Singularly perturbed parabolic problem with oscillating initial condition

The aim of this paper is to construct regularized asymptotics of the solution of a singularly perturbed parabolic problem with an oscillating initial condition. The presence of a rapidly oscillating function in the initial condition has led to the appearance of a boundary layer function in the solution, which has the rapidly oscillating character of the change. In addition, it is shown that the asymptotics of the solution contains exponential, parabolic boundary layer functions and their products describing the angular boundary layers. Continuing the ideas of works [1, 3] a complete regularized asymptotics of the solution of the problem is constructed.

On the Riemann–Hilbert Problem in the Domain with a Nonsmooth Boundary

The following Riemann–Hilbert problem is solved: find an analytical function Φ from the Smirnov class Ep (D), whose angular boundary values satisfy the condition Re[(a(t) + ib(t))Φ+ (t)] = ƒ(t). The boundary Γ of the domain D is assumed to be a piecewise smooth curve whose nonintersecting Lyapunov arcs form, with respect to D, the inner angles with values νkπ, 0 < νk ≤ 2.