Asymptotics of the solution of the main problem of the integral fire theory
The authors consider the main problem of the integral theory of fire in a room with a single opening. Precisely speaking the aim of the paper is to construct and analyze the solution \(\hat r(\tau)\) of the system of equation \[ {dr\over d\tau}=-\tau^2r+Q(1-r)^{3/2}\xi^{3/2}, \tag{1} \] \[ \tau^2+Q(1-r)^{1/2}(\xi^{3/2}-r^{-1/2}(1-\xi)^{3/2})=0, \tag{2} \] with the initial condition \[ \hat r(0)=1,\tag{3} \] where: \(r=r(\tau)\) -- is the relative density of the gas medium and is related to the volume-average temperature \(T_m\) in the room by the formula \(T_m=T_0/r(\tau)\) where \(T_0\) is the absolute emperature of the external air, \(\tau\) is the time since the beginning of the fire in the conventional units, \(Q\) is a positive parameter specified by the size of the opening and other conditions of the problem, \(\xi=\xi(\tau,r)\) has the meaning of the relative height of the equal pressure plane which separates the opposite flows of gases in the opening. The quantity \(\xi^0:=\xi(0,1)\) is the initial height of the equal pressure plane. The problem of an appropriate choice of the value \(\xi^0\) is an importance problem of the mathematical theory of fire. The authors obtain the asymptotic representations for the solution \(\hat r(\tau)\) both as \(\tau\to0\) and as \(\tau\to\infty\), computed the quantity \(\xi^0\) and derived conditions under which the one-side operation mode of the opening is possible. In the proof, they use the strong calculation, Picard approximations, Taylor formula and some standard theorems of mathematical analysis.
Cauchy problem for ordinary differential equations
scientific article; zbMATH DE number 5729803