scholarly journals On the spatially inhomogeneous particle coagulation-condensation model with singularity

2021 ◽  
Author(s):  
Debdulal Ghosh ◽  
Jayanta Paul ◽  
Jitendra Kumar
Keyword(s):  
Continuous Solution ◽  
Condensation Process ◽  
Particle Coagulation ◽  
Interesting Topic ◽  
Spatially Inhomogeneous ◽  
Uniqueness Of The Solution ◽  
Condensation Model ◽  
Coagulation Kernel

The spatially inhomogeneous coagulation-condensation process is an interesting topic of study as the phenomenon’s mathematical aspects mostly undiscovered and has multitudinous empirical applications. In this present exposition, we exhibit the existence of a continuous solution for the corresponding model with the following \emph{singular} type coagulation kernel: \[K(x,y)~\le~\frac{\left( x + y\right)^\theta}{\left(xy\right)^\mu}, ~~\text{for} ~x, y \in (0,\infty), \text{where}~ \mu \in \left[0,\tfrac{1}{2}\right] \text{ and } ~\theta \in [0, 1].\] The above-mentioned form of the coagulation kernel includes several practical-oriented kernels. Finally, uniqueness of the solution is also investigated.

A new weighted fraction Monte Carlo method for particle coagulation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Xiao Jiang ◽  
Tat Leung Chan
Keyword(s):  
Monte Carlo ◽  
Design Methodology ◽  
Computational Cost ◽  
Particle Coagulation ◽  
Content Type ◽  
Aerosol Dynamics ◽  
Stochastic Error ◽  
Adjustable Weight ◽  
Initial Distributions ◽  
Coagulation Kernel

Purpose The purpose of this study is to investigate the aerosol dynamics of the particle coagulation process using a newly developed weighted fraction Monte Carlo (WFMC) method. Design/methodology/approach The weighted numerical particles are adopted in a similar manner to the multi-Monte Carlo (MMC) method, with the addition of a new fraction function (α). Probabilistic removal is also introduced to maintain a constant number scheme. Findings Three typical cases with constant kernel, free-molecular coagulation kernel and different initial distributions for particle coagulation are simulated and validated. The results show an excellent agreement between the Monte Carlo (MC) method and the corresponding analytical solutions or sectional method results. Further numerical results show that the critical stochastic error in the newly proposed WFMC method is significantly reduced when compared with the traditional MMC method for higher-order moments with only a slight increase in computational cost. The particle size distribution is also found to extend for the larger size regime with the WFMC method, which is traditionally insufficient in the classical direct simulation MC and MMC methods. The effects of different fraction functions on the weight function are also investigated. Originality Value Stochastic error is inevitable in MC simulations of aerosol dynamics. To minimize this critical stochastic error, many algorithms, such as MMC method, have been proposed. However, the weight of the numerical particles is not adjustable. This newly developed algorithm with an adjustable weight of the numerical particles can provide improved stochastic error reduction.

scholarly journals A non-local macroscopic model for traffic flow

2021 ◽  
Author(s):  
Ioana Ciotir ◽  
Rim Fayad ◽  
Nicolas Forcadel ◽  
Antoine Tonnoir
Keyword(s):  
Traffic Flow ◽  
Existence And Uniqueness ◽  
Numerical Scheme ◽  
Continuous Solution ◽  
Microscopic Model ◽  
Macroscopic Model ◽  
Estimate Error ◽  
Uniqueness Of The Solution ◽  
Non Local

In this work we propose a non-local Hamilton-Jacobi model for traffic flow and we prove the existence and uniqueness of the solution of this model. This model is justified as the limit of a rescaled microscopic model. We also propose a numerical scheme and we prove an estimate error between the continuous solution of this problem and the numerical one. Finally, we provide some numerical illustrations.

Calculation of spatially inhomogeneous electron distribution functions

1998 ◽  
Vol 08 (PR7) ◽  
pp. Pr7-33-Pr7-42
Author(s):  
L. L. Alves ◽  
G. Gousset ◽  
C. M. Ferreira
Keyword(s):  
Electron Distribution ◽  
Distribution Functions ◽  
Spatially Inhomogeneous

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

Gluing Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Stress Tensor ◽  
Compact Support ◽  
Smooth Solution ◽  
Continuous Solution ◽  
Total Momentum ◽  
Frame Of Reference ◽  
Galilean Transformation ◽  
Hölder Continuous ◽  
Pressure Form ◽  
Original Frame

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

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