1. No category

Applied Mathematics and Computation 219 (2012) 1093–1107
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
On Bitsadze–Samarskii type nonlocal boundary value problems
for elliptic differential and difference equations: Well-posedness
Allaberen Ashyralyev a,⇑, Elif Ozturk b
a
b
Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey
Department of Mathematics, Uludag University, 16340 Bursa, Turkey
a r t i c l e
i n f o
a b s t r a c t
The well-posedness of the Bitsadze–Samarskii type nonlocal boundary value problem in
Hölder spaces with a weight is established. The coercivity inequalities for the solution of
the nonlocal boundary value problem for elliptic equations are obtained. The stable second
order of accuracy difference scheme for the approximate solution of the problem is presented. The well-posedness of this difference scheme in difference analogue of Hölder
spaces is established. For applications, the almost coercivity and the coercivity estimates
for solutions of difference schemes for elliptic equations are obtained.
Ó 2012 Elsevier Inc. All rights reserved.
Keywords:
Elliptic equations
Nonlocal boundary value problems
Difference schemes
Stability
1. Introduction
The role played by coercive inequalities in the study of local boundary-value problems for elliptic and parabolic differential equations is well known (see, e.g., [1,2]).
In the present paper, we consider the Bitsadze–Samarskii type nonlocal boundary value problem
8 2
d uðtÞ
>
>
> dt2 þ AuðtÞ ¼ f ðtÞ ð0 < t < 1Þ;
>
>
>
>
J
X
>
>
< uð0Þ ¼ u; uð1Þ ¼
aj uðkj Þ þ w;
j¼1
>
>
>
>
J
X
>
>
>
>
jaj j 6 1
0 < k1 < k2 < < kJ < 1;
>
:
ð1:1Þ
j¼1
for the differential equation of elliptic type in a Hilbert space H with self-adjoint positive definite operator A with a closed
domain DðAÞ H is considered. Here f(t) be a given abstract continuous function defined on ½0; 1 with values in H; u and w
are elements of DðAÞ and kj are numbers from the set ½0; 1. It is known (see, for example, [3–7]) that various nonlocal
boundary-value problems for elliptic equations can be reduced to the nonlocal boundary-value problem (1.1). The simply
nonlocal boundary value problem was presented and investigated for the first time by Bitsadze and Samarskii [8]. Further
in [9–27] the Bitsadze–Samarskii type nonlocal boundary value problems were investigated for the various differential
and difference equations of elliptic type.
Methods of the solutions of the abstract elliptic differential and difference equations have been studied extensively by
many researchers (see [28–34] and the references therein).
⇑ Corresponding author.
E-mail address: [email protected] (A. Ashyralyev).
0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.amc.2012.07.016
1094
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
A function u(t) is called a solution of problem (1.1) if following conditions are satisfied:
(i) u(t) is twice continuously differentiable on the segment ½0; 1. The derivatives at the endpoints of the segment are
understood as the appropriate unilateral derivatives.
(ii) The element u(t) belongs to DðAÞ for all t 2 ½0; 1 and the function AuðtÞ is continuous on the segment ½0; 1.
(iii) u(t) satisfies the equation and nonlocal boundary conditions (1.1).
A solution of problem (1.1) defined in this manner will from now on be referred to as a solution of problem (1.1) in the
space Cð½0; 1; HÞ of all continuous functions uðtÞ defined on ½0; 1 with values in H, equipped with the norm
kukCð½0;1;HÞ ¼ maxkuðtÞkH :
06t61
The well-posedness in Cð½0; 1; HÞ of the boundary value problem (1.1) means that coercive inequality
h
i
ku00 kCð½0;1;HÞ þ kAukCð½0;1;HÞ 6 M kf kCð½0;1;HÞ þ kAukH þ kAwkH
ð1:2Þ
is true for its solution uðtÞ 2 Cð½0; 1; HÞ with some M, does not depend on u, w and f ðtÞ 2 Cð½0; 1; HÞ.
In this paper, positive constants, which can differ in time (hence: not a subject of precision) will be indicated with an M.
On the other hand Mða; b; . . .Þ is used to focus on the fact that the constant depends only on a; b; . . ..
In fact, inequality (1.2) does not, generally speaking, hold in an arbitrary Hilbert space H and for the general unbounded
self-adjoint positive definite operator A. Therefore, the problem (1.1) is not well-posed in Cð½0; 1; HÞ. The well-posedness of
the boundary value problem (1.1) can be established if one considers this problem in certain spaces Fð½0; 1; HÞ of smooth Hvalued functions on ½0; 1.
A function u(t) is said to be a solution of problem (1.1) in Fð½0; 1; HÞ if it is a solution of this problem in Cð½0; 1; HÞ and the
functions u00 ðtÞ and AuðtÞ belong to Fð½0; 1; HÞ.
As in the case of the space Cð½0; 1; HÞ, we say that the problem (1.1) is well-posed in Fð½0; 1; HÞ, if the following coercivity
inequality is satisfied:
h
i
ku00 kFð½0;1;HÞ þ kAukFð½0;1;HÞ 6 MðdÞ kf kFð½0;1;HÞ þ kAukH0 þ kAwkH0 ;
where H0 H.
Let us denote by C a0;1 ð½0; 1; HÞ, 0 < a < 1 the Banach spaces obtained by completion of the set of all smooth H-valued functions uðtÞ in the norm
kukC a
0;1 ð½0;1;HÞ
¼ kukCð½0;1;HÞ þ
sup
kuðt þ sÞ uðtÞkH
0<t<tþs<1
sa
ðt þ sÞa ð1 tÞa :
In the present paper, the well-posedness of the nonlocal boundary-value problem (1.1) in C a0;1 ð½0; 1; HÞ spaces is established. The stable second order of accuracy difference scheme for the approximate solution of this problem is presented.
The coercive inequalities for the solution of this difference scheme in difference analogue of C a0;1 ð½0; 1; HÞ spaces are established. For applications, the almost coercive stability and the coercive stability estimates for solutions of difference schemes
for approximate solutions of nonlocal boundary value problems for elliptic equations are achieved.
2. The differential elliptic equation
1
1
In this section, we denote B ¼ A2 . Then it is clear that B is a self-adjoint positive definite operator and B P d2 I. We now
state the following result, which will be called upon later in this work.
Lemma 2.1. The following estimates hold [12]:
kBa expðtBÞkH!H 6 t a
kðI e2B Þ1 kH!H
aa
e
6 MðdÞ:
;
0 6 a 6 e; t > 0;
Lemma 2.2. Let
D¼
J
X
aj ðI e2B Þ1 ðeð1kj ÞB eð1þkj ÞB Þ:
j¼1
Then the operator I D has an inverse
T ¼ ðI DÞ1
ð2:1Þ
ð2:2Þ
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A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
and the following estimate is satisfied:
kTkH!H 6 MðdÞ:
ð2:3Þ
Proof. The proof of estimate (2.3) is based on the estimate
kI DkH!H P 1 J
1
X
jaj jeð1kJ Þd2 :
ð2:4Þ
j¼1
Using the spectral representation of B, we get
kðI e2B Þ1 ðeð1kj ÞB eð1þkj ÞB ÞkE!E 6 sup jð1 e2l Þ1 ðeð1kj Þl eð1þkj Þl Þj
1
d2 6l<1
1
1
6 eð1kj Þd2 sup ð1 e2l Þ1 ð1 e2kj l Þ 6 eð1kJ Þd2 :
ð2:5Þ
1
d2 6
l<1
Then, applying the triangle inequality and estimate (2.5), we obtain (2.4). Thus, Lemma 2.2 is proved. h
Now, we will obtain the formula for solution of problem (1.1). It is clear that (see [12]) the boundary value problem for the
elliptic equation
2
d u
dt
0 < t < 1; uð0Þ ¼ u; uð1Þ ¼ x
þ AuðtÞ ¼ f ðtÞ;
2
ð2:6Þ
has a unique solution
Z 1
uðtÞ ¼ ðI e2B Þ1 ðetB eð2tÞB Þu þ ðeð1tÞB eð1þtÞB Þu1 ðeð1tÞB eð1þtÞB Þð2BÞ1
ðeð1sÞB eð1þsÞB Þf ðsÞ ds
þ ð2BÞ1
Z
0
1
ðejtsjB eðtþsÞB Þf ðsÞ ds:
ð2:7Þ
0
Using (2.7) and nonlocal boundary conditions, we get
Z 1
J
X
x ¼ aj ðI e2B Þ1 ðekj B eð2kj ÞB Þu þ ðeð1kj ÞB eð1þkj ÞB Þx ðeð1kj ÞB eð1þkj ÞB Þð2BÞ1 ðeð1sÞB eð1þsÞB Þf ðsÞds
0
j¼1
þð2BÞ1
Z
1
ðejkj sjB eðkj þsÞB Þf ðsÞds þ w:
0
From Lemma 2.2 it follows that
I e2B J
X
aj ðeð1kj ÞB eð1þkj ÞB Þ
j¼1
has an inverse. Therefore
"
Z 1
J
X
1
aj ðekj B eð2kj ÞB Þu ðeð1kj ÞB eð1þkj ÞB Þ B1 ðeð1sÞB eð1þsÞB Þf ðsÞ ds
2
0
j¼1
!)#
Z kj
Z 1
Z 1
eðkj sÞB f ðsÞ ds þ
eðskj ÞB f ðsÞ ds eðkj þsÞB f ðsÞ ds
:
uð1Þ ¼ x ¼ P ðI e2B Þw þ
1
þðI e2B Þ B1
2
0
kj
ð2:8Þ
0
Here,
P¼
2B
Ie
!1
J
X
ð1kj ÞB
aj ðe
ð1þkj ÞB
e
Þ
:
j¼1
Consequently, if the function f(t) continuously differentiable on ½0; 1, u; w 2 DðAÞ and formulas (2.7), (2.8) give a solution of
the problem (1.1).
Theorem 2.1. Suppose that u; w 2 DðAÞ. Then the boundary value problem (1.1) is well posed in a Hölder space C a0;1 ð½0; 1; HÞ and
the following coercivity inequality holds:
ku00 kC a
0;1 ð½0;1;HÞ
þ kAukC a
0;1 ð½0;1;HÞ
6 MðdÞ½kAukH þ kAwkH þ
Mðd; k1 ; kJ Þ
kf kC a ð½0;1;HÞ :
0;1
að1 aÞ
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A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
Proof. By [12], we have the following coercivity inequality
ku00 kC a
0;1 ð½0;1;HÞ
þ kAukC a
0;1 ð½0;1;HÞ
6
MðdÞ
að1 aÞ
kf kC a
0;1 ð½0;1;HÞ
þ MðdÞ½kAuð0ÞkH þ kAuð1ÞkH ð2:9Þ
for the solution of boundary value problem (2.6). Then, the proof of Theorem 2.1 is based on coercivity inequality (2.9) and
on the estimate
kAuð1ÞkH 6
Mðd; k1 ; kJ Þ
kf kC a ð½0;1;HÞ þ MðdÞ½kAukH þ kAwkH :
0;1
að1 aÞ
ð2:10Þ
Therefore, we will prove (2.10). First, applying formula (2.8), we can write
Auð1Þ ¼ P ðI e2B ÞAw þ
J
X
1
1
1
aj ðekj B eð2kj ÞB ÞAu eð1kj ÞB ðf ð1Þ f ðkj ÞÞ þ ekj B ðf ð0Þ f ðkj ÞÞ þ ðeð1þkj ÞB
2
2
2
j¼1
1
1
þeð2kj ÞB eð2þkj ÞB Þf ð1Þ þ ðeð2þkj ÞB þ eð3kj ÞB Þf ðkj Þ þ ðI eð1kj ÞB ÞðI þ eð1þkj ÞB Þf ð1Þ þ ðeð1þkj ÞB eð2kj ÞB
2
2
Z 1
Z
1 1 ð1þsÞB
ð1sÞB
ð3kj ÞB
ð1kj ÞB
ð1þkj ÞB 1
e
Þf ð0Þ ðe
e
Þ
Be
ðf ðsÞ f ð1ÞÞ ds Be
ðf ðsÞ f ð0ÞÞ ds
2 0
2 0
!
Z kj
Z 1
Z 1
1
ðkj sÞB
ðskj ÞB
ðsþkj ÞB
2B
Be
ðf ðsÞ f ðkj ÞÞ ds þ
Be
ðf ðsÞ f ðkj ÞÞ ds Be
ðf ðsÞ f ð0ÞÞds
þ ðI e Þ
2
kj
0
0
þðI e2B Þðf ðkj Þ f ð1ÞÞ ¼ J 1 þ J 2 þ J 3 þ J 4 þ J 5 þ J 6 þ J 7 :
Here
J1 ¼ P
!
J
X
aj ðekj B eð2kj ÞB ÞAu þ ðI e2B ÞAw ;
j¼1
J
X
1
aj e2B I þ ðI eð1kj ÞB ÞðI þ eð1þkj ÞB Þ ðeð1þkj ÞB þ eð1kj ÞB eð2kj ÞB þ eð2þkj ÞB Þ f ð1Þ
2
j¼1
1
1
þ I e2B þ ðeð1kj ÞB ekj B þ eð2þkj ÞB þ eð3kj ÞB Þ f ðkj Þ þ ðeð1þkj ÞB eð2kj ÞB eð3kj ÞB þ ekj B Þf ð0Þ ;
2
2
J2 ¼ P
J 3 ¼ P
J
X
aj ðeð1kj ÞB eð1þkj ÞB Þ
j¼1
J 4 ¼ P
J
X
ð1kj ÞB
aj ðe
e
ð1þkj ÞB
j¼1
J5 ¼ P
J
X
2B
aj ðI e
j¼1
J6 ¼ P
J
X
aj ðI e2B Þ
j¼1
J 7 ¼ P
1
Þ
2
J
X
aj ðI e
1
2
2B
j¼1
Z
1
2
1
Þ
2
Z
!
1
Beð1sÞB ðf ðsÞ f ð1ÞÞ ds ;
0
Z
!
1
Be
ð1þsÞB
ðf ðsÞ f ð0ÞÞ ds ;
0
!
kj
ðkj sÞB
Be
ðf ðsÞ f ðkj ÞÞ ds ;
0
Z
1
Þ
2
!
1
Beðskj ÞB ðf ðsÞ f ðkj ÞÞ ds ;
kj
Z
!
1
Be
ðsþkj ÞB
ðf ðsÞ f ð0ÞÞ ds :
0
Let us estimate J k for k ¼ 1; . . . ; 7 separately. Using estimates (2.1) and (2.3), we obtain
kJ 1 kH 6 kPkH!H
J
X
!
jaj jkekj B eð2kj ÞB kH!H kAukH þ kI e2B kH!H kAwkH
6 M 1 ðdÞ½kAukH þ kAwkH :
j¼1
Let us estimate J 2 . Using estimates (2.1), (2.3) and the definition of the norm of the space C a0;1 ð½0; 1; HÞ, we get
kJ 2 kH 6 M 2 ðdÞmaxkf ðtÞkH 6 M 2 ðdÞkf kCa
06t61
0;1 ð½0;1;HÞ
:
Now we will estimate J 3 . Using estimates (2.1), (2.3) and the definition of the norm of space C a0;1 ð½0; 1; HÞ, we obtain
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A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
kJ 3 kH 6 kPkH!H
6
Z 1
Z
J
X
1 1
ds
jaj j
kBðeð1kj ÞB eð1þkj B Þeð1sÞB kH!H kf ðsÞ f ð1ÞkH ds 6 M 3 ðdÞ
kf kC a ð½0;1;HÞ
0;1
2
2
ðs
þ kj Þ
0
0
j¼1
M 3 ðdÞ
kf kC a ð½0;1;HÞ :
0;1
1 kJ
In a similar manner, we can show that
kJ 4 kH 6 kPkH!H
Z
Z 1
J
X
1 1 ð1kj ÞB
ds
jaj j
kB e
eð1þkj ÞB eð1þsÞB kH!H kf ðsÞ f ð0ÞkH ds 6 M 4 ðdÞ
kf kCa ð½0;1;HÞ
0;1
2
2
þ
s kj
0
0
j¼1
6 M 4 ðdÞkf kC a
0;1 ð½0;1;HÞ
:
Using the estimates (2.1), (2.3) and the definition of the norm of space C a0;1 ð½0; 1; HÞ, we get
kJ 5 kH 6 kPkH!H
6 M 5 ðdÞ
Z kj
J
X
1
jaj j kI e2B kH!H
kBeðkj sÞB kH!H kf ðsÞ f ðkj ÞkH ds
2
0
j¼1
Z
J
X
jaj j
0
j¼1
6 M 5 ðdÞ
kj
1
að1 kJ Þa
ds
a
ð1 sÞ ðkj kf kC a
0;1 ð½0;1;HÞ
sÞ1a kaj
kf kC a
0;1 ð½0;1;HÞ
6 M 5 ðdÞ
J
X
j¼1
jaj j
ð1 kj Þa
Z
kj
ds
ðkj sÞ1a kaj
0
kf kC a
0;1 ð½0;1;HÞ
:
Now we will estimate J 6 . Using estimates (2.1), (2.3) and the definition of the norm of space C a0;1 ð½0; 1; HÞ, we get
kJ 6 kH 6 kPkH!H
6 M 6 ðdÞ
Z 1
J
X
1
jaj j kI e2B kH!H
kBeðskj ÞB kH!H dskf ðsÞ f ðkj ÞkH
2
kj
j¼1
J
X
j¼1
6
jaj j
ð1 kj Þa
Z
kj
1
ds
ðs kj
Þ1a sa
kf kC a
0;1 ð½0;1;HÞ
6 M 6 ðdÞ
J
X
j¼1
jaj j
ð1 kj Þa kaj
Z
kj
1
ds
ðs kj Þ1a
kf kC a
0;1 ð½0;1;HÞ
M 6 ðdÞ
kf kC a ð½0;1;HÞ :
0;1
aka1
Finally, we will estimate J 7 . Using estimates (2.1), (2.3) and the definition of the norm of space C a0;1 ð½0; 1; HÞ, we obtain
kJ 7 kH 6 kPkH!H
6 M 7 ðdÞ
Z
Z 1
J
J
X
X
1 1
ds
jaj j
kBðI e2B Þeðsþkj ÞB kH!H dskf ðsÞ f ð0ÞkH 6 M 7 ðdÞ jaj j
kf kC a ð½0;1;HÞ
0;1
2 0
0 s þ kj
j¼1
j¼1
1
kf kC a ð½0;1;HÞ :
0;1
k1
Combining the estimates in H for J k , k ¼ 1; . . . ; 7, we get (2.10). Hence, Theorem 2.1 is proved. h
Remark. We can obtain the well-posedness the boundary value problem (1.1) in a Hölder space C a0;1 ð½0; 1; EÞ with the positive operator A in an arbitrary Banach space E under the additional assumptions that the operator
I
J
X
aj ðI e2B Þ1 ðeð1kj ÞB eð1þkj ÞB Þ:
j¼1
has an inverse T and the following estimate is satisfied:
kTkE!E 6 MðdÞ:
Now, we consider the applications of Theorem 2.1. First, we consider the nonlocal boundary value problem for the elliptic
equation
8
utt ðaðxÞux Þx þ du ¼ f ðt; xÞ; 0 < t < 1; 0 < x < 1;
>
>
>
>
J
>
X
>
>
>
uð0;
xÞ
¼
u
ðxÞ;
uð1;
xÞ
¼
aj uðkj ; xÞ þ wðxÞ;
0 6 x 6 1;
>
>
<
j¼1
J
>
X
>
>
>
jaj j 6 1;
0 < k1 < k2 < < kJ < 1;
>
>
>
>
j¼1
>
>
:
uðt; 0Þ ¼ uðt; 1Þ; ux ðt; 1Þ ¼ ux ðt; 0Þ;
0 6 t 6 1;
ð2:11Þ
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A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
where aðxÞ, uðxÞ, wðxÞ and f ðt; xÞ are given sufficiently smooth functions aðxÞ P a > 0, d ¼ const > 0. The problem has a unique solution uðt; xÞ. This allows us to reduce the nonlocal problem (2.11) to the nonlocal boundary value problem (1.1) in a
Hilbert space H ¼ L2 ½0; 1 with a self-adjoint positive definite operator A defined by (2.11).
Theorem 2.2. The solutions of nonlocal boundary value problem (2.11) satisfy the coercivity inequality
kutt kC a
0;1 ð½0;1;L2 ½0;1Þ
þ kukC a
2
0;1 ð½0;1;W 2 ½0;1Þ
6
Mðd; k1 ; kJ Þ
kf kC a ð½0;1;L2 ½0;1Þ þ MðdÞ½kukW 2 ½0;1 þ kwkW 2 ½0;1 :
0;1
2
2
að1 aÞ
The proof of Theorem 2.2 is based on the abstract Theorem 2.1 and the symmetry properties of the space operator A generated by problem (2.11).
Second, let X be the unit open cube in Rn fx ¼ ðx1 ; . . . ; xn Þ : 0 < xk < 1; 1 6 k 6 ng with boundary S, X ¼ X [ S. In ½0; 1 X,
the Dirichlet–Bitsadze–Samarskii type mixed boundary value problem for the multidimensional elliptic equation
8
n
X
>
>
>
utt ðar ðxÞuxr Þxr ¼ f ðt; xÞ; x 2 X; 0 < t < 1;
>
>
>
r¼1
>
>
>
>
J
>
X
>
>
< uð0; xÞ ¼ uðxÞ; uð1; xÞ ¼
aj uðkj ; xÞ þ wðxÞ;
x 2 X;
j¼1
>
>
>
J
>
X
>
>
>
jaj j 6 1;
0 < k1 < < kJ < 1;
>
>
>
>
j¼1
>
>
:
uðt; xÞjxS ¼ 0; 0 6 t 6 1;
ð2:12Þ
x 2 X;
is considered. The problem has unique smooth solution uðt; xÞ for the smooth ar ðxÞ P a > 0, f ðt; xÞ, uðxÞ and wðxÞ functions.
This allows us to reduce the Dirichlet–Bitsadze–Samarskii type mixed boundary value problem to the nonlocal boundary
problem in a Hilbert space H ¼ L2 ðXÞ of the all integrable functions defined on X, equipped with norm
kf kL2 ðXÞ ¼
Z
Z
jf ðxÞj2 dx1 dxn
12
:
x2X
with a self adjoint positive definite operator A defined by (2.12).
Theorem 2.3. The solutions of nonlocal boundary value problem (2.12), satisfy the coercivity inequality
kutt kC a
0;1 ð½0;1;L2 ðXÞÞ
þ kukCa
2
0;1 ð½0;1;W 2 ðXÞÞ
6
Mðk1 ; kJ Þ
þ M½kukW 2 ðXÞ þ kwkW 2 ðXÞ :
kf k a
2
2
að1 aÞ C0;1 ð½0;1;L2 ðXÞÞ
The proof of Theorem 2.3 is based on the abstract Theorem 2.1, the symmetry properties of the space operator A generated by problem (2.12) and the following theorem on the coercivity inequality for the solution of the elliptic differential
problem in L2 ðXÞ.
Theorem 2.4. For the solutions of the elliptic differential problem
n
X
ðar ðxÞuxr Þxr ¼ xðxÞ;
x 2 X;
uðxÞ ¼ 0;
x 2 S;
r¼1
the following coercivity inequality holds [12]:
kukW 2 ðXÞ 6 MkxkL2 ðXÞ :
2
3. The second order of accuracy difference scheme
Let us associate to the nonlocal boundary value problem (1.1) the corresponding difference problem
8 2
s ðukþ1 2uk þ uk1 Þ þ Auk ¼ uk ; uk ¼ f ðtk Þ;
>
>
>
>
< tk ¼ ks; 1 6 k 6 N 1; Ns ¼ 1; u0 ¼ u;
J
X
>
>
>
aj u½ksr þ ðu½ksr u½ksr 1 Þ ksr ksr þ w:
>
: uN ¼
j¼1
It is a well-known that B ¼ 12
sA þ
ð3:1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4A þ s2 A2 is self-adjoint positive definite operator and R ¼ ðI þ sBÞ1 which is de-
fined on the whole space H is a bounded operator. Here, I is the identity operator.
1099
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
Now, let us give some lemmas that will be needed below.
Lemma 3.1. The following estimates hold [12]:
8
1
s
>
ksA2
>
Rk kH#H 6 MðdÞ
;
>
ks
< ke
k
kðI R2N Þ1 kH!H 6 MðdÞ;
k P 1;
k
k
kR kH#H 6 MðdÞð1 þ dsÞ ; kskBR kH#H 6 MðdÞ; k P 1; d > 0;
>
>
a
>
: kBb ðRkþr Rk Þk
6 MðdÞ ðrsaÞþb ; 1 6 k < k þ r 6 N; 0 6 a; b 6 1:
H#H
ð3:2Þ
ðksÞ
Lemma 3.2. The following estimate holds [12]:
N1
X
skBRj kH!H 6 MðdÞYðs; BÞ;
ð3:3Þ
j¼1
where
1
Yðs; BÞ ¼ min ln ; 1 þ j ln kBkH!H j :
s
Here and in future.
Lemma 3.3. The operator
I R2N J
X
aj RN
kj s
kj kj kj kj
kj
ðR IÞ RN s þ RNþ s 1
þ RNþ s þ
s
j¼1
s
has an inverse
Ts ¼
I R2N kj kj kj kj !1
J
X
k
k
aj RN s þ RNþ s þ j j ðR IÞ RN s þ RNþ s 1
s
j¼1
s
and the following estimate is satisfied:
kT s kH#H 6 Mðd; k1 ; kJ Þ:
ð3:4Þ
Proof. We have that
1
T s T ¼ T s T e2A2 R2N þ
k 1
kj kj kj !
kj 1
J
kj
X
j
k
k
:
aj e 1 s s A2 RN s eð1þ s sÞA2 þ RNþ s þ j j ðR IÞ RN½ s þ RNþ s 1
s
j¼1
s
ð3:5Þ
Using estimate (3.2), we obtain
k k k k k 1
k 1
J
X
kj
kj
2A12
2N
N sj
Nþ sj
N sj
Nþ sj 1
ð1 sj sÞA2
ð1þ sj sÞA2
ðR I Þ R
R þ
aj e
R
þR
e
þ
þR
e
s
s
j¼1
6 M 1 ðdÞs:
H!H
ð3:6Þ
Estimate (3.4) is derived from formula (3.5) and estimate (3.6). Therefore, Lemma 3.3 is proved. h
Theorem 3.1. For any uk , 1 6 k 6 N 1 the solution of the problem (3.1) exists and the following formula holds
(
2N 1
uk ¼ ðI R Þ
k
ðR R
2Nk
Þu þ ðR
Nk
R
Nþk
Nk
ÞuN ðR
R
Nþk
N1
X
ÞS ðRN1i RN1þi Þui s
)
i¼1
N1
X
þ S ðRjkij1 Rkþi1 Þui s for k ¼ 0; . . . ; N 1;
ð3:7Þ
i¼1
8
0
0 kj
1
½ s 1
J
N 1
N1
N1
< kj
X
X ½kj1i
X
X
X
kj
kj
kj
kj
kj
uN ¼ T s @ a j
R½ s R2N½ s u RN½ s RNþ½ s S ðRN1i RN1þi Þui s þ ðI R2N ÞS@
Rs
ui s þ Ri½ s 1 ui s R½ s þi1 ui sA
:
kj
i¼1
j¼1
i¼1
i¼1
i¼½ s 8
0 kj
199
1
½ s 1
< kj
N 1
N1
N1
==
X
X
X
X
kj
kj
kj
kj
kj
kj
kj
kj
½ s 1
2N½ s N½ s Nþ½ s 1
Ni1
Nþi1
2N @
½ s 2i
i½ s 1
½ s þi2
þ
S ðR
þR
uþ R
þR
R
Þui s þ SðI R Þ
R
ui s R
ui s R
ui sA
ðR IÞ R
þ ðI R2N ÞwA:
:
;
;
s s
kj
i¼1
i¼1
i¼1
i¼½ s 1100
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
Here
S ¼ ð2I þ sBÞ1 B1 :
Proof. The second order of accuracy difference scheme
(
1 6 k 6 N 1; Ns ¼ 1;
s12 ½ukþ1 2uk þ uk1 þ Auk ¼ uk ;
u0 ¼ u;
ð3:8Þ
uN ¼ x
has a solution and the following formula holds:
(
2N 1
uk ¼ ðI R Þ
k
ðR R
2Nk
Þu þ ðR
Nk
R
Nþk
Nk
Þx ðR
R
Nþk
N 1
X
ÞS ðRN1i RN1þi Þui s
i¼1
)
þS
N1
X
ðRjkij1
i¼1
Rkþi1 Þui s:
ð3:9Þ
Applying formula (3.9) and the nonlocal boundary condition
J
X
x¼
j¼1
aj u½kjs þ
kj
s
kj
u kj u½kj1
þ w;
s
½s
s
we obtain
uN ¼
(
J
X
2N 1
aj ðI R Þ
(
kj
½s
R
R
kj
2N½ s )
N1
X
kj
kj
kj
kj
N½ s Nþ½ s N½ s Nþ½ s N1i
N1þi
uþ R
R
x R
R
R
Þui s
S ðR
i¼1
j¼1
0 kj
1
½ s 1
N 1
N 1
X
X
X ½kj1i
i½kj
1
½kj
þi1
s
s
s
@
R
usþ R
us R
u sA
þS
i
i
i¼1
kj
i
i¼1
kj
i¼½ s n
kj
n kj
kj
kj
kj
ðI R2N Þ1 ðR IÞ R½ s 1 þ R2N½ s u RN½ s þ RNþ½ s 1 x
0 kj
1
½ s 1
N1 N1
X
X ½kji2
X
kj
kj
kj
N½ s Nþ½ s 1
Ni1
Nþi1
i½
1
þ R
þR
R
ui þ S@ R s
ðR IÞui s R s ðR IÞui sA
S s R
þ
s
s
i¼1
i¼1
kj
i¼½ s þ1
)
Su½kj s þ SR1 u½kj s S
s
s
N1
X
kj
R½ s þi2 ðR IÞui s
þ w:
i¼1
Since the operator
I R2N J
kj
kj
kj
kj
X
k
k
aj RN½ s þ RNþ½ s þ j ðR IÞ RN½ s þ RNþ½ s 1
s
j¼1
s
has an inverse T s , it follows that
uN ¼ x ¼ T s
J
X
aj
(
kj
kj
R½ s R2N½ s kj
kj
X
N 1
ðRNi1 RNþi1 Þui s
u RN½ s RNþ½ s S
j¼1
i¼1
1
0k
k
½ sj 1 k
N1
N1 k
kj
kj
X ½ j i1
X
X
j
j
kj
kj
2N B
i½ s 1
½ s þi1
s
ðR IÞ R½ s 1 þ R2N½ s u
þðI R ÞS@
R
ui s þ R
ui s R
u i sC
Aþ
i¼1
i¼1
k
i¼½ sj s
s
99
0k
1
1
>
X
½ sj 1 k
=>
=
N 1
N 1
N 1 k
kj
kj
kj
X
X
X
j
j
B
C
C
þ ðI R2N ÞwA:
R½ s 2i ui s Ri½ s 1 ui sA SðI R2N Þ R½ s þi2 ui s
þ RN½ s þ RNþ½ s 1 S ðRNi1 RNþi1 Þui s þ ðI R2N ÞS@
>
;>
;
kj
i¼1
i¼1
i¼1
i¼½ s Hence, Theorem 3.1 is established. h
with values in the Hilbert space H. The Banach spaces
Let Fð½0; 1s ; HÞ be the linear space of mesh functions us ¼ fuk gN1
1
Cð½0; 1s ; HÞ and C a0;1 ð½0; 1s ; HÞ, 0 < a < 1 have the norms
kus kCð½0;1s ;HÞ ¼ max kuk kH ;
16k6N1
kus kC a
0;1 ð½0;1s ;HÞ
¼ kus kCð½0;1s ;HÞ þ
ððN kÞsÞa ððk þ rÞsÞa
kukþr uk kH :
ðrsÞa
16k6kþr6N1
sup
1101
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
Theorem 3.2. The solution of difference scheme (3.1) in Cð½0; 1s ; HÞ obey the almost coercive inequality
2
N1 þ fAuk gN1
fs ðukþ1 2uk þ uk1 Þg1 1
Cð½0;1s ;HÞ
Cð½0;1s ;HÞ
h
i
6 Mðd; k1 ; kJ Þ Yðs; BÞkus kCð½0;1s ;HÞ þ kAukH þ kAwkH :
ð3:10Þ
Proof. By [12],
2
N1 fs ðukþ1 2uk þ uk1 Þg1 Cð½0;1s ;HÞ
þ fAuk gN1
1
Cð½0;1s ;HÞ
h
i
6 MðdÞ Yðs; BÞkus kCð½0;1s ;HÞ þ kAukH þ kAuN kH
ð3:11Þ
for the solution of difference scheme (3.8). Then, the proof of (3.10) is based on (3.11) and on the estimate
kAuN kH 6 MðdÞ Yðs; BÞkus kCð½0;1s ;HÞ þ kAukH þ kAwkH
ð3:12Þ
for the solution of difference scheme (3.1). Applying formula (3.7) and A ¼ B2 R, we get
AuN ¼ J 1 þ J 2 ;
ð3:13Þ
where
J1 ¼ T s
!
k
k
J
kj
kj
X
j
j
k
k
R½ s 1 þ R2N½ s ðR IÞAu þ ðI R2N ÞAw ;
aj R½ s R2N½ s Au þ j j
s
j¼1
s
ð3:14Þ
8
0k
1
>
X
½ sj 1
J
N1
N1
N1
<
kj
kj
kj
kj
kj
X
X
X
X
B
C
B
RN½ s þ RNþ½ s S B2 ðRNi RNþi Þui s þ SðI R2N Þ@
B2 R½ s i ui s þ
B2 Ri½ s ui s B2 R½ s þi ui sA
J 2 ¼ T s @ aj
>
:
k
i¼1
i¼1
j¼1
i¼1
j
i¼½ s 8
0k
1
j
>
X
½ s 1
N1
N1
<
kj
kj
kj
kj
X
X
kj
kj
B
C
ðR IÞ
RN½ s þ RNþ½ s 1 S B2 ðRNi RNþi Þui s þ ðI R2N ÞS@
B2 R½ s 1i ui s B2 Ri½ s ui sA
þ
>
s s
:
kj
i¼1
i¼1
i¼½ s )!
N1
kj
X
:
ð3:15Þ
ðI R2N ÞS B2 R½ s þi1 ui s
0
i¼1
To this end it suffices to show that
kJ 1 kH 6 Mðd; k1 ; kJ Þ½kAukH þ kAwkH ð3:16Þ
kJ 2 kH 6 Mðd; k1 ; kJ ÞYðs; BÞkus kCð½0;1s ;HÞ :
ð3:17Þ
and
Estimate (3.16) follows from formula (3.14), and estimates (3.2) and (3.4). Using formula (3.15) and estimates (3.2), (3.4), we
obtain
(
J
N1 X
X
N½kj Nþ½kj 1
R s s kð2I þ sBÞ kH!H
kJ2 kH 6 kT s kH!H
ja j j
þ
kBRNi kH!H þ kBRNþi kH!H kui kH s
R
H!H
j¼1
0
½s BX ½ksj iþ1 þkSkH!H @ BR
kj
i¼1
H!H
H!H
H!H
kj
i¼½ s þ1
"
kj
N½kj kj þ kR IkH!H R s
s s
H!H
0k
j
½ s 1
kj
BX
BR½ s i þkR IkH!H kSkH!H @
i¼1
i¼1
N 1 X
i½kj þ1 BR s kui kH s þ
Nþ½kj 1 s
þ
R
H!H
N1 X
½kj þiþ1 BR s
kui kH s þ
kð2I þ sBÞ1 kH!H
H!H
N1 X
i½kj þ1 BR s þ
k
i¼½ sj H!H
i¼1
N1 X
1
C
kui kH sA
kBRNi kH!H þ kBRNþi kH!H kui kH s
i¼1
N1 X
½kj þi BR s H!H i¼1
391
>
=
N1
X
C
7 C
i
Akui kH s5 A 6 Mðd; k1 ;kJ Þ skBR kH!H kui kH :
>
;
H!H
i¼1
1
From the last estimate and estimate (3.3) follows estimate (3.17). Thus, Theorem 3.2 is proved. h
Theorem 3.3. The difference scheme (3.1) is well posed in Hölder spaces C a0;1 ð½0; 1s ; HÞ and the following coercivity inequality
holds:
1102
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
2
N1 fs ðukþ1 2uk þ uk1 Þg1 C a0;1 ð½0;1s ;HÞ
6 Mðd; k1 ; kJ Þ
1
kus kCa ð½0;1s ;HÞ þ kAukH þ kAwkH :
0;1
að1 aÞ
ð3:18Þ
Proof. By [12],
2
N1 fs ðukþ1 2uk þ uk1 Þg1 C a0;1 ð½0;1s ;HÞ
6 MðdÞ
1
kus kC a ð½0;1s ;HÞ þ kAukH þ kAuN kH
0;1
að1 aÞ
ð3:19Þ
for the solution of difference scheme (3.8). Then, the proof of (3.18) is based on (3.19) and on the estimate
kAuN kH 6 Mðd; k1 ; kJ Þ
1
kus kC a ð½0;1s ;HÞ þ kAukH þ kAwkH :
0;1
að1 aÞ
Applying the triangle inequality, formula (3.13) and estimate (3.16), we get
kAuN kH 6 kJ 1 kH þ kJ 2 kH 6 kJ 2 kH þ Mðd; k1 ; kJ Þ½kAukH þ kAwkH :
To this end it suffices to show that
kJ 2 kH 6 Mðd; k1 ; kJ Þ
1
að1 aÞ
kus kC a
0;1 ð½0;1s ;HÞ
ð3:20Þ
:
Applying formula (3.15), we get
( J
X
kj
kj
X
N 1
aj RN½ s RNþ½ s S
J2 ¼ T s
k
N½ sj R
kj
kj
X
N1
i¼1
j¼1
sB2 RNi ðui uN1 Þ þ RN½ s RNþ½ s S
k
R
Nþ½ sj BS I R
N1
sB2 RNþi1 ðui u1 Þ
i¼1
uN1 R
k
N½ sj k
R
Nþ½ sj kj
½X
s 1
kj
N
2N1
2N
u1 þ I R S sB2 R½ s i ui u kj
BS R R
½s
i¼1
N1
X
kj
þ I R2N S sB2 Ri½ s ui u kj
½s
k
i¼½ sj N1
X
kj
kj
kj
þ I R2N BS I R½ s 1 þ R1 RN½ s 2 u kj I R2N S sB2 R½ s þi ðui u1 Þ
½s
(
kj kj
kj
ðR I Þ
I R2N BSR½ s I RN1 u1 þ
s
s
i¼1
X
N1
kj
kj
RN½ s þ RNþ½ s 1 S B2 RNi ðui uN1 Þs
i¼1
X
N1
kj
kj
kj
kj
kj
kj
RN½ s þ RNþ½ s 1 S B2 RNþi ðui u1 Þs þ RN½ s þ RNþ½ s 1 BS I RN1 uN1 RN½ s þ RNþ½ s 1 BS RN R2N1 u1
i¼1
N1
N1
½X
X
X
s 1
kj
kj
kj
þ I R2N S
B2 R½ s 1i ui u kj s I R2N S B2 Ri1½ s ui u kj s I R2N S B2 R½ s þi1 ðui u1 Þs
kj
½s
i¼1
kj þ I R2N BSR½ s I RN1 u1
½s
k
i¼½ sj i¼1
k
X
6
kj
j
J z2 ;
þ SBsR R½ s 1 RN½ s u kj
¼
½s
z¼1
where
J 12 ¼ T s
J
kj
kj
kj
kj
kj
X
aj RN½ s RNþ½ s BS I RN1 uN1 RN½ s RNþ½ s BS RN R2N1 u1 I R2N BS I R½ s 1 þ R1
j¼1
kj kj
kj
kj
kj
kj
ðR IÞ RN½ s þ RNþ½ s 1 BS I RN1 uN1
RN½ s 2 u kj I R2N BSR½ s I RN1 u1 þ
s
½s
s
k
kj kj
kj
kj
j
;
RN½ s þ RNþ½ s 1 BS RN R2N1 u1 þ I R2N BSR½ s I RN1 u1 þ I R2N SB2 s R½ s 1 RN½ s u kj
½s
J 22
¼ Ts
J
X
kj
Nþ½ s aj R
R
kj
N½ s þ
j¼1
J 32 ¼ T s
J
X
kj
kj
aj RN½ s RNþ½ s j¼1
kj
s
kj
s
!
X
N2
kj
kj
kj
N½ s Nþ½ s 1
2 Ni
ðR I Þ R
S sB R ðui uN1 Þ ;
R
s
i¼1
X
N1
kj
kj
kj
ðR IÞ RN½ s þ RNþ½ s 1 S sB2 RNþi ðui u1 Þ;
s
i¼2
1103
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
0
kj
kj
J
½X
½X
s 1
s 1
kj
kj
X
kj
kj
B
4
2N
2 ½ s i
2N
IR
J 2 ¼ T s aj @ I R S
sB R
ui u½kj þ
sB2 ðR IÞR½ s 1i ui u½kj ;
s
s
i¼1
j¼1
s
s
i¼1
0
0
J
N1
N1
kj
kj
X
X
X
k
k
B
2N B
¼ T s aj @ðI R ÞS@
sB2 Ri½ s ui u½kj þ j j ðI R2N ÞS
sB2 ðI RÞR½ s 1i ui u½kj ;
J 52
j¼1
s
s
k
i¼½ sj þ1
s
s
k
i¼½ sj þ1
!
X
J
N1
N1
X
k
k
X
kj
kj
2N
2 ½ sj þi
2 ½ sj þi1
S sB R
¼ T s aj I R S
sB R ðui u1 Þ ðR IÞðui u1 Þ :
J 62
s
i¼2
j¼1
Second let us estimate
Jm
2
s
i¼1
for any m ¼ 1; . . . ; 6 separately. We start with J 12 . Using estimates (3.2) and (3.4), we obtain
kJ 12 kH 6 Mðd; k1 ; kJ Þ max kuk kH 6 Mðd; k1 ; kJ Þkus kC a
0;1 ð½0;1s ;HÞ
16k6N1
:
For J 22 , applying (3.2), (3.4) and the definition of the norm of space C a0;1 ð½0; 1s ; HÞ, we obtain
kJ 22 kH 6 kT s kH!H
J
X
j¼1
2I þ sBÞ1 kH!H
Nþ½kj s jaj j R
N½kj s þ
R
H!H
H!H
!
N2
X
skBRNi kH!H kui uN1 kH
i¼1
kj
N½kj kj R s kR
Ik
þ H!H s
s
H!H
Nþ½kj 1 s
þ
R
k
H!H
N2
Mðd; k1 ; kJ Þ X
ððN 1Þs isÞa
6
s
kus kCa ð½0;1s ;HÞ :
a
0;1
ðN 1 iÞs
ððN 1ÞsÞ i¼1
The sum
N2
X
ððN 1Þs isÞa
s
ðN 1 iÞs
i¼1
is the lower Darboux integral sum for the integral
Z
1
0
ds
ð1 sÞ1a
:
It follows that
1
kJ 22 kH 6 M 1 ðd; k1 ; kJ Þ kus kC a
a
0;1 ð½0;1s ;HÞ
:
Now let us estimate J 32 . Using estimates (3.2) and (3.4) and the definition of the norm of space C a0;1 ð½0; 1s ; HÞ, we get
kJ 32 kH 6 kT s kH!H
J
X
j¼1
þsBÞ1 kH!H
N½kj s jaj j R
H!H
Nþ½kj s þ
R
H!H
N½kj kj
kj R s kR
Ik
þ H!H
s
!
N1
X
skBRNþi kH!H kui u1 kH 6 Mðd; k1 ; kJ Þ
s
H!H
N 1
X
s
i¼2
ðN þ iÞs
i¼2
kus kC a
0;1 ð½0;1s ;HÞ
Nþ½kj 1 s
þ
R
kð2I
H!H
6 M 1 ðd; k1 ; kJ Þkus kCa
0;1 ð½0;1s ;HÞ
:
Let us estimate J 42 . Using estimates (3.2), and (3.4) and the definition of the norm space C a0;1 ð½0; 1s ; HÞ, we obtain
kJ 42 kH
6 kT s kH!H
J
X
j¼1
kj
X kj þ
sBR½ s i ½ s 1
i¼1
H!H
0
kj
½X
1 kj
kj s
Bkj
BðR IÞR½ s 1i jaj jkI R kH!H kð2I þ sBÞ kH!H @ s
2N
1
s
s
1
kj
½ s 1
J
X
X
C
u u kj A 6 Mðd; k1 ; kJ Þ jaj j
h i
i
½ s kj
H
j¼1
i¼1
s
H!H
i¼1
s
h i
kj
s
s is
s is
s
H
a
h i a
kj
u u kj i
½ s a
s ð1 isÞ
kus kC a
0;1 ð½0;1s ;HÞ
:
By the lower Darboux integral sum for the integral, it follows
kJ 42 kH 6
Z
J
X
M 1 ðd; k1 ; kJ Þ kj
ds
M 2 ðd; k1 ; kJ Þ s
M 3 ðd; k1 ; kJ Þ s
jaj j a
kus kC a ð½0;1s ;HÞ 6
ku kCa ð½0;1s ;HÞ :
a
a ku kC a0;1 ð½0;1s ;HÞ 6
1a
0;1
0;1
a
ð1
k
Þ
a
ð1
k
Þ
k
ðk
sÞ
0
J
j
j
j
j¼1
Let us estimate J 52 . Estimates (3.2) and (3.4) and the definition of the norm space C a0;1 ð½0; 1s ; HÞ give
1104
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
0
1
h i a
kj
N1
s
i
s
s
X
s
B
C
kJ 52 kH 6 Mðd; k1 ; kJ Þm jaj j@
h i h i a Akus kCa0;1 ð½0;1s ;HÞ :
kj
kj
a
j¼1
kj
i
s
s
ð
s
Þ
1
s
i
s
s
i¼½ J
s
The sum enclosed in the right-hand side brackets is the lower Darboux integral sum for the integral
Z
1
ds
ð1 kj Þa sa ðs kj Þ1a
kj
:
Since
Z
kj
1
ds
ð1 kj
Þa sa ðs k Þ1a
6
j
1
að1 kj Þa
;
it follows that
kJ 52 kH 6 M 1 ðd; k1 ; kJ Þ
1
að1 kJ Þa
kus kC a
0;1 ð½0;1s ;HÞ
:
Lastly, let us estimate J 62 . In exactly the same manner using estimates (3.2) and (3.4) and the definition of the norm space
C a0;1 ð½0; 1s ; HÞ, we obtain
J
X
kj þ 1 s
kJ 62 kH 6 Mðd; k1 ; kJ Þ jaj j ln
ku kC a ð½0;1s ;HÞ :
0;1
kj
j¼1
It follows that
kJ 62 kH 6 M 1 ðd; k1 ; kJ Þkus kC a
0;1 ð½0;1s ;HÞ
:
Combining estimates in H for J m
h
2 , m ¼ 1; . . . ; 6, we get estimate (3.20). Thus, Theorem 3.3 is proved.
Now we consider applications of Theorems 3.1–3.3. First, the nonlocal boundary value problem (2.11) for the elliptic equation is considered. The discretization of problem (2.11) is carried out in two steps. In the first step, let us define the grid space
½0; 1h ¼ fx : xr ¼ rh; 0 6 r 6 K; Kh ¼ 1g:
We introduce the Hilbert space L2h ¼ L2 ð½0; 1h Þ of the grid functions uh ðxÞ ¼ fur g1K1 defined on ½0; 1h , equipped with the norm
kuh kL2h ¼
!1=2
K1
X
juh ðxÞj2 h
:
r¼1
To the differential operator A generated by the problem (2.11), we assign the difference operator Axh by the formula
Axh uh ðxÞ ¼ fðaðxÞuxx;r þ dur gM1
;
1
h
ð3:21Þ
M1
r g1
acting in the space of grid functions u ðxÞ ¼ fu
satisfying the conditions u0 ¼ uM , u1 u0 ¼ uM uM1 . With the help
of Axh we arrive at the nonlocal boundary value problem
8 2
x h
d uh ðt;xÞ
h
>
>
> dt2 þ Ah u ðt; xÞ ¼ f ðt; xÞ; 0 < t < 1; x 2 ½0; 1h ;
<
J
X
>
>
uh ð0; xÞ ¼ uh ðxÞ; uh ð1; xÞ ¼
aj uh ðkj ; xÞ þ wh ðxÞ; x 2 ½0; 1h
>
:
ð3:22Þ
j¼1
for an infinite system of ordinary differential equations.
In the second step, we replace problem (3.22) by the difference scheme below
8 h
u
ðxÞ2uhk ðxÞþuhk1 ðxÞ
>
kþ1
þ Axh uhk ðxÞ ¼ fkh ðxÞ; fkh ðxÞ ¼ f ðt k ; xÞ;
>
>
s2
>
>
< t ¼ ks; 1 6 k 6 N 1; Ns ¼ 1;
uh0 ðxÞ ¼ uh ðxÞ; x 2 ½0; 1h :
k
J
>
h i
X
>
>
h
>
aj uh½kj =s ðxÞ þ ksj ksj uh½kj =sþ1 ðxÞ uh½kj =s ðxÞ þ wh ðxÞ;
>
: uN ðxÞ ¼
ð3:23Þ
x 2 ½0; 1h :
j¼1
Theorem 3.4. Let s and jhj be sufficiently small positive numbers. The solutions of the difference scheme (3.23) satisfy the
following almost coercivity estimates:
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
max ks2 uhkþ1 2uhk þ uhk1 kL2h þ max kuhk kW 2 6 Mðd; k1 ; kJ Þ ln
16k6N1
16k6N1
2h
1105
1
max kfkh kL2h þ kuh kW 2 þ kwh kW 2 :
2h
2h
s þ jhj 16k6N1
The proof of Theorem 3.4 is based on the abstract Theorem 3.2 on the estimate
1
min ln ; 1 þ j ln kBxh kH!H j 6 M ln
s
1
s þ jhj
and the symmetry properties of the difference operator Axh defined by (3.21) in L2h .
Theorem 3.5. Let s and jhj be sufficiently small positive numbers. The solutions of the difference scheme (3.23) satisfy the
following coercivity estimates:
N1
k s2 uhkþ1 2uhk þ uhk1 1 kC a
0;1 ð½0;1s ;L2h Þ
6 Mðd; k1 ; kJ Þ kuh kW 2 þ kwh kW 2 þ
2h
2h
N1 þ uhk 1 C a0;1 ð½0;1s ;W 22h Þ
N1
1
k fkh 1 kCa ð½0;1 L2h Þ :
s
0;1
að1 aÞ
The proof of Theorem 3.5 is based on the abstract Theorem 3.3 and the symmetry properties of the difference operator Axh
defined by (3.21) in L2h .
Second, the nonlocal boundary value problem (2.11) is considered. The discretization of problem (3.22) is carried out in
two steps.
In the first step, let us define the grid sets
e h ¼ fx ¼ xm ¼ ðh1 m1 ; . . . ; hm mm Þ; m ¼ ðm1 ; . . . ; mm Þ; 0 6 ms 6 Ns ; hs Ns ¼ 1; j ¼ 1; . . . ; mg;
X
e h \ X;
Xh ¼ X
e h \ S:
Sh ¼ X
e h Þ of the grid functions uh ðxÞ ¼ fuðh1 m1 ; . . . ; hh mm Þg defined on X
e k , equipped with
We introduce the Hilbert space L2h ¼ L2 ð X
the norm
0
h
ku kL2h
¼@
X
11=2
2
ju ðxÞj h1 hn A
h
:
x2Xh
To the differential operator A generated by problem (3.22), we assign the difference operator Axh by the formula
n
X
Axh uh ¼ ar ðxÞuhxr x ;j
r¼1
ð3:24Þ
r r
acting in the space of grid functions uh ðxÞ, satisfying the conditions uh ðxÞ ¼ 0 for all x 2 Sh . It is known that Axh is a self-adjoint
positive definite operator in L2h . With the help of Axh we arrive at the nonlocal boundary value problem for an infinite system
of ordinary differential equations
8 2
h
>
e h;
>
d udtðt;xÞ
þ Axh uh ðt; xÞ ¼ f h ðt; xÞ; 0 6 t 6 1; x 2 X
2
>
<
J
X
>
>
uh ð0; xÞ ¼ uh ðxÞ; uh ð1; xÞ ¼
aj uh kj ; x þ wh ðxÞ;
>
:
ð3:25Þ
e h:
x2X
j¼1
In the second step, (3.25) is replaced by the difference scheme below, we get second order of accuracy difference scheme
8 uh ðxÞ2uh ðxÞþuh ðxÞ
k
k1
>
kþ1
þ Axh uhk ðxÞ ¼ fkh ðxÞ; fkh ðxÞ ¼ f h ðt k ; xÞ; x 2 Xh ;
>
s2
>
>
>
<
e h;
uh0 ðxÞ ¼ uh ðxÞ; x 2 X
t k ¼ ks; 1 6 k 6 N 1; Ns ¼ 1;
J
>
h
i
X
>
>
h
>
>
aj uh½kj =s ðxÞ þ ksj ksj uh½kj =sþ1 ðxÞ uh½kj =s ðxÞ þ wh ðxÞ;
: uN ðxÞ ¼
ð3:26Þ
e h:
x2X
j¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
Theorem 3.6. Let s and jhj ¼ h1 þ þ hn be sufficiently small positive numbers. The solutions of the difference scheme (3.26)
satisfy the following almost coercivity estimates:
max ks2 uhkþ1 2uhk þ uhk1 kL2h þ max kuhk kW 2 6 Mðd; k1 ; kJ Þ ln
16k6N1
16k6N1
2h
1
max kfkh kL2h þ kuh kW 2 þ kwh kW 2 :
2h
2h
s þ jhj 16k6N1
The proof of Theorem 3.6 is based on the abstract Theorem 3.2 on the estimate
1106
A. Ashyralyev, E. Ozturk / Applied Mathematics and Computation 219 (2012) 1093–1107
1
min ln ; 1 þ j ln kBxh kH!H j 6 M ln
s
1
s þ jhj
and the symmetry properties of the difference operator Axh defined by (3.24) in L2h and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h .
Theorem 3.7. For the solutions of the elliptic difference problem
Axh uh ðxÞ ¼ xh ðxÞ;
uh ðxÞ ¼ 0;
x 2 Xh ;
ð3:27Þ
x 2 Sh
the following coercivity inequality holds [34]:
kuh kW 2 6 MðdÞkxh kL2h :
2h
Theorem 3.8. Let s and jhj be a sufficiently small positive numbers. Then the solution of difference scheme (3.26) satisfy the following coercivity stability estimate:
N1
N1
k s2 uhkþ1 2uhk þ uhk1 1 kC a ð½0;1 ;L2h Þ þ k uhk 1 kCa ð½0;1 ;W 2 Þ
s
s 2h
0;1
0;1
1
N1
k fkh 1 kCa ð½0;1s L2h Þ :
6 Mðd; k1 ; kJ Þ kuh kW 2 þ kwh kW 2 þ
0;1
2h
2h
að1 aÞ
The proof of Theorem 3.8 is based on the abstract Theorem 3.3 and the symmetry properties of the difference operator Axh
defined by the formula (3.24) and on Theorem 3.7 on the coercivity inequality for the solution of the elliptic difference Eq.
(3.1) in L2h .
Acknowledgement
The authors would like to thank Prof. P.E. Sobolevskii for his helpful suggestions to the improvement of this paper.
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