scholarly journals MULTIVARIATE MASTER EQUATION AND IT'S WAVE SOLUTIONS

1983 ◽  
Vol 32 (3) ◽  
pp. 285
Author(s):  
CHEN ZHENG-XIONG ◽  
HUO YU-PING
Keyword(s):  
Master Equation ◽  
Wave Solutions ◽  
Multivariate Master Equation

On the diffusion operator of the multivariate master equation

1977 ◽  
Vol 89 (2) ◽  
pp. 339-352 ◽  
Author(s):  
C. Van Den Broeck ◽  
W. Horsthemke ◽  
M. Malek-Mansour
Keyword(s):  
Master Equation ◽  
Diffusion Operator ◽  
Multivariate Master Equation

scholarly journals An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence

2015 ◽  
Vol 15 (8) ◽  
pp. 12213-12240 ◽  
Author(s):  
L. Alfonso
Keyword(s):  
Master Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Correlation Coefficients ◽  
Cloud Droplet ◽  
Droplet Growth ◽  
Multivariate Master Equation ◽  
Smoluchowski Coagulation Equation ◽  
Cloud Modeling ◽  
Stochastic Coalescence

Abstract. In cloud modeling studies, the time evolution of droplet size distributions due to collision-coalescence events is usually modeled with the kinetic collection equation (KCE) or Smoluchowski coagulation equation. However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitude of correlations are significant, and must be taken into account when modeling the evolution of a finite volume coalescing system.

scholarly journals An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence

2015 ◽  
Vol 15 (21) ◽  
pp. 12315-12326 ◽  
Author(s):  
L. Alfonso
Keyword(s):  
Master Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Correlation Coefficients ◽  
Cloud Droplet ◽  
Droplet Growth ◽  
Multivariate Master Equation ◽  
Smoluchowski Coagulation Equation ◽  
Cloud Modeling ◽  
Stochastic Coalescence

Abstract. In cloud modeling studies, the time evolution of droplet size distributions due to collision–coalescence events is usually modeled with the Smoluchowski coagulation equation, also known as the kinetic collection equation (KCE). However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain types of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was seen by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitudes of correlations are significant and must be taken into account when modeling the evolution of a finite volume coalescing system.

Chapman-Enskog development of the multivariate master equation

1980 ◽  
Vol 101 (1) ◽  
pp. 167-184 ◽  
Author(s):  
C. Van den Broeck ◽  
J. Houard ◽  
M. Malek Mansour
Keyword(s):  
Master Equation ◽  
Multivariate Master Equation

Asymptotic Wave Solutions for the Model of a Medium with Van Der Pol Oscillators

2014 ◽  
Vol 59 (9) ◽  
pp. 932-938
Author(s):  
V.A. Danylenko ◽  
◽  
S.I. Skurativskyi ◽  
I.A. Skurativska ◽  
◽  
...  
Keyword(s):  
Van Der Pol ◽  
Wave Solutions ◽  
Van Der Pol Oscillators

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

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