Solution of generalized density evolution equation via a family of δ sequences

2008 ◽  
Vol 43 (6) ◽  
pp. 781-796 ◽  
Author(s):  
Wenliang Fan ◽  
Jianbing Chen ◽  
Jie Li
Keyword(s):  
Evolution Equation ◽  
Density Evolution

Improvement to the discretized initial condition of the generalized density evolution equation

2021 ◽  
pp. 107999
Author(s):  
Gang Liu ◽  
Kai Gao ◽  
Qingshan Yang ◽  
Wei Tang ◽  
S.S. Law
Keyword(s):  
Evolution Equation ◽  
Initial Condition ◽  
Density Evolution

PROBABILITY DENSITY EVOLUTION EQUATIONS — A HISTORICAL INVESTIGATION

2009 ◽  
Vol 03 (03) ◽  
pp. 209-226 ◽  
Author(s):  
LI JIE ◽  
CHEN JIANBING
Keyword(s):  
Evolution Equation ◽  
Probability Density ◽  
Evolution Equations ◽  
Planck Equation ◽  
Point Of View ◽  
Lagrangian Description ◽  
Density Evolution ◽  
Physical Point ◽  
Probability Density Evolution ◽  
Generalized Probability

The paper aims at clarifying the essential relationship between traditional probability density evolution equations and the generalized probability density evolution equation which is developed by the authors in recent years. Using the principle of preservation of probability as a uniform fundamental, the probability density evolution equations, including the Liouville equation, Fokker–Planck equation and the Dostupov–Pugachev equation, are derived from the physical point of view. It is pointed out that combining with Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle and dynamical systems are viewed from Lagrangian description, we are led to the generalized probability density evolution equation.

scholarly journals Age–depth correlation, grain growth and dislocation-density evolution, for three ice cores

2009 ◽  
Vol 55 (190) ◽  
pp. 345-352 ◽  
Author(s):  
L.W. Morland
Keyword(s):  
Dislocation Density ◽  
Evolution Equation ◽  
Grain Growth ◽  
Ice Cores ◽  
Density Evolution ◽  
Depth Profiles ◽  
Growth Profiles ◽  
The Given ◽  
Theoretical Analyses

AbstractTwo previous theoretical analyses of data from the GRIP, Vostok and Byrd ice cores, presenting age–depth correlations, grain growth and dislocation-density evolution, are re-examined. It is found that the age–depth correlations are inconsistent with the idealized flow with unchanging history adopted, but that good correlations can be obtained by relaxing those restrictions. A modified grain-growth relation is proposed, consistent with the distinct growth profiles of the Vostok and other two cores, which can be solved simultaneously with the given dislocation-density evolution equation. These are solved for all three cores with the given parameters, and the depth profiles of grain diameter and dislocation density at the present time are determined with the new age–depth correlation and with that shown empirically in the papers. The varying flow history influences the age–depth correlation, and hence the depth profiles, which is important both for the interpretation of core data, and for the determination of constitutive variables at each depth at the present time.

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