scholarly journals Classical equations of an electron from the special form of the Dirac equation

2021 ◽  
Vol 10 (4) ◽  
Author(s):  
Miroslav Pardy ◽  
Keyword(s):  
Dirac Equation ◽  
Special Form ◽  
Wkb Approximation ◽  
System Of Equations ◽  
Equivalent System ◽  
Tensor Equation ◽  
Lorentz Equation

The equivalent system of equations corresponding to the Dirac equation is derived and the WKB approximation of this system is found. Similarly, the WKB approximation for the equivalent system of equation corresponding to the squared Dirac equation is found and it is proved that the Lorentz equation and the Bargmann-Michel-Telegdi iquations follow from the new Dirac-Pardy system. The new tensor equation with sigma matrix is derived for the verification by adequate laboratories.

scholarly journals An efficient convergent method for the delay reaction-diffusion equations

2021 ◽  
Author(s):  
Marzieh Heidari ◽  
Mehdi Ghovatmand ◽  
Mohammad Hadi Noori Skandari
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
System Of Equations ◽  
Equivalent System ◽  
Numerical Examples ◽  
Delay Function ◽  
Mild Conditions ◽  
Convergent Method

In this manuscript, we consider the delay reaction-diffusion equation and implement an efficient spectral collocation method to approximate the solution of this equation. We first replace the delay function in the delay reaction-diffusion equation and achieve an equivalent system of equations. We then utilize the Legendre-Gauss-Lobatto and two-dimensional interpolating polynomial to approximate the solution of obtained system. Moreover, we prove the convergent of method under some mild conditions. Finally, the capability and efficiency of the method is illustrated by providing several numerical examples and comparing them with others

THE LORENTZ–DIRAC EQUATION, I

2000 ◽  
Vol 12 (04) ◽  
pp. 657-686 ◽  
Author(s):  
BERNHARD RUF ◽  
P.N. SRIKANTH
Keyword(s):  
Electromagnetic Field ◽  
Dirac Equation ◽  
Potential Barrier ◽  
Global Bifurcation ◽  
Transmission Time ◽  
Maximum Point ◽  
Qualitative Properties ◽  
Lorentz Equation

The Lorentz–Dirac equation (LDE) [Formula: see text] models the point limit of the Maxwell–Lorentz equation describing the interaction of a charged extended particle with the electromagnetic field. Since (LDE) admits solutions which accelerate even if they are outside the zone of interaction, Dirac proposed to study so-called "non runaway" solutions satisfying the condition x″ (t)→0 as t → +∞. We study the scattering of particles for a localized potential barrier V(x). We show, using global bifurcation techniques, that for every T>T0 there exists a reflection solution with "returning time" T, and for every T>0 there exists a transmission solution with "transmission time" T. Furthermore, some qualitative properties of the solutions are proved; in particular, it is shown that for increasing T, these solutions spend more and more time near the maximum point s0 of V.

WKB approximation for the Dirac equation at Z > 137

1978 ◽  
Vol 80 (1-2) ◽  
pp. 68-72 ◽  
Author(s):  
V.S. Popov ◽  
V.L. Eletsky ◽  
V.D. Mur ◽  
D.N. Voskresensky
Keyword(s):  
Dirac Equation ◽  
Wkb Approximation

scholarly journals SELF-ACCELERATED UNIVERSE

2005 ◽  
Vol 20 (11) ◽  
pp. 2459-2464 ◽  
Author(s):  
B. P. KOSYAKOV
Keyword(s):  
Dirac Equation ◽  
Free Electron ◽  
Vacuum Energy ◽  
Alternative Explanation ◽  
Equations Of Motion ◽  
Accelerated Expansion ◽  
System Of Equations ◽  
Entire System ◽  
Expansion Of The Universe ◽  
The Universe

It is widely believed that the large redshifts for distant supernovae are due to the vacuum energy dominance, which is responsible for the anti-gravitation effect. A tacit assumption is that particles move along geodesics for the background metric. This is in the same spirit as the consensus regarding the uniform Galilean motion of a free electron. However, apart from the Galilean solution, there is a self-accelerated solution to the Lorentz–Dirac equation governing the behavior of a radiating electron. Likewise, a runaway solution to the entire system of equations, both gravitation and matter equations of motion including, may exist, which provides an alternative explanation for the accelerated expansion of the Universe.

scholarly journals Invariant submodel of rank 1 and two families of exact solutions of gas dynamics equations with an equation of state of the special form

2021 ◽  
Vol 2099 (1) ◽  
pp. 012017
Author(s):  
D Siraeva
Keyword(s):  
Equation Of State ◽  
Exact Solutions ◽  
Special Form ◽  
Gas Dynamics ◽  
Three Dimensional ◽  
Inhomogeneous Deformation ◽  
System Of Equations ◽  
Gas Dynamics Equations ◽  
Linear Velocity Field ◽  
Invariant Submodel

Abstract In this article, the gas dynamics equations with an equation of state of the special form are considered.The equation of state is the pressure which is equal to the sum of two functions, with one being a function of a density, and the other one being a function of an entropy. The system of equations is invariant under the action of 12-parameter transformations group. For three-dimensional subalgebra 3.32 of the 12-dimensional Lie algebra invariants are calculated, an invariant submodel of rank 1 is constructed, and two families of exact solutions are obtained. The obtained solutions specify the motion of particles in space with a linear velocity field with inhomogeneous deformation. The first family of solutions has two moments of time of particles collapse. The second family of solutions has one moment of time of particles collapse on the plane. In the simplest case of second family of solutions, a surface consisting of particle trajectories is constructed.

scholarly journals A Superlinearly Convergent Method for the Generalized Complementarity Problem over a Polyhedral Cone

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fengming Ma ◽  
Gang Sheng ◽  
Ying Yin
Keyword(s):  
Complementarity Problem ◽  
Polyhedral Cone ◽  
System Of Equations ◽  
Equivalent System ◽  
Type Method ◽  
Linear System Of Equations ◽  
Generalized Complementarity Problem ◽  
Convergence Results ◽  
Convergent Method ◽  
Weaker Conditions

Making use of a smoothing NCP-function, we formulate the generalized complementarity problem (GCP) over a polyhedral cone as an equivalent system of equations. Then we present a Newton-type method for the equivalent system to obtain a solution of the GCP. Our method solves only one linear system of equations and performs only one line search at each iteration. Under mild assumptions, we show that our method is both globally and superlinearly convergent. Compared to the previous literatures, our method has stronger convergence results under weaker conditions.

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