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Ordinary Differential Equations 213
By refining the above inequalities, we can further investigate the behavior of pǫ,n near
the boundary layer.
In figures 9.7.1 and 9.7.2, we plot the polynomial pǫ,n for n = 16 and n = 22, cor-
responding to the approximation by the tau method of problem (9.7.1) in the Legendre
case (ν = 0) with ǫ = 10 −4 .
Figure 9.7.1 - Tau-Legendre Figure 9.7.2 - Tau-Legendre
approximation of problem (9.7.1) approximation of problem (9.7.1)
for n = 16 and ǫ = 10 −4 . for n = 22 and ǫ = 10 −4 .
For higher values of n, we have enough resolution to control the violent deviation of Uǫ
near the point x = 1. Then, the oscillations disappear. This happens approximately
when n is larger than C/ √ ǫ, where the constant C > 0 does not depend on ǫ.
Similar conclusions follow for the collocation method. Again, we consider the ul-
traspherical case, where the polynomial pǫ,n, n ≥ 2, ǫ > 0, satisfies: