Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 904824, 16 pages
doi:10.1155/2008/904824
Research Article
Nonlocal Boundary Value Problems for
Elliptic-Parabolic Differential and Difference
Equations
Allaberen Ashyralyev1 and Okan Gercek2
1
2
Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey
Vocational School, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey
Correspondence should be addressed to Okan Gercek, ogercek@fatih.edu.tr
Received 30 June 2008; Accepted 17 September 2008
Recommended by Yong Zhou
The abstract nonlocal boundary value problem −d2 ut/dt2 Aut gt, 0 < t < 1, dut/dt −
Aut ft, 1 < t < 0, u1 u−1 μ for differential equations in a Hilbert space H with
the self-adjoint positive definite operator A is considered. The well-posedness of this problem in
Hölder spaces with a weight is established. The coercivity inequalities for the solution of boundary
value problems for elliptic-parabolic equations are obtained. The first order of accuracy difference
scheme for the approximate solution of this nonlocal boundary value problem is presented. The
well-posedness of this difference scheme in Hölder spaces is established. In applications, coercivity
inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained.
Copyright q 2008 A. Ashyralyev and O. Gercek. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
It is known that various problems in fluid mechanics and other areas of engineering, physics,
and biological systems lead to partial differential equations of variable types. Methods of
solutions of nonlocal boundary value problems for partial differential equations of variable
type have been studied extensively by many researchers see, e.g., 1–4 and the references
given therein.
The nonlocal boundary value problem
−
d2 ut
Aut gt,
dt2
dut
− Aut ft,
dt
0 < t < 1,
−1 < t < 0,
u1 u−1 μ
1.1
2
Discrete Dynamics in Nature and Society
for differential equations in a Hilbert space H with the self-adjoint positive definite operator
A is considered.
α
−1, 1, H, 0 < α < 1 the Banach space obtained by completion
Let us denote by C0,1
of the set of all smooth H-valued function ϕt on −1, 1 in the norm
α
ϕC0,1
−1,1,H ϕC−1,1,H −tα ϕt τ − ϕtH
τα
−1<t<tτ<0
sup
1 − tα t τα ϕt τ − ϕtH
sup
,
τα
0<t<tτ<1
1.2
α
0, 1, H, 0 < α < 1 the Banach space obtained by completion of the set
and denote by C0,1
of all smooth H-valued function ϕt on 0, 1 in the norm
α
ϕC0,1
0,1,H ϕC0,1,H 1 − tα t τα ϕt τ − ϕtH
,
τα
0<t<tτ<1
sup
1.3
finally denote by C0α −1, 0, H, 0 < α < 1 the Banach space obtained by completion of the
set of all smooth H-valued function ϕt on −1, 0 in the norm
ϕC0α −1,0,H ϕC−1,0,H −tα ϕt τ − ϕtH
.
τα
−1<t<tτ<0
sup
1.4
Here Ca, b, H stands for the Banach space of all continuous functions ϕt defined on
a, b with values in H equipped with the norm
||ϕ||Ca,b,H maxϕtH .
a≤t≤b
1.5
A function ut is called a solution of problem 1.1 if the following conditions are satisfied.
i ut is twice continuously differentiable on the segment 0, 1 and continuously
differentiable on the segment −1, 1; the derivatives at the endpoints of the segment
are understood as the appropriate unilateral derivatives.
ii The element ut belongs to the domain DA of A for all t ∈ −1, 1, and the
function Aut is continuous on the segment −1, 1.
iii ut satisfies the equations and the nonlocal boundary condition 1.1.
A solution of problem 1.1 defined in this manner will henceforth be referred to as a
solution of problem 1.1 in the space CH C−1, 1, H.
We say that problem 1.1 is well-posed in CH, if there exists a unique solution ut
in CH of problem 1.1 for any gt ∈ C0, 1, H, ft ∈ C−1, 0, H, and μ ∈ DA, and
the following coercivity inequality is satisfied:
u C0,1,H u C−1,0,H AuCH ≤ M gC0,1,H fC−1,0,H AμH ,
where M is independent of μ, ft, and gt.
1.6
A. Ashyralyev and O. Gercek
3
Problem 1.1 is not well-posed in CH 5. The well-posedness of the boundary
value problem 1.1 can be established if one considers this problem in certain spaces FH
of smooth H-valued functions on −1, 1.
A function ut is said to be a solution of problem 1.1 in FH if it is a solution of this
problem in CH and the functions u t t ∈ 0, 1, u t t ∈ −1, 1 and Aut t ∈ −1, 1
belong to FH.
As in the case of the space CH, we say that problem 1.1 is well-posed in FH, if
the following coercivity inequality is satisfied:
u F0,1,H u F−1,0,H AuFH ≤ M gF0,1,H fF−1,0,H AμH ,
1.7
where M is independent of μ, ft, and gt.
α
α
If we set FH equal to C0,1
H C0,1
−1, 1, H 0 < α < 1, then we can establish
the following coercivity inequality.
Theorem 1.1. Suppose μ ∈ DA. Then the boundary value problem 1.1 is well-posed in a Hölder
α
H and the following coercivity inequality holds:
space C0,1
α
α
u C0,1
0,1,H u C0α −1,0,H AuC0,1 H
1
fCα −1,0,H gCα 0,1,H AμH .
≤M
0
0,1
α1 − α
1.8
Here M is independent of ft, gt, and μ.
The proof of this assertion follows from the scheme of the proof of the theorem on
well-posedness of paper 5 and is based on the following formulas:
1/2 −1
1/2
1/2
e−tA − e−−t2A u0
ut I − e−2A
1/2
1/2
1/2 −1
e−1−tA − e−t1A u1 I − e−2A
1
1/2
1/2
1/2
1/2
× e−1−tA − e−t1A
A−1/2 2−1 e−1−sA − e−s1A gsds
−
1
0
A−1/2 2−1 e−tsA
0
ut etA u0 t
− e−|t−s|A
1/2
1/2
et−sA fsds,
0 ≤ t ≤ 1,
gsds,
−1 ≤ t ≤ 0,
0
−1
1/2
1/2
1/2
− 2e−A A
u0 I e−2A A1/2 I − e−2A
1
−1
1/2
1/2
1/2
1/2
× e−A 2
e−1sA fsds A−1/2 e−1−sA − e−s1A gsds 2e−A μ
0
0
I e−2A A1/2 I − e−2A
I − e−2A
1
1/2
× − A−1/2 f0 A−1/2 e−sA gsds
1/2
1/2
0
1/2
− 2e−A
1/2
A
−1
1.9
4
Discrete Dynamics in Nature and Society
for the solution of problem 1.1 and on the estimates
1/2 −1 I − e−2A
H →H
≤ M,
−1 1/2
1/2
1/2
− 2e−A A I e−2A A1/2 I − e−2A
H →H
≤ M,
−1 1/2
1/2
1/2
1/2
I e−2A A1/2 I − e−2A
− 2e−A A A
1/2 α −tA1/2 A e
H →H
H →H
≤ t−α ,
Aα e−tA ||H → H ≤ t−α ,
≤ M,
1.10
t > 0, 0 ≤ α ≤ 1,
t > 0, 0 ≤ α ≤ 1.
Remark 1.2. The nonlocal boundary value problem for the elliptic-parabolic equation
−
dut
Aut ft,
dt
0 < t < 1,
d2 ut
Aut gt,
dt2
−1 < t < 0,
1.11
u1 u−1 μ
in a Hilbert space H with a self-adjoint positive definite operator A is considered in paper
6. The well-posedness of this problem in Hölder spaces Cα H without a weight was
established under the strong condition on μ.
Now, the applications of this abstract results are presented.
First, the mixed boundary value problem for the elliptic-parabolic equations
ga − utt − axux x δu gt, x,
ut axux x − δu ft, x,
ut, 0 ut, 1,
0 < t < 1, 0 < x < 1,
−1 < t < 0, 0 < x < 1,
ux t, 0 ux t, 1,
u1, x u−1, x μx,
−1 ≤ t ≤ 1,
1.12
0 ≤ x ≤ 1,
u0, x u0−, x, ut 0, x ut 0−, x,
0≤x≤1
is considered. Problem 1.12 has a unique smooth solution ut, x for ax ≥ a > 0 x ∈
0, 1, and gt, x t ∈ 0, 1, x ∈ 0, 1, ft, x t ∈ −1, 0, x ∈ 0, 1 the smooth functions and
δ const > 0. This allows us to reduce the mixed problem 1.12 to the nonlocal boundary
value problem 1.1 in the Hilbert space H L2 0, 1 with a self-adjoint positive definite
operator A defined by 1.12.
A. Ashyralyev and O. Gercek
5
Theorem 1.3. The solutions of the nonlocal boundary value problem 1.12 satisfy the coercivity
inequality
α
α
utt C0,1
0,1,L2 0,1 ut C0α −1,0,L2 0,1 uC0,1
−1,1,W22 0,1
1
α
α
gC0,1
μ
≤M
f
2
0,1,L2 0,1
C0 −1,0,L2 0,1
W2 0,1 ,
α1 − α
1.13
where M is independent of ft, x, gt, x, and μx.
The proof of Theorem 1.3 is based on the abstract Theorem 1.1 and the symmetry
properties of the space operator are generated by problem 1.12.
Second, let Ω be the unit open cube in the n-dimensional Euclidean space Rn 0 < xk <
1, 1 ≤ k ≤ n with boundary S, Ω Ω ∪ S. In −1, 1 × Ω, the boundary value problem for the
multidimensional elliptic-parabolic equation
−utt −
ut n
ar xuxr xr gt, x,
r1
n
ar xuxr xr ft, x,
0 < t < 1, x ∈ Ω,
−1 < t < 0, x ∈ Ω,
1.14
r1
ut, x 0,
x ∈ S, − 1 ≤ t ≤ 1;
u0, x u0−, x,
u1, x u−1, x μx,
ut 0, x ut 0−, x,
x ∈ Ω,
x∈ Ω
is considered. Problem 1.14 has a unique smooth solution ut, x for ar x ≥ a > 0 x ∈ Ω
and gt, x t ∈ 0, 1, x ∈ Ω, ft, x t ∈ −1, 0, x ∈ Ω, the smooth functions. This allows
us to reduce the mixed problem 1.14 to the nonlocal boundary value problem 1.1 in the
Hilbert space H L2 Ω of all the integrable functions defined on Ω, equipped with the
norm
fL2 Ω 2
···
|fx| dx1 · · · dxn
1/2
1.15
x∈Ω
with a self-adjoint positive definite operator A defined by 1.14.
Theorem 1.4. The solution of the nonlocal boundary value problem 1.14 satisfies the coercivity
inequality
utt Cα
ut Cα −1,0,L2 Ω uCα −1,1,W 2 Ω
2
0
0,1
1
gCα 0,1,L2 Ω fCα −1,0,L2 Ω μW 2 Ω ,
≤M
2
0
0,1
α1 − α
0,1 0,1,L2 Ω
where M is independent of ft, x, gt, x, and μx.
1.16
6
Discrete Dynamics in Nature and Society
The proof of Theorem 1.4 is based on the abstract Theorem 1.1 and the symmetry
properties of the space operator generated by problem 1.14 and the following theorem on
the coercivity inequality for the solution of the elliptic differential problem in L2 Ω.
Theorem 1.5. For the solution of the elliptic differential problem
n
ar xuxr xr ωx,
x ∈ Ω,
1.17
r1
ux 0,
x ∈ S,
1.18
the following coercivity inequality holds [7]:
n
r1
uxr xr L2 Ω ≤ M||ω||L2 Ω .
1.19
2. The first order of accuracy difference scheme
Let us associate the boundary-value problem 1.1 with the corresponding first order of
accuracy difference scheme
−τ −2 uk1 − 2uk uk−1 Auk gk ,
gk gtk ,
tk kτ, 1 ≤ k ≤ N − 1,
τ −1 uk − uk−1 − Auk−1 fk ,
tk−1 k − 1τ,
fk ftk−1 ,
2.1
− N 1 ≤ k ≤ 0,
uN u−N μ,
u1 − u0 u0 − u−1 .
A study of discretization, over time only, of the nonlocal boundary value problem also
permits one to include general difference schemes in applications if the differential operator
in space variables, A, is replaced by the difference operators Ah that act in the Hilbert spaces
Hh and are uniformly self-adjoint positive definite in h for 0 < h ≤ h0 .
Let P P τA I τA−1 . Then the following estimates are satisfied 8:
P k H → H ≤ M1 δτ−k ,
Aβ P kr − P k H → H ≤ M
kτAP k H → H ≤ M,
rτα
kταβ
,
k ≥ 1, δ > 0,
1 ≤ k < k r ≤ N, 0 ≤ α, β ≤ 1.
2.2
2.3
Furthermore, for a self-adjoint positive definite operator A it follows that the operator R I
τB−1 is defined on the whole space H, it is a bounded operator, and the following estimates
A. Ashyralyev and O. Gercek
7
hold:
Rk H → H ≤ M1 δτ−k ,
Bβ Rkr − Rk H → H ≤ M
Here B 1/2τA kτBRk H → H ≤ M,
rτα
2.4
1 ≤ k < k r ≤ N, 0 ≤ α, β ≤ 1.
,
kταβ
k ≥ 1, δ > 0,
2.5
A4 τ 2 A. From 2.2 and 2.4, it follows that
||I − R2N −1 ||H → H ≤ M,
2.6
I I τAI 2τA−1 R2N−1 B−1 AI 2τA−1 I − R2N−1 −2I τBI 2τA−1 RN P N−1 −1 ≤ M.
2.7
H →H
Theorem 2.1. For any gk , 1 ≤ k ≤ N − 1 and fk , − N 1 ≤ k ≤ 0, the solution of problem 2.1
exists and the following formulas hold:
2N −1
uk I − R
Rk − R2N−k u0
R
N−k
−R
Nk
0
P u0 − τ
N
P
sN
fs μ
s−N1
− R
N−k
−R
Nk
−1
I τB2I τB B
−1
N−1
R
N−s
−R
Ns
2.8
gs τ
s1
I τB2I τB−1 B−1
N−1
|k−s|
− Rks gs τ,
R
1 ≤ k ≤ N,
s1
uk P −k u0 − τ
0
P s−k fs ,
sk1
−1
u0 Tτ I 2τA I τA
−N ≤ k ≤ 0,
2.9
2 τBR
N
−τ
0
P
sN
fs μ
s−N1
−R
N−1
B
−1
N−1
R
−R
N−s
Ns
gs τ
s1
I − R
2N
B
−1
N−1
s−1
R
s1
gs τ − I − R
2N
−1
I τBB P f0 ,
2.10
where
−1
Tτ I I τAI 2τA−1 R2N−1 B−1 AI 2τA−1 I −R2N−1 −2I τBI 2τA−1 RN P N−1 .
2.11
8
Discrete Dynamics in Nature and Society
Proof. By 8, 9,
2N −1
uk I − R
Rk − R2N−k ξ RN−k − RNk ψ
− R
N−k
−R
Nk
−1
I τB2I τB B
−1
N−1
R
N−s
−R
Ns
gs τ
2.12
s1
I τB2I τB−1 B−1
N−1
R|k−s| − Rks gs τ,
1 ≤ k ≤ N,
s1
is the solution of the boundary value difference problem
−τ −2 uk1 − 2uk uk−1 Auk gk ,
gk gtk ,
tk kτ,
u0 ξ,
2.13
uN ψ,
0
uk P −k ξ − τ
1 ≤ k ≤ N − 1,
P s−k fs ,
−N ≤ k ≤ 0
2.14
sk1
is the solution of the inverse Cauchy problem
τ −1 uk − uk−1 − Auk−1 fk ,
tk−1 k − 1τ,
fk ftk−1 ,
2.15
− N 1 ≤ k ≤ 0, u0 ξ.
Exploiting 2.12, 2.14, and the formulas
ψ u−N μ,
ξ u0 ,
2.16
we obtain formulas 2.8 and 2.9. For u0 , using 2.8, 2.9, and the formula
u1 − u0 u0 − u−1 ,
2.17
we obtain the operator equation
2N −1
I − R
R − R2N−1 u0 RN−1 − RN1 × P u0 − τ
N
0
P
sN
fs μ
s−N1
− R
N−1
−R
N1
−1
I τB2I τB B
−1
N−1
R
N−s
−R
Ns
gs τ
s1
I τB2I τB−1 B−1
N−1
Rs−1 − R1s gs τ 2u0 − P u0 τP f0 .
s1
2.18
A. Ashyralyev and O. Gercek
9
The operator
I I τAI 2τA−1 R2N−1 B−1 AI 2τA−1 I − R2N−1 − 2I τBI 2τA−1 RN P N−1
2.19
has an inverse
−1
Tτ II τAI 2τA−1 R2N−1 B−1 AI 2τA−1 I −1R2N−1 −2I τBI 2τA−1 RN P N−1 ,
2.20
and the following formula
u0 Tτ I τAI 2τA
−1
2 τBR
0
−τ
N
P
sN
fs μ
s−N1
−R
N−1
B
−1
N−1
R
N−s
−R
Ns
2.21
gs τ
s1
I − R
2N
B
−1
N−1
s−1
R
gs τ − I − R
2N
−1
I τBB P f0
s1
is satisfied. This concludes the proof of Theorem 2.1.
b
Let Fτ H Fa, bτ , H be the linear space of mesh functions ϕτ {ϕk }N
Na defined on
a, bτ {tk kh, Na ≤ k ≤ Nb , Na τ a, Nb τ b} with values in the Hilbert space H.
α
α
−1, 1τ , H, C0,1
−1, 0τ , H, C0α 0, 1τ ,
Next on Fτ H we denote by Ca, bτ , H and C0,1
H0 < α < 1 Banach spaces with the norms
ϕτ Ca,bτ ,H max ϕk H ,
Na ≤k≤Nb
τ
α
ϕτ C0,1
−1,1τ ,H ϕ C−1,1τ ,H sup
−N≤k<kr≤0
ϕkr − ϕk E
1≤k<kr≤N−1
ϕτ C0α −1,0τ ,H ϕτ C−1,0τ ,H τ
α
ϕτ C0,1
0,1τ ,H ϕ C0,1τ ,H ϕkr − ϕk E
sup
k rτα N − kα
,
rα
−k
,
rα
ϕkr − ϕk E
k rτα N − kα
.
rα
−N≤k<kr≤0
1≤k<kr≤N−1
2.22
α
ϕkr − ϕk E
sup
sup
−kα
rα
The nonlocal boundary value problem 2.1 is said to be stable in F−1, 1τ , H if we have
the inequality
uτ F−1,1τ ,H ≤ Mf τ F−1,0τ ,H g τ F0,1τ ,H μH ,
where M is independent of not only f τ , g τ , μ but also τ.
2.23
10
Discrete Dynamics in Nature and Society
Theorem 2.2. The nonlocal boundary value problem 2.1 is stable in C−1, 1τ , H norm.
Proof. By 9,
{uk }0−N C−1,0τ ,H
≤ Mf τ C−1,0τ ,H u0 H 2.24
for the solution of the inverse Cauchy difference problem 2.15 and
{uk }N−1
1
C0,1τ ,H
≤ Mg τ C0,1τ ,H u0 H uN H 2.25
for the solution of the boundary value problem 2.13. The proof of Theorem 2.2 is based on
the stability inequalities 2.24, 2.25, and on the estimates
u0 H ≤ Mf τ C−1,0τ ,H g τ C0,1τ ,H μH ,
uN H ≤ Mf τ C−1,0τ ,H g τ C0,1τ ,H μH 2.26
for the solution of the boundary value problem 2.1. Estimates 2.26 are derived from
formula 2.10 and estimates 2.2, 2.4, 2.7. This concludes the proof of Theorem 2.2.
The nonlocal boundary value problem 2.1 is said to be coercively stable well-posed
in F−1, 1τ , H if we have the coercive inequality
−2
{τ uk1 − 2uk uk−1 }N−1 1
F0,1τ ,H
−1
0
{τ uk − uk−1 }−N1 F−1,0
τ
N−1 }
{Au
k
−N ,H
F−1,1τ ,H
2.27
≤ Mf τ F−1,0τ ,H g τ F0,1τ ,H AμH ,
where M is independent of not only f τ , g τ , μ but also τ.
Since the nonlocal boundary value problem 1.1 in the space C0, 1, H of
continuous functions defined on −1, 1 and with values in H is not well-posed for the general
positive unbounded operator A and space H, then the well-posedness of the difference
nonlocal boundary value problem 2.1 in C−1, 1τ , H norm does not take place uniformly
with respect to τ > 0. This means that the coercive norm
N−1
uτ Kτ E {τ −2 uk1 − 2uk uk−1 }1
0
C0,1τ ,H
{τ −1 uk − uk−1 }−N1 C−1,0τ ,H {Auk }N−1
−N C−1,1τ ,H
2.28
tends to ∞ as τ → 0 . The investigation of the difference problem 2.1 permits us to establish
the order of growth of this norm to ∞.
A. Ashyralyev and O. Gercek
11
Theorem 2.3. Assume that μ ∈ DA and f0 ∈ DI τB. Then for the solution of the difference
problem 2.1 we have the almost coercivity inequality
uτ Kτ E ≤ MAμH I τBf0 H
1
min ln , 1 | ln AH → H | f τ C−1,0τ ,H g τ C0,1τ ,H ,
τ
2.29
where M is independent of not only f τ , g τ , μ but also τ.
Proof. By 9,
0
{τ −1 uk − uk−1 }−N1 C−1,0τ ,H {Auk }0−N C−1,0τ ,H
1
τ
≤ M min ln , 1 | ln AH → H | f C−1,0τ ,H Au0 H
τ
2.30
for the solution of the inverse Cauchy difference problem 2.15 and
N−1 −2
{Auk }N−1
{τ uk1 − 2uk uk−1 }1 1
C0,1τ ,H
C0,1τ ,H
1
≤ M min ln , 1 | ln AH → H | g τ C0,1τ ,H Au0 H AuN H
τ
2.31
for the solution of the boundary value problem 2.13. Then the proof of Theorem 2.3 is based
on the almost coercivity inequalities 2.30, 2.31, and on the estimates
Au0 H ≤ MAμH I τBf0 H
1
min ln , 1 | ln AH → H | f τ C−1,0τ ,H g τ C0,1τ ,H ,
τ
AuN H ≤ MAμH I τBf0 H
1
min ln , 1 | ln AH → H | f τ C−1,0τ ,H g τ C0,1τ ,H τ
2.32
for the solution of the boundary value problem 2.1. The proof of these estimates follows the
scheme of papers 8, 9 and relies on formula 2.10 and on estimates 2.2, 2.4, and 2.7.
This concludes the proof of Theorem 2.3.
Theorem 2.4. Let the assumptions of Theorem 2.3 be satisfied. Then the boundary value problem 2.1
α
−1, 1τ , H and the following coercivity inequality holds:
is well-posed in a Hölder space C0,1
N−1
α
{τ −2 uk1 − 2uk uk−1 }1 C0,1
0,1τ ,H
0
{Auk }N−1
{τ −1 uk − uk−1 }−N1 C0α −1,0τ ,H
−N α
C0,1 −1,1τ ,H
1
τ
τ
α
α
f C0 −1,0τ ,H g C0,1 0,1τ ,H ,
≤ M AμH I τBf0 H α1 − α
where M is independent of not only f τ , g τ , μ but also τ and α.
2.33
12
Discrete Dynamics in Nature and Society
Proof. By 8, 9,
0
−1
{τ uk − uk−1 }−N1 C0α −1,0τ ,H
{Auk }0−N Cα −1,0
1
f τ C0α −1,0τ ,H Au0 H
≤M
α1 − α
τ ,H
0
2.34
for the solution of the inverse Cauchy difference problem 2.15 and
N−1 −2
{τ uk1 − 2uk uk−1 }1 α
C0,1
0,1τ ,H
{Auk }N−1
1
α
C0,1
0,1τ ,H
1
α
g τ C0,1
≤M
0,1τ ,H Au0 H AuN H
α1 − α
2.35
for the solution of the boundary value problem 2.13. Then the proof of Theorem 2.4 is based
on the coercivity inequalities 2.34, 2.35, and on the estimates
1
α
f τ C0α −1,0τ ,H g τ C0,1
,
0,1τ ,H
α1 − α
1
τ
τ
α
α
≤ M AμH I τBf0 H f C0 −1,0τ ,H g C0,1 0,1τ ,H
α1 − α
2.36
Au0 H ≤ M AμH I τBf0 H AuN H
for the solution of the boundary value problem 2.1. Estimates 2.36 are derived from the
formulas
Au0 Tτ I 2τA−1 I τA
×
2 τBR
N
−τ
−R
AB
−2
AP
sN
fs − f−N1 Aμ
s−N1
N−1
0
N−1
N−s
BR
gs − gN−1 τ s1
I − R
2N
AB
−2
N−1
s−1
BR
N−1
Ns
BR
g1 − gs τ
s1
gs − g1 τ
s1
Tτ I 2τA−1 I τA
2 τBRN P N − If−N1
− RN−1 AB−2 I − RN−1 gN−1 − RN−2 − R2N−1 g1
I −R2N AB−2 I −RN−1 g1 −I −R2N I τBB−1 AP f0 ,
A. Ashyralyev and O. Gercek
13
AuN P N Tτ I 2τA−1 I τA
×
2 τBR
N
−τ
−R
AB
−2
AP
sN
fs − f−N1 Aμ
s−N1
N−1
0
N−1
N−s
BR
gs − gN−1 τ s1
I − R
2N
AB
−2
N−1
N−1
s−1
BR
Ns
BR
g1 − gs τ
s1
gs − g1 τ
s1
−τ
0
AP sN fs − f−N1 Aμ P N − If−N1
s−N1
P N {Tτ I 2τA−1 I τA{{2 τBRN P N − If−N1
− RN−1 AB−2 {I − RN−1 gN−1 − RN−2 − R2N−1 g1 }}
I − R2N AB−2 I − RN−1 g1 − I − R2N I τBB−1 AP f0 }}
2.37
for the solution of problem 2.1 and estimates 2.2, 2.4, and 2.7. This concludes the proof
of Theorem 2.4.
Now, the applications of this abstract result to the approximate solution of the
mixed boundary value problem for the elliptic-parabolic equation 1.14 are considered. The
discretization of problem 1.14 is carried out in two steps. In the first step, the grid sets
h {x xm h1 m1 , . . . , hn mn , m m1 , . . . , mn , 0 ≤ mr ≤ Nr , hr Nr 1, r 1, . . . , n},
Ω
h ∩ Ω,
Ωh Ω
h ∩ S
Sh Ω
2.38
are defined. To the differential operator A generated by problem 1.14 we assign the
difference operator Axh by the formula
Axh uhx −
n r1
ar xuhxr
xr ,jr
2.39
acting in the space of grid functions uh x, satisfying the conditions uh x 0 for all x ∈ Sh .
With the help of Axh we arrive at the nonlocal boundary-value problem
−
d2 uh t, x
Axh uh t, x g h t, x,
dt2
duh t, x
− Axh uh t, x f h t, x,
dt
0 < t < 1, x ∈ Ωh ,
−1 < t < 0, x ∈ Ωh ,
14
Discrete Dynamics in Nature and Society
h,
x∈Ω
uh −1, x uh 1, x μh x,
duh 0, x duh 0−, x
,
dt
dt
uh 0, x uh 0−, x,
h
x∈Ω
2.40
for an infinite system of ordinary differential equations.
In the second step problem 2 is replaced by the difference scheme 2.1:
−
uhk1 x − 2uhk x uhk−1 x
τ2
gkh x g h tk , x,
Axh uhk x gkh x,
tk kτ, 1 ≤ k ≤ N − 1,
uhk x − uhk−1 x
τ
fkh x f h tk , x,
Nτ 1, x ∈ Ωh ,
− Axh uhk−1 x fkh x,
tk−1 k − 1τ,
2.41
−N 1 ≤ k ≤ −1, x ∈ Ωh ,
uh−N x uhN x μh x,
h,
x∈Ω
h.
x∈Ω
uh1 x − uh0 x uh0 x − uh−1 x,
Based on the number of corollaries of the abstract theorems given above, to formulate the
h of all the grid functions ϕh x result, one needs to introduce the space L2h L2 Ω
{ϕh1 m1 , . . . , hn mn } defined on Ωh , equipped with the norm
ϕ L2 Ω h h
1/2
2
|ϕ x| h1 · · · hn
h
2.42
.
x∈Ωh
Theorem 2.5. Let τ and |h| h21 · · · h2n be sufficiently small numbers. Then the solutions of
the difference scheme 2.41 satisfy the following stability and almost coercivity estimates:
h N−1 {uk }−N C−1,1τ ,L2h −1
≤ M {fkh }−N1 C−1,0τ ,L2h N−1 −2 h
τ uk1 − 2uhk uhk−1 1 C0,1τ ,L2h 0
τ −1 uhk − uhk−1 −N1 C−1,0τ ,L2h ≤ M μh W 2 τf0h W 1 ln
2h
2h
N−1 {gkh }1 C0,1τ ,L2h μh L2h ,
N−1 {uhk }−N 2
C−1,1τ ,W2h
1 N−1 h −1 ,
{gkh }1 {fk }−N1 C−1,0τ ,L2h C0,1τ ,L2h τ |h|
where M is independent of τ, h, μh x, and gkh x, 1 ≤ k ≤ N − 1, fkh , −N 1 ≤ k ≤ 0.
2.43
A. Ashyralyev and O. Gercek
15
The proof of Theorem 2.5 is based on the abstract Theorems 2.2, 2.3, on the estimate
min
1
ln , 1 | ln Axh L2h → L2h |
τ
≤ M ln
1
τ |h|
2.44
as well as the symmetry properties of the difference operator Axh defined by formula 2.39
in L2h , along with the following theorem on the coercivity inequality for the solution of the
elliptic difference problem in L2h .
Theorem 2.6. For the solution of the elliptic difference problem,
Axh uh x ωh x,
uh x 0,
x ∈ Ωh ,
2.45
x ∈ Sh ,
2.46
≤ M||ωh ||L2h .
2.47
the following coercivity inequality holds [7]:
n h
u xr xr ,jr r1
L2h
Theorem 2.7. Let τ and |h| be sufficiently small numbers. Then the solutions of the difference scheme
2.41 satisfy the following coercivity stability estimates:
N−1 −2 h
τ uk1 − 2uhk uhk−1 1 α
C0,1
0,1τ ,L2h 0
τ −1 uhk − uhk−1 −N1 C0α −1,0τ ,L2h ≤ M μh W 2 τf0h W 1 2h
2h
N−1 {uhk }−N α
2
C0,1
−1,1τ ,W2h
1
N−1 h −1 ,
{gkh }1 α
{fk }−N1 α
C0 −1,0τ ,L2h C0,1 0,1τ ,L2h α1 − α
2.48
where M is independent of τ, h, μh x, and gkh x, 1 ≤ k ≤ N − 1, fkh , −N 1 ≤ k ≤ 0.
The proof of Theorem 2.7 is based on the abstract Theorem 2.4, the symmetry
properties of the difference operator Axh defined by formula 2.39, and on Theorem 2.6 on
the coercivity inequality for the solution of the elliptic difference equation 2.45 in L2h .
Note that in a similar manner the difference schemes of the first order of accuracy with
respect to one variable for approximate solutions of the boundary value problem 1.12 can
be constructed. Abstract theorems given above permit us to obtain the stability, the almost
stability, and the coercive stability estimates for the solution of these difference schemes.
Acknowledgment
The authors would like to thank Professor P. E. Sobolevskii Jerusalem, Israel for his helpful
suggestions to the improvement of this paper.
16
Discrete Dynamics in Nature and Society
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