scholarly journals Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 101-102
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Variation Of Parameters ◽  
First Approximation ◽  
The One ◽  
Equivalent Method

For the solution of mechanical problems, two methods in general present themselves: the one furnished by the variation of parameters, or constants, which complete the integral obtained by the first approximation,—the other furnished by the integration of the differential equations by means of indeterminate coefficients, or some equivalent method. Each of these methods is applicable to the theory of the perturbations of the heavenly bodies, and they lead to expressions which are of course substantially identical, but which do not appear in the same shape except after certain transformations. The object of the author in the present paper is to effect transformations, by which their identity is established, making use of the developments given in his former papers, published in the Philosophical Transactions. The identity of the results obtained by either methods affords a confirmation of the exactness of those expressions.

scholarly journals VIII.— Researches in physical astronomy

1832 ◽  
Vol 122 ◽  
pp. 229-236
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Mechanical Problem ◽  
Variation Of Parameters ◽  
First Approximation ◽  
The One ◽  
Equivalent Method

In general, two methods present themselves of solving any mechanical problem: the one furnished by the variation of parameters or constants, which complete the integral obtained by the first approximation; the other furnished by the integration of the differential equations by means of indeterminate coefficients, or some equivalent method. Each of these methods may be applied to the theory of the perturbations of the heavenly bodies; and they lead to expressions which are, of course, substantially identical, but which do not appear in the same shape except after certain transformations. My object in the following pages is to effect these transformations, by which their identity is established, making use of the developments of R and r d R /d r given in the Philosophical Transactions for 1831, p. 295. The identity of the results obtained by either method serves to confirm the exactness of those expressions.

scholarly journals Algebraic approach for the one-dimensional Dirac–Dunkl oscillator

2020 ◽  
Vol 35 (31) ◽  
pp. 2050255
Author(s):  
D. Ojeda-Guillén ◽  
R. D. Mota ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Energy Spectrum ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Representation Theory ◽  
Algebraic Approach ◽  
The Other ◽  
One Dimensional ◽  
The One

We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.

A Novel Method for Solving Consistency-Order Initial Value of Ordinary Differential Equations

2014 ◽  
Vol 543-547 ◽  
pp. 1844-1847
Author(s):  
Si Min Zhu ◽  
Hai Yun Deng ◽  
Kai Zheng ◽  
Hua Mei Li ◽  
Xiao Zhou Chen
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Integral Formula ◽  
Absolute Error ◽  
The Other ◽  
It Use ◽  
Initial Value ◽  
Novel Method ◽  
The One ◽  
Legendre Quadrature

It is known that the level of the consistency-order of initial value problem is an important standard to determine whether the constructed methods for solving initial value problem of ODEs is suitable or not. There are two methods to solve the consistency-order of initial value problem in general. The one is using the remainder of integral formula as local truncated error, and the other one is using absolute error as local truncated error. In the paper, we propose a novel method based on Gauss-Legendre quadrature formula. It use the method of the remainder of integral formula as local truncated error exists in most of the literatures, and it will be solved once again for the consistency-order of the constructed methods that exist in currently literatures by using absolute error as local truncated error, and then draw a conclusion that is differ from what has been proved correspondingly.

scholarly journals Differential Equations: Between the Theoretical Sublimation and the Practical Universalization

2019 ◽  
Vol 21 (1) ◽  
Author(s):  
Juan Eduardo Nápoles Valdez
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
The Other ◽  
The One

In this paper, we present, briefly, the bifront character of the ordinary differential equations (ODE): on the one hand the theoretical specialization in different areas and on the other, the multiplicity of applications of the same, as well as some reflections on the development of a course of ode in this context.

An introduction to numerical methods for stochastic differential equations

Acta Numerica ◽  
1999 ◽  
Vol 8 ◽  
pp. 197-246 ◽  
Author(s):  
Eckhard Platen
Keyword(s):  
Numerical Analysis ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
The Other ◽  
Weak Approximation ◽  
Approximation Methods ◽  
Unified Framework ◽  
Stochastic Nature ◽  
The One

This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

scholarly journals The vapour-pressure curve

1941 ◽  
Vol 179 (977) ◽  
pp. 194-201 ◽  
Keyword(s):  
Free Energy ◽  
Condensed Phase ◽  
Vapour Pressure ◽  
Virial Coefficients ◽  
The Other ◽  
Pressure Curve ◽  
Pressure Equation ◽  
First Approximation ◽  
Vapour Pressure Curve ◽  
The One

By an extension of Mayer’s theory of condensation, exact equations are derived for the vapour pressure and the condensation volume of the gas in terms of the virial coefficients on the one hand and the free energy of the condensed phase on the other. The formula for the vapour pressure reduces in first approximation to Stem’s vapour-pressure equation.

scholarly journals Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 128-129
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Disturbing Function ◽  
Direct Integration ◽  
Planetary Theory ◽  
The Moon ◽  
Disturbing Force ◽  
Other Hand ◽  
The Stability ◽  
The One

The present paper contains some further developments of the theory of the moon, which are given at length, in order to save the trouble of the calculator, and to avoid the danger of mistake. The author remarks, that while it seems desirable, on the one hand, to introduce into the science of physical astronomy a greater degree of uniformity, by bringing to perfection a theory of the moon founded on the integration of the equations employed in the planetary theory, it is also no less important, on the other hand, to complete, in the latter, the method hitherto applied solely to the periodic inequalities. Hi­therto those terms in the disturbing function which give rise to the secular inequalities, have been detached, and the stability of the system has been inferred by means of the integration of certain equations, which are linear when the higher powers of the eccentri­cities are neglected and from considerations founded on the varia­tion of the elliptic constants. But the author thinks that the stability of the system may be inferred also from the expressions which result at once from the direct integration of the differential equations. The theory, he states, may be extended, without any analytical difficulty, to any power of the disturbing force, or of the eccentricities, ad­mitting the convergence of the series; nor does it seem to be limited by the circumstance of the planet’s moving in the same direction.

scholarly journals Symmetries and the inverse problem of Lagrangian dynamics for linear systems

1983 ◽  
Vol 25 (2) ◽  
pp. 232-243 ◽  
Author(s):  
W. Sarlet
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
The Other ◽  
Time Dependent ◽  
Lagrangian Mechanics ◽  
Lagrangian Dynamics ◽  
The One ◽  
General Time

AbstractWe discuss general, time-dependent, linear systems of second-order ordinary differential equations. A study is made of the similarities and discrepancies between the inverse problem of Lagrangian mechanics on the one hand, and the search for linear dynamical symmetries on the other hand.

scholarly journals Reconstruction of isotropic conductivities from non smooth electric fields

2018 ◽  
Vol 52 (3) ◽  
pp. 1173-1193 ◽  
Author(s):  
Marc Briane
Keyword(s):  
Differential Equations ◽  
Electric Fields ◽  
Sobolev Spaces ◽  
Bounded Function ◽  
Gradient Flow ◽  
The Other ◽  
Gradient Field ◽  
The One ◽  
Derivatives Of ◽  
Uniformly Continuous

In this paper we study the isotropic realizability of a given non smooth gradient field ∇u defined in ℝd, namely when one can reconstruct an isotropic conductivity σ > 0 such that σ∇u is divergence free in ℝd. On the one hand, in the case where ∇u is non-vanishing, uniformly continuous in ℝd and Δu is a bounded function in ℝd, we prove the isotropic realizability of ∇u using the associated gradient flow combined with the DiPerna, Lions approach for solving ordinary differential equations in suitable Sobolev spaces. On the other hand, in the case where ∇u is piecewise regular, we prove roughly speaking that the isotropic realizability holds if and only if the normal derivatives of u on each side of the gradient discontinuity interfaces have the same sign. Some examples of conductivity reconstruction are given.

XV. — Researches in physical astronomy

1831 ◽  
Vol 121 ◽  
pp. 231-282
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Lunar Theory ◽  
System Of Equations ◽  
Planetary Theory ◽  
The Moon ◽  
Celestial Bodies ◽  
Independent Variable ◽  
The Mean ◽  
The One

The method pursued by Clairaut in the solution of this important problem of Physical Astronomy, consists in the integration of the differential equations furnished by the principles of dynamics, upon the hypothesis that in the gravitation of the celestial bodies the force varies inversely as the square of the distance, and in which the true longitude of the moon is the independent variable ; the time is thus obtained in terms of the true longitude, and by the reversion of series the longitude is afterwards obtained in terms of the time, which is necessary for the purpose of forming astronomical tables. But while on the one hand this method possesses the advantage, that the disturbing func­tion can be developed with somewhat greater facility in terms of the true lon­gitude of the moon than in terms of the mean longitude, yet on the other hand, the differential equations in which the true longitude is the independent variable are far more complicated than those in which the time is the inde­pendent variable. The latter equations are used in the planetary theory ; so that the method of Clairaut has the additional inconvenience, that while the lunar theory is a particular case of the problem of the three bodies, one system of equations is used in this case, and another in the case of the planets. The method of Clairaut has been adopted, however, by Mayer, by Laplace, and by M. Damoiseau. The last-mentioned author has arranged his results with remarkable clearness, so that any part of his processes may be easily verified by any one who does not shrink from this gigantic undertaking; and the immense labour which this method requires, when all sensible quantities are retained, may be seen in his invaluable memoir. Mr. Brice Bronwin has recently communicated to the Society a lunar theory, in which the same method is adopted.

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