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A regularized asymptotics of the solution to the time-dependent Schrödinger equation in which the spatial derivative is multiplied by a small Planck constant is constructed. It is shown that the asymptotics of the solution contains a rapidly oscillating boundary layer function. -
Keywords: singularly perturbed time-dependent Schrödinger equation regularized asymptotics solutions
Cтроится регуляризованная асимптотика решения временнoго уравнения Шрёдингера, когда малая константа Планка стоит перед пространственной производной. Показано, что асимптотика решения содержит погранслойную функцию, имеющую быстро осциллирующий характер изменения
A regularized asymptotic expansion of the solution to a singularly perturbed two-dimensional parabolic problem in domains with boundaries containing corner points is constructed. The asymptotics of solutions to such problems contain ordinary boundary-layer functions, parabolic boundary-layer functions, and their products, which describe a corner boundary layer. -
Keywords: singularly perturbed parabolic problems parabolic boundary layer regularized asymptotics
Строится регуляризованная асимптотика решения сингулярно возмущенной двумерной параболической задачи в областях с угловыми точками границы. Асимптотика решения таких задач содержит как обыкновенные погранслойные функции, так и параболические погранслойные функции и их произведения, которые описывают угловой пограничный слой.
The work is devoted to the construction of the asymptotic behavior of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The asymptotics of the solution of such problems contains boundary layer functions.
The regularized asymptotics of the solution of the first boundary-value problem for a two-dimensional differential equation of parabolic type is constructed in the case where the phase derivative vanishes at a single point. It is shown that angular and multidimensional boundary-layer functions appear in problems of this kind parallel with other types of boundary layers.
The aim of this paper is to construct regularized asymptotics of the solution of a two-dimensional partial differential equation of parabolic type with a small parameter for all spatial derivatives and a rapidly oscillating free term.
The case when the first derivative of the phase of the free term at the initial point vanishes is considered. The two-dimensionality of the equation leads to the emergence of a two-dimensional boundary layer. The presence in the free term of a rapidly oscillating factor leads to the inclusion in the asymptotic of the boundary layer with a rapidly oscillating na ...Более
The first boundary value problem is studied for an n-dimensional parabolic linear system of differential equations with a small parameter multiplying the spatial derivative. A complete regularized asymptotics of the solution is constructed for the case in which the system is uniformly Petrovskii parabolic. The asymptotics contains 2n parabolic boundary layer functions described by the complementary error function.
Изучается первая краевая задача для -мерной линейной системы дифференциальных уравнений параболического типа с малым параметром при пространственной производной. Построена полная регуляризованная асимптотика решения в случае, когда система является равномерно параболической в смысле Петровского. Построенная асимптотика содержит параболических погранслойных функций, описываемых “дополнительным интегралом вероятности”.
The Cauchy problem with a rapidly oscillating initial condition for the homogeneous Schrodinger equation was studied in [5]. Continuing the research ideas of this work and [3], in this paper we construct the asymptotic solution to the following mixed problem for the nonstationary Schrodinger equation:
L(h)u ih partial derivative(t)u + h(2)partial derivative(2)(x)u - b(x,t)u = f(x,t) (x,t) is an element of Omega = (0,1) x (0,t],
u vertical bar(t=0) = g(x), u vertical bar(x=0) = u vertical bar(x=1) = 0
where h > 0 is a Planck constant, u = u(x,t,h). b(x,t), f (x,t) is an element of C-infinity ...Более
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 regularize ...Более
We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudar ...Более