The angular boundary layer in mixed singularly perturbed problems for hyperbolic equations

V. F. Butuzov

Sbornik. Mathematics, Tome 33 (1977) no. 3, pp. 403-425

Résumé

We obtain an asymptotic expansion in the small parameter $\varepsilon$ of the solution of a mixed boundary value problem for the equation $$ \varepsilon^2\biggl(\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}\biggr)+\varepsilon^ka(x,t)\frac{\partial u}{\partial t}+b(x,t)u=f(x,t)\qquad(0,\quad0\leqslant T) $$ in the two cases $k=1$ and $k=1/2$. The asymptotics of the solution contains a regular part, consisting of ordinary boundary functions, which play a role in a neighborhood of the sides $t=0$, $x=0$, and $x=l$, and the so-called angular boundary functions, which come into play in a neighborhood of the angular points $(0,0)$ and $(l,0)$. When $k=1$, these angular boundary functions are determined from hyperbolic equations with constant coefficients; when $k=1/2$, they are determined from parabolic equations with constant coefficients. Bibliography: 7 titles.