First Integrals of a Cubic System of Differential Equations

We have considered a stationary outflowing envelope accelerated by the radiative force in arbitrary optical depth case. Introduced approximations provide satisfactory description of the behavior of the matter flux with partially separated radiation at arbitrary optical depths. The obtained systemof differential equations provides a continuous transition of the solution between optically thin and optically thick regions. We analytically derivedapproximate representation of the solution at the vicinity of the sonic point. Using this representation we numerically integrate the system of equations from the critical point to the infinity. Matching the boundary conditions we obtain solutions describing the problem system of differential equations. The theoretical approach advanced in this work could be useful for self-consistent simulations of massive star evolution with mass loss.

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Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽

10.3390/sym13030501 ◽

2021 ◽

Vol 13 (3) ◽

pp. 501

Author(s):

Ahmed Boudaoui ◽

Khadidja Mebarki ◽

Wasfi Shatanawi ◽

Kamaleldin Abodayeh

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Metric Space ◽

Fixed Point Theorems ◽

Coupled Fixed Point ◽

Sufficient Conditions ◽

Impulsive Differential Equations ◽

System Of Differential Equations ◽

Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

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A New Model for the Stochastic Point Reactor: Development and Comparison with Available Models

Energies ◽

10.3390/en14040955 ◽

2021 ◽

Vol 14 (4) ◽

pp. 955

Author(s):

Alamir Elsayed ◽

Mohamed El-Beltagy ◽

Amnah Al-Juhani ◽

Shorooq Al-Qahtani

Keyword(s):

Kinetic Model ◽

Differential Equations ◽

Test Cases ◽

System Of Differential Equations ◽

Reactor Model ◽

Time System ◽

New Model ◽

Reactor Dynamics ◽

Mean And Variance ◽

The Mean

The point kinetic model is a system of differential equations that enables analysis of reactor dynamics without the need to solve coupled space-time system of partial differential equations (PDEs). The random variations, especially during the startup and shutdown, may become severe and hence should be accounted for in the reactor model. There are two well-known stochastic models for the point reactor that can be used to estimate the mean and variance of the neutron and precursor populations. In this paper, we reintroduce a new stochastic model for the point reactor, which we named the Langevin point kinetic model (LPK). The new LPK model combines the advantages, accuracy, and efficiency of the available models. The derivation of the LPK model is outlined in detail, and many test cases are analyzed to investigate the new model compared with the results in the literature.

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The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-2006-018-7 ◽

2006 ◽

Vol 49 (2) ◽

pp. 170-184

Author(s):

Richard Atkins

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Shortest Paths ◽

Equivalence Problem ◽

Second Order ◽

System Of Differential Equations ◽

Lagrange Equations ◽

Invariant Functions ◽

Underlying Geometry ◽

The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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