Numerical Integration Of Differential Equations Research Articles

Supersonic laminar boundary layer near the plane of symmetry of a cone at incidence

Supersonic laminar boundary-layer equations near the plane of symmetry of a cone at incidence are treated by the similarity method. Numerical integration of differential equations governing such a flow is performed, taking into consideration the temperature dependence of the Prandtl numberPrand viscosity μ throughout the boundary layer. On the leeward side, a detailed consideration of the solutions shows the existence of two solutions up to a critical incidence beyond which it appears that no solution may be found. Calculations carried out for a set of values of the external flow Mach number show up a significant effect of this parameter on the behaviour of the boundary layer.

40. Investigation of minor meteor streams

The paper consists of three sections: (1)In continuation of a previous paper by the author, on 154 minor meteor streams, elements of orbits and other data are presented for an additional 95 minor streams (most of them less active). These streams have been found both by the studies of the photographic orbits of meteor bodies known before, and from the visual radiants of faint showers.(2)The problems of a possible family of six minor meteor streams associated with the Lexell comet 1770-I, and the connection between nine other minor streams and long-period comets, are examined. The assumption is made that nearly-parabolic comets may be accompanied by meteor streams of considerable width.(3)Radiants and elements of the orbits for 30 meteor bodies of the Cyclids are described, as were established by photographic data. Perturbed motion of one such meteor body is investigated by numerical integration of differential equations of motion on the electronic computer BESM-2, using Cowell's method of quadratures and taking account of perturbations from six planets (Venus–Uranus) and of high-order terms through to the 4th order. Over the time interval studied of 45 years the orbit of the Cyclids has been stable. Perturbing action of the Earth does not lead to any substantial changes in the elements of the orbits. Even at close approaches of the order of 0–003 AU the changes in the angular elements are not greater than about 1°.

Comments on a Floating-Point Version of Nordsieck's Scheme for the Numerical Integration of Differential Equations

Comments on a Floating-Point Version of Nordsieck's Scheme for the Numerical Integration of Differential Equations

Comments on a floating-point version of Nordsieck’s scheme for the numerical integration of differential equations

One of the important problems in numerical analysis which arises in scientific and engineering research is that of numerical integration of a system of differential equations. Because of the frequency with which this problem occurs, it is valuable to have a general-purpose numerical scheme with which the integration of a large class of differential equations can be reliably performed. One numerical scheme which is useful as such a general-purpose method, and well adapted for efficient use with digital computers, has been proposed by A. Nordsieck [1], [2]. This scheme is designed to solve a system of first-order equations,

Computer methods of network analysis

In this tutorial paper, the influence of the computer not only on the modus operandi of circuit design, but also on network theory itself, is discussed. The topological properties of linear graphs are reviewed and a matrix-topological formulation of the network problem is described. In addition to the classical mesh, node, and cutset methods, a mixed method of analysis is described which is applicable to dc, ac, and transient problems. Numerical methods of solving linear and nonlinear dc network problems are discussed and a new approach to ac analysis, using the mixed method and a numerical solution of the matrix eigenvalue problem, is described. The extension of this method to the transient analysis of linear networks is also explained. Finally, the problem of instability in the numerical integration of differential equations is discussed and several means of solving the problem are outlined.

A One-Step Method for the Numerical Integration of the Differential Equation y'' = f(x)y + g(x)

1. The numerical integration of ordinary differential equations by the use of Gaussian quadrature methods was introduced into the literature by Hammer and Hollingsworth (1955), for subsequent developments, see Morrison and Stoller (1958), Korganoff (1958), Kuntzmann (1961), Henrici (1962). In this paper we develop a one-step method for the numerical integration of the ordinary differential equation y = f(x)y + g{x), y(o) — .Vo> y(*o) — yo based on the Gauss two-point rule (see Hildebrand, 1956). Theoretical and computational comparison of the new method with other methods is given.

A New Method of Numerical Integration of Differential Equations

A New Method of Numerical Integration of Differential Equations

A new method of numerical integration of differential equations

(1) y= f(x, y), y(xo) = yo is Heun's method. However, this method requires two substitutions into the differential equation (1) for each step forward. A method described by Lotkin [1] achieves comparable accuracy but requires only one substitution in (1). This can be quite a saving in time when (1) is a complicated function. This article illustrates another method which also requires only one substitution in (1).

PREDICTION OF MOVEMENT OF RADIONUCLIDES IN SOLUTION THROUGH POROUS MEDIA.

The transport of a cationic radionuclide through the earth was determined by hydrodynamic convection, hydraulic dispersion, and ion-exchange sorption. If the manner of water movement is first asdertained by tritiated water tracer measurements and the exchange properties are determined by laboratory tests, it is possible to predict the arrival of individual radioisotopes at some distant point of concern. For the prediction theory to be fully applicable, the formation should be reasonably homogeneous, the isotope distribution factor should remain constant, and exchange equilibria should prevail at all points. It is also desirable that the hydraulic flow net remain unchanged. The validity of the prediction theory was tested with laboratory columns and with a field investigation. The field test involved several arrays of injections and relief wells and twenty-three observation wells penetrating a confined aquifer 100 ft below the earth's surface. The results demonstrated the theoretical validity and practical utility of the theory. (auth) H03 H0237001 Boiling Water Reactors. Zero-power experiments were carried out on the central superheater core CSH-1 of BORAX V. The chemistry of the reactor cooling water was followed. The maintenance and development involved in BORAX V are described. Liquid-Metal-Cooled Reactors. Central reactivity coefficient and Doppler coefficient measurements aremore » reported for a 4000-liter core in Assembly 43 of ZPR- III. Development of fast reactor fuels is described. The operation of EBR-1, Mark IV and EBR-II is described. The fuel cycle facility and process development of EBR-II are also considered. The design studies and other development work on FARET are discussed. General Reactor Technology. Nuclear physics studies which were made included Doppler effect calculations, numerical integration of differential equations, and development of a solid-state fast neutron spectrometer. Corrosion studies on reactor materiais are described. The development of ceramic and particulate metal fuel elements is discussed, and the irradiation and extrusion studies are described. Among the chemical studies carried out were the formation of Tc/sub 2/ Zn/sub 11/in Tc-- Zn, dissolution of U in flowing liquid Cd, thermal decomposition of UO/sub 2/Cl/sub 2/ in molten LiCi--KCl, solubility of U in Cd--Mg--Zn alloys, fluidization and volatility separation processes, preparation of UC and US, analysis of PuF/sub 6/, solubility of Ar in Na, etc. In the Plutonium Recycle Program, the cross-section correction factors for the Pu region and the building of Sm/sup 149/ in the reactor zones are discussed. Advanced Systems Research and Development. The operation of a thermionic flim-boiling liquid-metal diode with cesium is described. The design of a nuclear magnetohydrodynamic system (fast reactor coupled with liquid-sodium MHD generator) was analyzed, and the system characteristics are tabulated. Nuclear Safety. Oxidation studies were carried out on Pu and Pu--2 at.% Al alloy. The Al--U/sub 3/O/sub 8/ thermite reaction and the vapor-phase burning of Al in water were studied. In-pile oxidation studies in air were made on one sample of U and one sample of a U-Fs alloy. (D.L.C.)« less

Numerical integration of differential equations of the structure of shockwaves in plasma

Numerical integration of differential equations of the structure of shockwaves in plasma

Periodic Solutions of van der Pol's Equations with Large Damping Coefficientlambda = 0 sim 10

A method of computing a periodic solution of van der Pol's equation is devised reducing the problem to the solution of a certain equation by means of Newton's method. For computing the value of the derivative necessary to apply Newton's method, the properties of variation of the orbit in the phase plane are used and, for step-by-step numerical integration of differential equations, a somewhat new method based on Stirling's interpolation formula combined with an ordinary Adams' extrapolating integration formula is used. The periodic solutions are actually computed for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda = 0 \sim 10</tex> and the minute but important change of the amplitude described by van der Pol's equation is found.

Numerical Integration of Differential Equations. By A. A. Bennett, W. E. Milne and H. Bateman. Pp. 108. $1.35. 1956. (Dover Publications, New York)

Numerical Integration of Differential Equations. By A. A. Bennett, W. E. Milne and H. Bateman. Pp. 108. $1.35. 1956. (Dover Publications, New York)

Numerical Integration of Differential Equations.

Numerical Integration of Differential Equations.

Numerical integration of differential equations on the 704 EDPM

To provide a convenient means for solving complex and nonlinear ordinary differential equations, a system has been devised which causes a 704 EDPM to behave like an analogue differential analyzer. Use of this system involves the interconnection of appropriate subroutines in a manner analogous to the interconnection of operational amplifiers in an analogue machine. The subroutines are designed so that the user may compile the required connections automatically, without much knowledge of digital computer coding. Solving differential equations by this method takes no more setup time than that required by analogue machines and in many cases may be a good deal faster. Integration is done by using Milne's formulas. The system automatically adjusts the integration interval to take advantage of portions of the solution which can be computed with large intervals at no sacrifice of accuracy. Nonlinear functions may be expressed by a table of values in storage.

Osculatory extrapolation and a new method for the numerical integration of differential equations

This chapter presents the problem of numerical integration of differential equations and the techniques available for the derivation of numerical algorithms. It presents a unified and direct development of many of the equations of interest in the solution of initial value ordinary differential equations (ODES). In addition to the first-order scalar equation, it is possible to consider a set of simultaneous first-order equations or an equivalent high-order single equation. To lead into the development of certain equations of major importance in numerically solving ODES, a number of interpolation formulas are defined in the chapter. These two formulas are termed “Newton's forward formula” and “Newton's backward formula” respectively. Each is obtained by fitting a polynomial to the sequence of points (yn) at equally spaced (xn).

A method for the numerical integration of differential equations of second order without explicit first derivatives

A method for the numerical integration of differential equations of second order without explicit first derivatives

A note on the numerical integration of differential equations

A note on the numerical integration of differential equations

A NOTE ON THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Journal Article A NOTE ON THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS Get access E. M. WILSON E. M. WILSON Royal Naval Scientific ServiceAdmiralty, London, S.W.1 Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mechanics and Applied Mathematics, Volume 2, Issue 2, 1949, Pages 208–211, https://doi.org/10.1093/qjmam/2.2.208 Published: 01 January 1949 Article history Received: 19 February 1948 Published: 01 January 1949

Note on the Numerical Integration of Differential Equations

Note on the Numerical Integration of Differential Equations

Formulae for Numerical Integration

Numerical integration, including the numerical integration of differential equations, is becoming increasingly necessary in technical applications. For such purposes, when a very high order of accuracy is not required, formulae which use successive values of the function to be integrated are more convenient than those using differences. Individual authors have given formulae of this type from time to time but a systematic list is not known to the present writer, and would seem to be of service.The present list gives formulae of four types : (I) formulae for evaluating definite integrals ; (II) formulae for forward integration ; (III) formulae for integrating over a subrange ; and (IV) formulae involving values of both the function and its integral. The leading term in the error series is given. Finally, some comments are made upon the formulae and their uses.