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.

Abstract linear second order differential equations with two small parameters and depending on time operators

In a real Hilbert space H consider the following singularly perturbed Cauchy problem. We study the behavior of solutions uεδ to this problem in two different cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0. We show the boundary layer and boundary layer function in both cases.

Effective Approximation for Elliptic Partial Differential Equation with Periodical Boundary Value Problem

Elliptic partial differential equation with periodical boundary value problem was considered. The equation would degenerate to parabolic partial differential equation when small parameter tends to zero. This is a multi-scale problem. Firstly, the property of boundary layer was discussed. Secondly, the boundary layer function was presented. The smooth component was constructed according to the boundary layer function. Thirdly, finite difference scheme for the smooth component is proposed according to transition point in time direction. Finally, experiment was proposed to illustrate that our presented method is an effective computational method.

Bending Analysis of Thick Laminated Rectangular Plates Using a Boundary Layer Function

In this article, the governing bending equations of thick laminated transversely isotropic rectangular plates are derived based on third-order shear deformation theory (TSDT). Using a new function, called the boundary layer function, the three coupled governing equations are converted to two decoupled equations. These equations are in terms of the deflection of the plate and the mentioned boundary layer function, which are written in invariant form. By solving the decoupled equations, a Levy-type analytical solution is presented for bending of a transversely isotropic plate. Finally, numerical results are presented for boundary layer phenomenon and its effects in TSDT. It is shown that all of the boundary layer effects in Mindlin—Reissner theory appear in this theory. However, it is shown that the intensity of the boundary layer effects in TSDT exceeds that of the Mindlin—Reissner theory.

On the Stress Singularities and Boundary Layer in Moderately Thick Functionally Graded Sectorial Plates

In this paper, the stress analysis of moderately thick functionally graded (FG) sector plate is developed for studying the singularities in vicinity of the vertex. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method and defining some new functions, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations called edge-zone and interior equations. These equations are solved analytically for the sector plate with the simply supported radial edges and arbitrary boundary condition along the circular edge. The singularities of shear force and moment resultants are discussed for both salient and re-entrant sectorial plates. Also, the effects of power of the FGM, thickness to length ratio on the stress singularities of the FG sector plates are investigated.