings, which depend on the numerical solution of the partial differential equations of two-dimensional gas dynamics, are developed. The main contribution is a computer program for the design of shockless transonic airfoils using the hodograph transformation and analytic continuation into the complex domain. The mathematical theory is described, and a manual for users of the programs is provided. Numerical examples are given and computational results are discussed, and the computer programs themselves are listed. The analysis routine can be used to ascertain whether the profiles behave well at off-design conditions, or to smooth coordinates and obtain a desirable shape more quickly when perfectly shockless flow is not essential.
Bauer, F.↗
On the tidal effects in the motion of earth satellites and the love parameters of the earth
The tidal effects in the motion of artificial satellites are studied to determine the elastic properties of the earth as they are observed from extraterrestrial space. Considering Love numbers, the disturbing potential is obtained as the analytical continuation of the tidal potential from the surface of the earth into-outer space, with parameters which characterize the earth's elastic response to tidal attraction by the moon and the sun. It is concluded that the tidal effects represent a superposition of a large number of periodic terms, and the rotation of the lunar orbital plane produces a term of 18 years period in tidal perturbations of the ascending node of the satellite's orbit.
Musen, P.↗
Control law synthesis and sensor design for active flutter suppression.
Methods are presented for representing unsteady aerodynamic loadings, valid for arbitrary motion, for the two-dimensional typical section with a trailing-edge control surface in incompressible flow and a three-dimensional lifting surface with leading- or trailing-edge control surfaces in subsonic compressible flow. Loads for the two-dimensional incompressible case are obtained via analytic continuation of the Theodorsen function into the complex plane, with guidance from a time-domain approximation of the Wagner indicial function. For the three-dimensional case, it is proposed that oscillatory generalized aerodynamic forces be approximated by Pade fractions, thereby permitting an obvious continuation into the complex plane. A theoretical justification for this procedure is briefly outlined, and some practical aspects of its implementation are discussed.
Lyons, M. G.↗
A representation of Jacchia's thermospheric models in spherical harmonic functions
The Jacchia models are represented in terms of spherical harmonic functions. This representation has the advantage of ease of comparison with other global theoretical and empirical models that use this mathematical form. Furthermore, it is analytic, continuous, and has continuous derivatives all over the globe. The representation of the exospheric temperatures shows clearly the amplitudes of the various periodic terms and uses relatively few constants. An example of a similar representation for the total mass density at a particular height and level of solar activity is given as well.
Blum, P.↗
Hypersonic flows generated by parabolic and paraboloidal shock waves
A computer algorithm has been developed to determine the blunt-body flowfields supporting symmetric parabolic and paraboloidal shock waves at infinite free-stream Mach number. Solutions are expressed in an analytic form as high-order power series, in the coordinate normal to the shock, whose coefficients can be determined exactly. Analytic continuation is provided by the use of Pade approximations. Test cases provide solutions of very high accuracy. In the axisymmetric case for gamma equals 715 the solution has been found far downstream, where it agrees with the modified blast-wave results. For plane flow, on the other hand, a limit line appears within the shock layer, a short distance past the sonic line, suggesting the presence of an imbedded shock. Local solutions in the downstream limit are discussed.
Schwartz, L. W.↗