scholarly journals A Series-Solution Method for Cometary Orbits

1972 ◽  
Vol 45 ◽  
pp. 43-51
Author(s):  
P. E. Nacozy
Keyword(s):  
Series Solution ◽  
Solution Method ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Reference Solution ◽  
Successive Interval ◽  
First Order ◽  
Order Solution ◽  
The Mean ◽  
The Masses

A series-solution method for highly-eccentric perturbed orbits using a modified form of Hansen's method of partial anomalies is presented. Series in Chebyshev polynomials in the eccentric anomaly of a comet and the mean anomaly at an epoch of a planet provide a theory valid to first order with respect to the masses. The first-order solution becomes a reference solution about which higher-order perturbations are obtained by the method of successive approximations. The first-order solutions are valid approximations for long durations of time, whereas the higher orders are valid only over the interval of time that is selected for the Chebyshev expansions. The method is somewhat similar to Encke's method of special perturbations except that for each successive interval of time perturbations about a first-order solution are calculated instead of perturbations about a conic solution.

A Successive Approximation Approach to Problems in Radiative Transfer with a Differential Formulation

1980 ◽  
Vol 102 (1) ◽  
pp. 86-91 ◽  
Author(s):  
W. W. Yuen ◽  
C. L. Tien
Keyword(s):  
Radiative Transfer ◽  
Solution Method ◽  
Practical Importance ◽  
Approximate Solutions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Dimensional Problem ◽  
Weighted Residuals ◽  
First Order ◽  
Differential Formulation

The radiation intensity in a gray participating medium is expressed in a differential form. The energy equation for radiative transfer becomes an infinite-order differential equation. Utilizing the method of weighted residuals and introducing some appropriate formulations for the intensity boundary conditions, a method of successive approximations is developed. The solution method is applied to a one-dimensional problem with linear-anisotropic scattering. This problem is chosen because of its practical importance and the availability of exact solutions. A first-order closed-form result, which has never been derived analytically before, is obtained and shown to have good accuracy. Successive higher-order approximate solutions are also presented. These solutions are easily attainable algebraically and converge quickly to the exact result. To illustrate the possible applicability of the solution method for multidimensional problems, the first-order solution to a simple two-dimensional problem is presented. Results show that based on the present approach, reasonably accurate approximate solutions can be generated with some simple mathematical developments.

scholarly journals Spin-Dependent Coupled Altarelli-Parisi Equations in the NLO and the Method of Successive Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Ranjit Choudhury ◽  
D. K. Choudhury
Keyword(s):  
Gluon Distribution ◽  
Quark Distribution ◽  
Distribution Functions ◽  
Integrodifferential Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Leading Order ◽  
Order Linear ◽  
First Order ◽  
Analytic Expressions

The coupled Altarelli-Parisi (AP) equations for polarized singlet quark distribution and polarized gluon distribution, when considered in the small x limit of the next to leading order (NLO) splitting functions, reduce to a system of two first order linear nonhomogeneous integrodifferential equations. We have applied the method of successive approximations to obtain the solutions of these equations. We have applied the same method to obtain the approximate analytic expressions for spin-dependent quark distribution functions with individual flavour and polarized structure functions for nucleon.

First Order Partial Functional Differential Equations with Unbounded Delay

2003 ◽  
Vol 10 (3) ◽  
pp. 509-530
Author(s):  
Z. Kamont ◽  
S. Kozieł
Keyword(s):  
Differential Equations ◽  
Integral Functional ◽  
Functional Differential Equations ◽  
Generalized Solutions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
First Order ◽  
Unbounded Delay ◽  
The Cauchy Problem ◽  
Functional Differential

Abstract The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

Statistical hydromechanics of disperse systems. Part 2. Solution of the kinetic equation for suspended particles

1972 ◽  
Vol 52 (2) ◽  
pp. 345-355 ◽  
Author(s):  
Yu. A. Buyevich
Keyword(s):  
Kinetic Equation ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Mean Flow ◽  
Time And Space ◽  
Two Phase ◽  
The Mean ◽  
Derivatives Of ◽  
Dynamic Variables

To solve the kinetic equation for particles of a monodisperse two-phase mixture the method of successive approximations is developed; this resembles in its main features the well-known Chapman-Enskog method in the kinetic theory of gases. This method is applicable for a mixture whose state differs slightly from the equilibrium, i.e., when time and space derivatives of the dynamic variables describing the mean flow of both phases of the mixture are sufficiently small. Accordingly, the solution obtained is valid when the time and space scales of the mean flow exceed considerably those for random pseudo-turbulent motion of particles and a fluid. The conservation equations for determination of all the dynamic variables are formulated in approximations which have the same meaning as those of Euler and Navier & Stokes in hydromechanics of one-phase media.

New classes of integrals inherent in the mathematical structure of extended equations describing superconducting systems

2015 ◽  
Vol 29 (17) ◽  
pp. 1550117 ◽  
Author(s):  
Ryszard Gonczarek ◽  
Mateusz Krzyzosiak ◽  
Adam Gonczarek ◽  
Lucjan Jacak
Keyword(s):  
Mathematical Structure ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Superconducting Gap ◽  
S Wave ◽  
First Order ◽  
Order Expansion ◽  
Quantitative Characteristics ◽  
Fundamental Equations ◽  
New Generation

In this paper, we discuss the mathematical structure of the s-wave superconducting gap and other quantitative characteristics of superconducting systems. In particular, we evaluate and discuss integrals inherent in fundamental equations describing superconducting systems. The results presented here extend the approach formulated by Abrikosov and Maki, which was restricted to the first-order expansion. A few infinite families of integrals are derived and allow us to express the fundamental equations by means of analytic formulas. They can be then exploited in order to find some quantitative characteristics of superconducting systems by the method of successive approximations. We show that the results can be applied to some modern formalisms in order to study high-Tc superconductors and other superconducting materials of the new generation.

Predicting Daily Maintenance Dose of Fluindione, an Oral Anticoagulant Drug

1996 ◽  
Vol 75 (05) ◽  
pp. 731-733 ◽  
Author(s):  
V Cazaux ◽  
B Gauthier ◽  
A Elias ◽  
D Lefebvre ◽  
J Tredez ◽  
...  
Keyword(s):  
Effective Dose ◽  
Maintenance Dose ◽  
Successive Approximations ◽  
Hemorrhagic Complication ◽  
Simple Method ◽  
Individual Variations ◽  
Anticoagulant Drug ◽  
The Mean ◽  
Mean Time ◽  
Daily Maintenance Dose

SummaryDue to large inter-individual variations, the dose of vitamin K antagonist required to target the desired hypocoagulability is hardly predictible for a given patient, and the time needed to reach therapeutic equilibrium may be excessively long. This work reports on a simple method for predicting the daily maintenance dose of fluindione after the third intake. In a first step, 37 patients were delivered 20 mg of fluindione once a day, at 6 p.m. for 3 consecutive days. On the morning of the 4th day an INR was performed. During the following days the dose was adjusted to target an INR between 2 and 3. There was a good correlation (r = 0.83, p<0.001) between the INR performed on the morning of day 4 and the daily maintenance dose determined later by successive approximations. This allowed us to write a decisional algorithm to predict the effective maintenance dose of fluindione from the INR performed on day 4. The usefulness and the safety of this approach was tested in a second prospective study on 46 patients receiving fluindione according to the same initial scheme. The predicted dose was compared to the effective dose soon after having reached the equilibrium, then 30 and 90 days after. To within 5 mg (one quarter of a tablet), the predicted dose was the effective dose in 98%, 86% and 81% of the patients at the 3 times respectively. The mean time needed to reach the therapeutic equilibrium was reduced from 13 days in the first study to 6 days in the second study. No hemorrhagic complication occurred. Thus the strategy formerly developed to predict the daily maintenance dose of warfarin from the prothrombin time ratio or the thrombotest performed 3 days after starting the treatment may also be applied to fluindione and the INR measurement.

First passage time for the state of exact chemical equilibrium

1980 ◽  
Vol 45 (3) ◽  
pp. 777-782 ◽  
Author(s):  
Milan Šolc
Keyword(s):  
Chemical Equilibrium ◽  
First Passage Time ◽  
Passage Time ◽  
The State ◽  
Recurrence Time ◽  
First Passage ◽  
First Order ◽  
The Mean ◽  
Passage Times ◽  
Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

The Structure and Photochromism of 2,4,4,6-Tetraphenyl-4H-thiopyran

1992 ◽  
Vol 57 (6) ◽  
pp. 1326-1334 ◽  
Author(s):  
Jaroslav Vojtěchovský ◽  
Jindřich Hašek ◽  
Stanislav Nešpůrek ◽  
Mojmír Adamec
Keyword(s):  
Solid State ◽  
Uv Light ◽  
Order Reaction ◽  
Boat Conformation ◽  
X Rays ◽  
Exponential Time ◽  
First Order ◽  
Double Plane ◽  
The Mean ◽  
Independent Molecules

2,4,4,6-Tetraphenyl-4H-thiopyran, C29H22S, orthorhombic, Pna21, a = 17.980(4), b = 6.956(2), c = 34.562(11) Å, V = 4323(2) Å3, Z = 8, Dx = 1.237 g cm-3, F(000) = 1696, λ(CuKα) = 1.54184 A, μ = 1.372 mm-2, T = 294 K. The final R was 0.050 for the unique set of 3103 observed reflections. The central 4H-thiopyran ring forms a boat conformation for both symmetrically independent molecules with average boat angles 4.4(3) and 6.8(3)° at S and C(sp3), respectively. The mean planes of phenyls at the position 2 and 6 are turned from the double plane of 4H-thiopyran by 42.5(5) and 35.8(3)°, respectively. The investigated material undergoes a photochromic change in the solid state after irradiation with UV light or X-rays. The maximum of the new absorption band is situated at 564 nm. The non-exponential time dependence of photochromic bleaching is analysed in terms of a dispersive first-order reaction.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

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