2007 | OriginalPaper | Chapter
Fractional spaces generated by the positive differential and difference operators in a Banach space
Author : Allaberen Ashyralyev
Published in: Mathematical Methods in Engineering
Publisher: Springer Netherlands
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The structure of the fractional spaces
E
α,q
,(
L
q
[0, 1],
A
x
) generated by the positive differential operator
A
x
defined by the formula
A
x
u
= −
a
(
x
)
d
2
u
/
dx
2
+
δu
, with domain
D
(
A
x
) = {
u
∈
C
(2)
[0, 1] :
u
(0) =
u
(1),
u
′(0) =
u
′(1)} is investigated. It is established that for any 0 <
α
< 1/2 the norms in the spaces
E
α,q
(
L
q
[0, 1],
A
x
) and
W
q
2
α
[0, 1] are equivalent. The positivity of the differential operator
A
x
in
W
q
2α
[0, 1](0 ≤
α
< 1/2) is established. The discrete analogy of these results for the positive difference operator
A
h
x
a second order of approximation of the differential operator
A
x
, defined by the formula
$$ A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }} {{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1 $$
with
u
0
=
u
M
and −
u
2
+ 4
u
1
− 3
u
0
=
u
M
−2
− 4
u
M
−1
+ 3
u
M
is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equation and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.