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.

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

The effect of weak shear on finite-amplitude internal solitary waves

1999 ◽  
Vol 395 ◽  
pp. 125-159 ◽  
Author(s):  
S. R. CLARKE ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Evolution Equation ◽  
Solitary Waves ◽  
Vries Equation ◽  
Physical Space ◽  
Finite Amplitude ◽  
Initial Condition ◽  
Weak Current ◽  
Inflection Points ◽  
Wide Range ◽  
De Vries

A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg–de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of the Korteweg–de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Korteweg–de Vries equation. The two types of solution are then used to qualititatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.

The Homogeneous Flow

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Second Order ◽  
Initial Condition ◽  
Homogeneous Flow ◽  
Short Time

In this chapter, a second-order nonlinear evolution equation is constructed that starts with any parallelism as initial condition and flows in the direction of a parallelism with vanishing curvature. The existence of unique short-time solutions is proved.

Solutions of evolution equations associated to infinite-dimensional Laplacian

2016 ◽  
Vol 14 (04) ◽  
pp. 1640018 ◽  
Author(s):  
Habib Ouerdiane
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Dimensional Space ◽  
Generalized Function ◽  
Initial Condition ◽  
Distribution Space ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Main Technique ◽  
The One

We study an evolution equation associated with the integer power of the Gross Laplacian [Formula: see text] and a potential function V on an infinite-dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg [K. J. Hochberg, Ann. Probab. 6 (1978) 433.] in the one-dimensional case with [Formula: see text], as well as by Barhoumi–Kuo–Ouerdiane for the case [Formula: see text] (See Ref. [A. Barhoumi, H. H. Kuo and H. Ouerdiane, Soochow J. Math. 32 (2006) 113.]).

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