scholarly journals Gluon dynamics from an ordinary differential equation

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
A. C. Aguilar ◽  
M. N. Ferreira ◽  
J. Papavassiliou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Gluon Propagator ◽  
Dyson Equation ◽  
Kinetic Term ◽  
Practical Implementation ◽  
Gluon Vertex ◽  
Central Hypothesis ◽  
Approximate Form ◽  
Novel Method

AbstractWe present a novel method for computing the nonperturbative kinetic term of the gluon propagator from an ordinary differential equation, whose derivation hinges on the central hypothesis that the regular part of the three-gluon vertex and the aforementioned kinetic term are related by a partial Slavnov–Taylor identity. The main ingredients entering in the solution are projection of the three-gluon vertex and a particular derivative of the ghost-gluon kernel, whose approximate form is derived from a Schwinger–Dyson equation. Crucially, the requirement of a pole-free answer determines the initial condition, whose value is calculated from an integral containing the same ingredients as the solution itself. This feature fixes uniquely, at least in principle, the form of the kinetic term, once the ingredients have been accurately evaluated. In practice, however, due to substantial uncertainties in the computation of the necessary inputs, certain crucial components need be adjusted by hand, in order to obtain self-consistent results. Furthermore, if the gluon propagator has been independently accessed from the lattice, the solution for the kinetic term facilitates the extraction of the momentum-dependent effective gluon mass. The practical implementation of this method is carried out in detail, and the required approximations and theoretical assumptions are duly highlighted.

scholarly journals A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Yong Xu ◽  
Qixian Liu ◽  
Jike Liu ◽  
Yanmao Chen
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Separation Of Variables ◽  
Computational Cost ◽  
Diffusion Equations ◽  
Novel Method ◽  
High Computational Efficiency

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

scholarly journals The Solution ofSO(3) through a Single Parameter ODE

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Chein-Shan Liu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
General Solution ◽  
Governing Equation ◽  
Orthogonal Transformation ◽  
Nonlinear Ordinary Differential Equation ◽  
Single Parameter ◽  
Novel Method

In many applications we need to solve an orthogonal transformation tensorQ∈SO(3) from a tensorial equationQ˙ = WQunder a given spin historyW. In this paper, we address some interesting issues about this equation. A general solution ofQis obtained by transforming the governing equation into a new one in the space ofℝP3. Then, we develop a novel method to solveQin terms of a single parameter, whose governing equation is a single nonlinear ordinary differential equation (ODE).

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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