Вычислительная математика - ИСЭМ СО РАН

Obviously function F x (η) is positive for η > α.
Taking into account notation (4), integral equation (3) has the following form
F x (η)dη
α
x(t) = a + bt + (c − a − bβ)
β∫
F x (η)dη
∫ t
α
≡ P t (x), α < t < β. (5)
Obviously for α t β and ∀x ∈ C [α,β] we have inequality 0
Hence for ∀x ∈ C [α,β] we have estimation
∫ t
α Fx(η)dη
∫ β
α
Fx(η)dη
1.
||P t (x)|| |a| + |b||β| + (c − a − bβ) = r. (6)
Thus integral operator P t : C [α,β] → C [α,β] maps sphere ||x|| r into itself.
Let us prove that operator P t is completely continuous. We exploit the Arzela‘ theorem [8] on
precompact sets in C [α,β] . In consequence of estimation (6) image of set ||x|| r by mapping P t (x) is
uniformly bounded by r.
Farther
∫
|c − a − bβ|
t 2
|P t1 (x) − P t (x)| |b||t 1 − t 2 | + | F x (η)dη|. (7)
β∫
| F x (η)dη| t 1
Let us note, that there are exist constants C 1 , C 2 such that
As a result of inequalities (8), (9) gives such estimate correction (7):
where
|
∫ β
α
α
F x (η)dη | C 1 > 0, (8)
0 < F x (η) C 2 . (9)
|P t1 (x) − P t2 (x)| l|t 1 − t 2 |,
l =| b | + | c − a − bβ | C 2
C 1
.
At the same time l is the same for any (x, t 1 , t 2 ) ∈ D. It has been shown that image P t (x) of the sphere
S(0, r) is uniformly bounded and equicontinual in C [α,β] . Therefore, following the Arzela‘ theorem [8]
it is precompact set in C [α,β] . Hence operator P is completely continuous, for which all the conditions
of the Shauder theorem [8] are fulfilled and equivalent integral equations (3) has the solution in space
C [α,β] . It’s to be noted that di P
, i = 0, 1, 2, 3 for any x ∈ C
dt i [α,β] are continuous due to the operator P
structure. Hence the solution of equation (3) will be three times continuously differentiable. Thus the
theorem is proved.
Presented here result is an extension of our results [7], [9] on the investigation of boundary layer
problem in the theory of mathematical modelling of the melt spinning process [1], [2], [5].
References
[1] Glauert M.B., Lighthill M.J., The axisymmetric boundary layer on a long thin cylinder, Proc. R.
Soc. London, 1955, 320, 188–203.
[2] Schlichting H. Boundary Layer Theory, 7 th edition, McGraw Hill, (1951).
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