Asymptotical Solutions for a Vibrationally Relaxing Gas

Mathematical Modelling and Analysis

Kriauzienė

2011

We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein -Gordon, Hirota -Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro -differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro -differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

Section: Resultsmentioning

confidence: 99%

“…The periodical problems with quadratic non-linearity are reduced to analogical averaged integro -differential systems [2,20,21,24]. A general form of non-linearity requires special analysis.…”

Section: Introductionmentioning

confidence: 99%

Asymptotical Analysis of Some Coupled Nonlinear Wave Equations

Mathematical Modelling and Analysis

Kriauzienė

2011

“…Let us note that systems like (23), (24) appear while applying the method of averaging in models of the resonance interaction of nonlinear waves. In many cases such systems are left unsolved as a separate problem for finding asymptotics [2,[8][9][10][11]. In [5][6][7], problems similar to (23), (24) were solved by numerical methods.…”

Section: Methods Of Averagingmentioning

confidence: 99%

“…The nonperturbated system (11), i.e., when ε = 0, describes two independent waves r − 0 (x + t) and r + 0 (x − t) moving in two different directions. Here, r ± (x) are smoothly differentiable functions describing the initial conditions of the problem (11). While trying to construct the direct asymptotic approximation…”

Section: State Of Problemmentioning

confidence: 99%

Asymptotic solution of the mathematical model of nonlinear oscillations of absolutely elastic inextensible weightless string

Lavcel-Budko

Miškinis

2010

NAMC

“…Sometimes such averaging is called internal [15], and this is its essential difference as compared with various partial derivatives averaging schemes [23,24] which may be called the external averaging. The systems obtained by internal averaging often remain without a subsequent study and left in the literature as a certain theoretical result of the asymptotic analysis [3,28]. Let us note that system (3.4) does not directly depend on the small parameter ε and thus has no problems of asymptotic integration.…”

Section: When All Numbersmentioning

confidence: 99%

Uniformly Valid Asymptotics for Carrier’s Mathematical Model of String Oscillations

Miškinis

Mathematical Modelling and Analysis

Lavcel-Budko

2017

In the paper, an asymptotic analysis of G.F. Carrier’s mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the system’s solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.