scholarly journals Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis

Nonlinearity ◽  
2020 ◽  
Vol 33 (10) ◽  
pp. 5111-5141
Author(s):  
Chiun-Chang Lee ◽  
Zhi-An Wang ◽  
Wen Yang
2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.


1957 ◽  
Vol 61 (557) ◽  
pp. 361-361
Author(s):  
G. V. Lachmann

The method referred to in Dr. Coleman's notes was developed with the collaboration of my colleague Mr. J. B. Edwards of Handley Page Research Department. The purpose was to obtain a rational estimate of suction quantities and suction distribution, linked up with measurements of boundary layer profiles and suction quantities on wind tunnel models, and also to assess the effect of a certain degree of roughness of the order to be expected on actual wings. Existing theoretical methods ignore roughness which, however, is a most important parameter not only in wind tunnel tests, but also in flight at higher values of the unit Reynolds number; surface roughness obviously limits the intensity of suction which can be applied at a spanwise suction strip.It was intuitively assumed that the removal of fluid by suction was equivalent to cutting off the lower portion of the boundary layer profile at the upstream edge of the suction strip and that a rapid re-adjustment of the boundary layer profile within a short distance took place.


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