scholarly journals On the Cauchy problem for a quantum kinetic equation linked to the Compton effect

2006 ◽  
Vol 43 (7-8) ◽  
pp. 838-853 ◽  
Author(s):  
Elisa Ferrari ◽  
Anne Nouri
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Compton Effect ◽  
Quantum Kinetic Equation ◽  
The Cauchy Problem

THE NUMERICAL SOLUTION OF THE BOLTZMANN KINETIC EQUATION FOR GRANULAR GASES

2019 ◽  
pp. 25-30
Author(s):  
I. Gapyak
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Numerical Solution ◽  
Dimensional Space ◽  
Boltzmann Kinetic Equation ◽  
Granular Gases ◽  
Two Dimensional ◽  
System Of Particles ◽  
The Cauchy Problem

For a system of particles with a dissipative interaction we consider the Boltzmann type kinetic equation for granular gases. A numerical solution of the Cauchy problem for the Boltzmann type kinetic equation is constructed in two dimensional space and its stability is investigated.

Transport-collapse scheme for heterogeneous scalar conservation laws

2018 ◽  
Vol 15 (01) ◽  
pp. 119-132
Author(s):  
Darko Mitrović ◽  
Andrej Novak
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Conservation Laws ◽  
Admissible Solution ◽  
Scalar Conservation Laws ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Scalar Conservation

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

scholarly journals The generalized kinetic equation for symmetric particle systems

2012 ◽  
Vol 110 (1) ◽  
pp. 140
Author(s):  
Halyna M. Hubal
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Pair Potential ◽  
Bbgky Hierarchy ◽  
Particle Systems ◽  
Symmetric System ◽  
The Cauchy Problem ◽  
Generalized Kinetic Equation ◽  
Many Particles

The generalized kinetic equation is obtained for symmetric system of many particles interacting via a pair potential. A representation of a solution of the Cauchy problem for the BBGKY hierarchy is used in the form of an expansion over particle groups whose evolution is governed by the cumulants (semi-invariants).

scholarly journals Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2995
Author(s):  
Alexander V. Shapovalov ◽  
Anton E. Kulagin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Kinetic Equation ◽  
Linear Partial Differential Equation ◽  
Metal Vapor ◽  
Problem Solution ◽  
Semiclassical Approach ◽  
Active Media ◽  
The Cauchy Problem

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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