scholarly journals Boltzmann’s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell-Auzhan Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
A. Sakabekov ◽  
Y. Auzhani
Keyword(s):  
Boundary Conditions ◽  
A Priori ◽  
Initial Value ◽  
One Dimensional ◽  
Moment System ◽  
Uniqueness Of The Solution ◽  
System Equations ◽  
Particular Solution ◽  
Integral Equality ◽  
A Priori Estimation

We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We have managed to obtain a particular solution of this equation in an explicit form.

scholarly journals The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

2022 ◽  
Author(s):  
Auzhan Sakabekov ◽  
Yerkanat Auzhani
Keyword(s):  
Distribution Function ◽  
Nonlinear System ◽  
Boundary Conditions ◽  
Moving Boundary ◽  
Collision Operator ◽  
Maxwell Distribution ◽  
Moment Equations ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
The Moment

AbstractThe paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation $${f}_{k}(t,x,c)$$ f k ( t , x , c ) , where $${f}_{k}(t,x,c)$$ f k ( t , x , c ) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions $$C\left(\left[0,T\right];{L}^{2}\left[-a,a\right]\right)$$ C 0 , T ; L 2 - a , a are proved.

scholarly journals Analysis of Different Boundary Conditions on Homogeneous One-Dimensional Heat Equation

2021 ◽  
Vol 7 (1) ◽  
pp. 15-21
Author(s):  
Norazlina Subani ◽  
Muhammad Aniq Qayyum Mohamad Sukry ◽  
Muhammad Arif Hannan ◽  
Faizzuddin Jamaluddin ◽  
Ahmad Danial Hidayatullah Badrolhisam
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Initial Value ◽  
Partial Differential ◽  
Different Boundary Conditions ◽  
One Dimensional ◽  
Profile Changes ◽  
The Right

Partial differential equations involve results of unknown functions when there are multiple independent variables. There is a need for analytical solutions to ensure partial differential equations could be solved accurately. Thus, these partial differential equations could be solved using the right initial and boundaries conditions. In this light, boundary conditions depend on the general solution; the partial differential equations should present particular solutions when paired with varied boundary conditions. This study analysed the use of variable separation to provide an analytical solution of the homogeneous, one-dimensional heat equation. This study is applied to varied boundary conditions to examine the flow attributes of the heat equation. The solution is verified through different boundary conditions: Dirichlet, Neumann, and mixed-insulated boundary conditions. the initial value was kept constant despite the varied boundary conditions. There are two significant findings in this study. First, the temperature profile changes are influenced by the boundary conditions, and that the boundary conditions are dependent on the heat equation’s flow attributes.

LINEAR TRANSPORT OF PARTICLES ON NETWORKS

1996 ◽  
Vol 06 (02) ◽  
pp. 279-294 ◽  
Author(s):  
LUIGI BARLETTI
Keyword(s):  
Boundary Conditions ◽  
Initial Value Problem ◽  
Connected Graph ◽  
Well Posedness ◽  
Linear Boundary ◽  
Initial Value ◽  
One Dimensional ◽  
Linear Transport ◽  
Streaming Transport

We introduce a model of transport of particles in a network, which is represented by a connected graph with m vertices and n edges. Each edge represents a one-dimensional conductor of particles, whose behavior is described by means of a linear Boltzmann-like equation. In graph vertices, a system of linear boundary conditions is given which takes into account the exchanges of particles between the edges. The well-posedness of the initial value problem is studied into an abstract L1-like setting and the structure of the solution is given for simplest case of pure streaming transport.

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