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TMF, 2016, Volume 189, Number 2, Pages 239–255 (Mi tmf9108)  

Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave

A. Kh. Khachatryana, Kh. A. Khachatryanb

a Armenian State Agrarian University, Yerevan, Armenia
b Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, Armenia

Abstract: We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space $L_1[-r,r]$ for all finite $r<+\infty$. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.

Keywords: nonlinearity, shock wave, integral equation, bounded solution, iteration, pointwise convergence

Funding Agency Grant Number
State Committee on Science of the Ministry of Education and Science of the Republic of Armenia SCS 15T-1A033
This research is supported by the State Science Committee, Ministry of Education and Science, Republic of Armenia (Project No. SCS 15T-1A033).


DOI: https://doi.org/10.4213/tmf9108

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English version:
Theoretical and Mathematical Physics, 2016, 189:2, 1609–1623

Bibliographic databases:

Received: 14.12.2015
Revised: 29.01.2016

Citation: A. Kh. Khachatryan, Kh. A. Khachatryan, “Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave”, TMF, 189:2 (2016), 239–255; Theoret. and Math. Phys., 189:2 (2016), 1609–1623

Citation in format AMSBIB
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