ISSN:
1432-0673
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Conclusions In the preceding sections we have applied mollifiers to prove the validity of an asymptotic approximation in the case where the approximating functions are non-differentiable. If γ + is C δ[x 1, x 2] with 0〈δ≦1 (and piecewise continuously differentiable in the open interval (x 1, x 2)) we can try to prove a result analogous to Theorems I and II. It appears that γ + has to be regularized, resulting in γ ɛ+. Now we take in the boundary layer (x, η) as local coordinates with ɛη=y−γ ɛ+. The coefficients of L 1 and L 2 remain C 3 in ɛη under this transformation, and we can expand as in §3 into powers of ɛη, resulting in the operators M 0, M 1 and M 2. The coefficients of δ 2/δη 2 in M 0 will be dependent on γ ɛ+ ′ ; hence also v 0 will contain a factor γ ɛ+ ′ , so M 2 v 0 will contain a factor γ ɛ ‴ , and we see that (ɛL 1+L 2)(w+v 0+ɛv 1) is at most of order O(ɛ; λ(δ−3)+1). On the other hand, in order that v 0+ɛv 1 gives a good correction along the upper boundary, exp1ɛ(γ ɛ+(x)−γ +(x)) has to approximate 1 within a certain order of ɛ. From this it follows that γ ɛ+−γ +=0(ɛ;λδ) has to be of an order smaller than O(ɛ) and so λδ has to be greater than 1. It is obvious that we cannot have both λδ〉1 and λ(δ−3)+1〉0, hence regularization does not work in this case. A long calculation shows that, if γ + is C n+δ with n+δ〉3/2, this method works and results in an approximation of order O(ɛ; min1, 2/3(n+δ)−1).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00250483
