The method of boundary integral equations for determining boundary layer asymptotics of the solution to the nonlinear singularly perturbed heat conduction boundary value problem (Q1594246)

Consider the following singularly perturbed heat conduction equation \[ \frac {\partial T} {\partial t}= \varepsilon \frac{\partial^2 T} {\partial x^2}, \quad (x,t)\in M_t \equiv \{(x,t)\mid kt< x< N_2(t), 0<t\leq 1\}\tag{*} \] with nonlinear Stefan-Boltzmann boundary conditions: \(T(x,t)=T_0\) in \(t=+0\), \(0\leq x\leq N_2(0)\); \(\partial T/\partial x=\gamma T^4 (x,t)\) in \(x=kt\), \(0\leq t\leq 1\); \(\partial T/\partial x=0\) in \(x=N_2(t)\), \(0\leq t\leq 1\). Such a problem arises, for example, in plasma-coating processes in a vacuum. An approach based on the method of boundary integral equations and a modification of the Laplace method, obtaining a boundary-layer asymptotic expansion in Poincaré sense for the solution of the problem (*) is described. It is established that \[ T(x,t){\underset{\varepsilon\to 0}\sim}T_0-\frac\gamma{k}T_0^4 \exp\left\{-\frac{(x-kt)k}\varepsilon\right\}\sum_{i=0}^\infty a_i \varepsilon^{i+1}, \] where \(a_0=1\), \(a_1=4\gamma T_0^3/k\), \(a_2=22\gamma^2 T_0^6/k^2\).