Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 454831, 22 pages
doi:10.1155/2012/454831
Research Article
FDM for Elliptic Equations with
Bitsadze-Samarskii-Dirichlet Conditions
Allaberen Ashyralyev1, 2 and Fatma Songul Ozesenli Tetikoglu1
1
2
Department of Mathematics, Fatih University, 34500 Istanbul, Turkey
Department of Mathematics, ITTU, 74400 Ashgabat, Turkmenistan
Correspondence should be addressed to Fatma Songul Ozesenli Tetikoglu, ftetikoglu@fatih.edu.tr
Received 8 April 2012; Accepted 6 May 2012
Academic Editor: Ravshan Ashurov
Copyright q 2012 A. Ashyralyev and F. S. Ozesenli Tetikoglu. 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.
A numerical method is proposed for solving nonlocal boundary value problem for the
multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet
condition. The first and second-orders of accuracy stable difference schemes for the approximate
solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity,
and almost coercivity inequalities for solution of these schemes are established. The theoretical
statements for the solutions of these nonlocal elliptic problems are supported by results of
numerical examples.
1. Introduction
Many problems in fluid mechanics, dynamics, elasticity, and other areas of engineering,
physics, and biological systems lead to partial differential equations of elliptic type. 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
−
d2 ut
Aut ft,
dt2
0 < t < 1,
u 0 ϕ,
u 1 βu λ ψ, β ≤ 1, 0 ≤ λ < 1
1.1
2
Abstract and Applied Analysis
for elliptic differential equations in a Hilbert space H with self-adjoint positive definite
operator A. It is known see, e.g., 3–11 that various nonlocal boundary value problems for
elliptic equations can be reduced to the boundary value problem 1.1. The simply nonlocal
boundary value problem was presented and investigated for the first time by Bitsadze
and Samarskii 12. Further, methods of solutions of Bitsadze-Samarskii nonlocal boundary
value problems for elliptic differential equations have been studied extensively by many
researchers see 13–21 and the references given therein.
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. Derivatives at the
endpoints of the segment are understood as the appropriate unilaterial derivatives.
ii The element ut belongs to DA for all t ∈ 0, 1, and the function Aut is
continuous on 0, 1.
iii ut satisfies the equation and nonlocal boundary condition in 1.1.
Let Ω be the open unit cube in Rn x x1 , . . . , xn : 0 < xk < 1, 1 ≤ k ≤ n with
boundary S, Ω Ω ∪ S. In present paper, we are interested in studying the stable difference
schemes for the numerical solution of the following nonlocal boundary value problem for the
multidimensional elliptic equation
−utt −
n
r1
ar xuxr xr δu ft, x,
0 < t < 1,
x x1 , . . . , xn ∈ Ω,
ut 0, x ϕx,
ut 1, x βut λ, x ψx,
ut, x 0,
1.2
x ∈ Ω, β ≤ 1, 0 ≤ λ < 1,
0 ≤ t ≤ 1, x ∈ S, S ∂Ω.
Here ψx, ϕx x ∈ Ω, and ft, x t ∈ 0, 1, x ∈ Ω are given smooth functions, δ is a
large positive constant and ar x ≥ a > 0.
In the present paper, the first and second-orders of accuracy difference schemes are
presented for the approximate solution of problem 1.2. The stability and coercive stability
estimates for the solution of these difference schemes are obtained. A numerical method is
proposed for solving nonlocal boundary value problem for the multidimensional elliptic
partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. A procedure of
modified Gauss elimination method is used for solving these difference schemes in the case
of two-dimensional elliptic partial differential equations.
Abstract and Applied Analysis
3
2. Difference Schemes: Well-Posedness
The discretization of problem 1.2 is carried out in two steps. In the first step let us define
the grid sets as follows:
h {x : x xm h1 m1 , . . . , hn mn , m m1 , . . . , mn ,
Ω
0 ≤ mr ≤ Nr , hr Nr L, r 1, . . . , n}
h ∩ Ω,
Ωh Ω
2.1
h ∩ S.
Sh Ω
h of the grid functions ϕh x {ϕh1 m1 , . . . ,
We introduce the Hilbert space L2h L2 Ω
hn mn } defined on Ωh , equipped with the norm
h
ϕ ⎞1/2
2
⎝
ϕh x h1 · · · hn ⎠
⎛
h
L2 Ω
2.2
x∈Ωh
h defined on Ω
h , equipped with the norm
and the Hilbert space W22 Ω
h
ϕ ⎛
h
W22 Ω
⎞1/2
2
⎝
ϕh x h1 · · · hn ⎠
h
x∈Ω
⎛
⎞1/2 ⎛
⎞1/2
n n 2
2
⎝
ϕhxr ,mr h1 · · · hn ⎠ ⎝
ϕhxr xr ,mr h1 · · · hn ⎠ .
h r1
x∈Ω
2.3
h r1
x∈Ω
Finally, we introduce the Banach spaces C0, 1τ , L2h and Cα 0, 1τ , L2h of grid abstract
N−1
function {ϕhk x}1 defined on 0, 1τ with values L2h , equipped with the following norms:
h N−1 ϕ
k 1 C0,1τ ,L2h h N−1 ϕ
k
1
Cα 0,1τ ,L2h max ϕhk 1≤k≤N−1
max ϕhk 1≤k≤N−1
L2h
L2h
sup
1≤k<kr≤N−1
,
ϕkr − ϕk L2h
rτα
2.4
.
To the differential operator A generated by the problem 1.2 we assign the difference
operator Axh by the formula
Axh uh −
n r1
ar xuhxr
xr ,mr
δuhxr
2.5
acting in the space of grid functions uh x, satisfying the condition uh x 0 for all x ∈ Sh .
h . With the help of
It is known that Axh is a self-adjoint positive definite operator in L2 Ω
4
Abstract and Applied Analysis
Axh , we arrive at the nonlocal boundary value problem for an infinite system of the following
ordinary differential equations:
−
d2 uh t, x
Axh uh t, x f h t, x,
dt2
uht 0, x
ϕ x,
h
uht 1, x
0 < t < 1, x ∈ Ωh ,
βuht λ, x
ψ x,
h
2.6
h.
x∈Ω
In the second step, we replaced problem 2.6 by the first-order of accuracy difference scheme
as follows:
−
uhk1 x − 2uhk x uhk−1 x
τ2
fkh x f h tk , x,
Axh uhk x fkh x,
tk kτ, 1 ≤ k ≤ N − 1, Nτ 1, x ∈ Ωh ,
uh1 x − uh0 x
ϕh x,
τ
uλ/τ1 x − uλ/τ x
uhN x − uhN−1 x
β
ψ h x,
τ
τ
h
2.7
h,
x∈Ω
h
h
x∈Ω
and the second order of accuracy difference scheme as follows:
−
uhk1 x − 2uhk x uhk−1 x
fkh x f h tk , x,
τ2
Axh uhk x fkh x,
tk kτ, 1 ≤ k ≤ N − 1, Nτ 1, x ∈ Ωh ,
−3uh0 x 4uh1 x − uh2 x
h,
ϕh x, x ∈ Ω
2τ
⎡
uhλ/τ−1 x − 4uhλ/τ x 3uhλ/τ1 x
uhN−2 x − 4uhN−1 x 3uhN x
⎣
β
2τ
2τ
⎤
λ ⎦
−
τ
τ
uhλ/τ1 x − 2uhλ/τ x uhλ/τ−1 x λ
ψ h x,
τ
h.
x∈Ω
2.8
Now, we will study the well-posedness of 2.7 and 2.8. We have the following theorem on
stability of 2.7 and 2.8.
Abstract and Applied Analysis
5
Theorem 2.1. Let τ and |h| be sufficiently small positive numbers. Then the solutions of difference
schemes 2.7 and 2.8 satisfy the following stability estimate:
max uhk L2h
1≤k≤N
≤ M1 ϕh L2h
ψ h L2h
max fkh L2h
1≤k≤N−1
2.9
,
where M1 does not depend on τ, h, ψ h x, ϕh x and fkh x, 1 ≤ k ≤ N − 1.
Proof. The proof of 2.9 is based on the following formula:
uhk x
I −R
2N
−1 Rk − R2N−k uh0 x RN−k − RNk uhN x
− R
N−k
−R
Nk
2I −1 x −1
Bh
τBhx
N−1
N−1−i
R
−R
N−1i
for k 1, . . . , N − 1,
i1
2.10
where
−1 x −1
uh0 x Pτ I τBhx 2I τBhx
Bh
N−1
RN−i − RNi fih xτ
× I RRN−2
i1
I R I R2N−2 − β RN−λ/τ−1 RNλ/τ
λ/τ−1
Ri−1 fih xτ − β RN RN−1
Rλ/τ−i fih xτ
N−1
i1
i1
N−1
Ri−λ/τ fih xτ
β RN−2 RN−1
iλ/τ1
β R R
N
N−1
N−1
h
Rλ/τi fih xτ − β RN−1 RN fλ/τ
xτ
i1
− Pτ I − R−1 I R2N−1 − β RN−λ/τ−1 RNλ/τ ϕh xτ
Pτ I − R−1 RN−1 RN ψ h xτ,
fih xτ
i1
−1 x −1 N−1
R|k−i|−1 − Rki−1 fih xτ
Bh
2I τBhx
×
6
Abstract and Applied Analysis
−1 x −1
uhN x Pτ I τBhx 2I τBhx
Bh
N−1
RN−i − RNi fih xτ
× R−1 I R − β RN−λ/τ−1 RNλ/τ−1
i1
λ/τ−1
Rλ/τ−i fih xτ β I R2N−1 R−1
− β I R2N−1
i1
N−1
Ri−λ/τ fih xτ
iλ/τ1
N−1
h
β I R2N−1
Rλ/τi fih xτ − β I R2N−1 fλ/τ
xτ
i1
N−1
Ri−1 fih xτ
I R RN RN−1 − β Rλ/τ R2N−λ/τ−1
i1
− Pτ I − R RN R2N−1 − β Rλ/τ R2N−λ/τ−1 ϕh xτ
−1
Pτ I − R−1 I R2N−1 ψ h xτ,
−1
Pτ I − R2N−2 − β RN−λ/τ−1 RNλ/τ−1
,
2.11
for 2.7, and
−1 x −1
uh0 x Dτ I τBhx 2I τBhx
Bh
N−2
RN−i − RNi fih xτ
× I R 4R − I − R2 I − 3RRN−4 I − R2N
− I R 4R − I − R2 I − R2N
i1
× 3I − R − R2N−2 I − 3R
λ
λ
−
− β RN−λ/τ−1 I − 3R 2
I − R
τ
τ
λ
λ
−RNλ/τ−1 3I − R 2
−
I − R
τ
τ
×
i2
×
Ri−2 fih xτ − βI RRN−2 4R − I − R2 I − R2N
N−1
λ/τ−1
λ
λ
−
Rλ/τ−i−1 fih xτ
I − 3R 2
I − R
τ
τ
i1
N−1
λ
λ
3I − R 2
−
Ri−λ/τ−1 fih xτ
I − R
τ
τ
iλ/τ2
N−1
λ
λ
λ/τi−1 h
−
R
fi xτ
− I − 3R 2
I − R
τ
τ
i1
Abstract and Applied Analysis
7
λ
λ
−
− β I − R2N I RRN−2 4R − R2 − I 4 4
τ
τ
h
h
× fλ/τ1
xτ − fλ/τ
xτ − I R4I − R I − R2N
× 3I − R − R2N−2 I − 3R
λ
λ
− β RN−λ/τ−1 3I − R 2
−
I − R
τ
τ
λ
λ
−
−RNλ/τ−1 I − 3R 2
I − R
τ
τ
× f1h xτ − I RRN−1 4R − R2 − I I − R2N 4I R2N−3 I − 3R
h
×fN−1
xτ
− Dτ I − R2N I − R−1 I RRN−2 4R − I − R2 2τψ h x
Dτ I − R2N I − R−1
× 3I − R − R2N−2 I − 3R
λ
λ
N−λ/τ−1
−β R
−
3I − R 2
I − R
τ
τ
λ
λ
Nλ/τ−1
−R
−
I − 3R 2
2τϕh x,
I − R
τ
τ
uhN x
−1 x −1
τBhx
Bh
2.12
2I Dτ I ×
I − R2N I R 4R − I − R2 R−2 R − 3I
τBhx
λ
λ
−
β R − 3I 3I − R 2
I − R RN−λ/τ−1
τ
τ
λ
λ
−
−3R − I I − 3R 2
I − R RNλ/τ−1
τ
τ
N−2
RN−i − RNi fih xτ I R I − R2N 4R − I − R2
×
i1
× I RRN−2 4R − I − R2
λ
λ
−
− β Rλ/τ−1 I − 3R 2
I − R
τ
τ
N−1
λ
λ
2N−λ/τ−1
−
−R
Ri−2 fih xτ
3I − R 2
I − R
τ
τ
i2
8
Abstract and Applied Analysis
β I − R2N R − 3I R2N−2 I − 3R
λ/τ−1
λ
λ
−
Rλ/τ−i−1 fih xτ
I − 3R 2
I − R
τ
τ
i1
×
N−1
λ
λ
3I − R 2
−
Ri−λ/τ−1 fih xτ
I − R
τ
τ
iλ/τ2
N−1
λ
λ
λ/τi−1 h
−
R
fi xτ
− I − 3R 2
I − R
τ
τ
i1
β R − 3I R2N−2 3I − R I − R2N
λ
λ
h
h
−
× 44
I − R fλ/τ1
xτ − fλ/τ
xτ
τ
τ
I R I − R2N
× I RRN−2 I − 4R R2
λ
λ
−
− β Rλ/τ−1 I − 3R 2
I − R
τ
τ
λ
λ
−R2N−λ/τ−1 3I − R 2
−
f1h xτ
I − R
τ
τ
R I − R2N−1 R − 4I R2N−4 I R2 − 3R 3I − R − R2 4I − R
λ
λ
−
− β RN−λ/τ−1 3I − R 2
I − R
τ
τ
× 3R − I R2N 3I − R
λ
λ
−R
−
I − 3R 2
I − R
τ
τ
2N
h
fN−1 xτ
× 3I − R R I − 3R
Nλ/τ−1
Dτ I − R2N I − R−1 R − 3I R2N I − 3R 2τϕh x
− Dτ I − R2N I − R−1
× RN−2 I R R2 − 4R I
λ
λ
−
I − 3R 2
−β R
I − R
τ
τ
λ
λ
2N−λ/τ−1
−R
−
2τψ h x,
3I − R 2
I − R
τ
τ
λ/τ−1
Abstract and Applied Analysis
9
−1
Dτ I − R2N
× −3I − R2 I − 3R2
λ
λ
Nλ/τ−3
−
− β −R
I − R
I − 3R I − 3R 2
τ
τ
−1
λ
λ
N−λ/τ−1
−
−R
,
I − R
3I − R 3I − R 2
τ
τ
−1
R I τBhx ,
2
1
τAxh 4Axh τ 2 Axh
Bhx 2
2.13
for 2.8, and the symmetry properties of the difference operator Axh defined by the formula
2.5.
Difference schemes 2.7 and 2.8 are ill-posed in C0, 1τ , L2h . We have the
following theorem on almost coercive stability.
Theorem 2.2. Let τ and |h| be sufficiently small positive numbers. Then the solutions of difference
schemes 2.7 and 2.8 satisfy the following almost coercive stability estimate:
h
u − 2uh uh k1
k
k−1 max 1≤k≤N−1
τ2
≤ M2 ϕh 2
W2h
max uhk 2
W2h
1≤k≤N−1
L2h
ψ h 2
W2h
1
max fkh ln
,
L2h
τ |h| 1≤k≤N−1
2.14
where M2 does not depend on τ, h, ψ h x, ϕh x, and fkh x, 1 ≤ k ≤ N − 1.
Proof. The proof of 2.14 is based on the formulas 2.11, 2.12, 2.13, the symmetry
properties of the difference operator Axh defined by the formula 2.5 and on the following
theorem on well-posedness of the elliptic difference problem.
Theorem 2.3. For the solutions of the elliptic difference problem
Axh uh x wh x,
uh x 0,
x ∈ Ωh ,
2.15
x ∈ Sh
the following coercivity inequality holds (see [21]):
n h u
xr xr , m r r1
where M does not depend on h and wh x.
L2h
≤ Mwh L2h
,
2.16
10
Abstract and Applied Analysis
Theorem 2.4. Let ϕh x ψ h x 0. Then the difference problems 2.7 and 2.8 are well-posed in
Hölder spaces Cα 0, 1τ , L2h and the following coercivity inequality holds:
N−1 h
uk1 − 2uhk uhk−1
max 2
1≤k≤N−1
τ
1
h N−1 u
k
Cα 0,1τ , L2h 1
2
Cα 0,1τ , W2h
h N−1 M3
f
max ≤
.
α1 − α 1≤k≤N−1 k 1 Cα 0,1τ ,L2h Here M3 does not depend on τ, h, and fkh x, 1 ≤ k ≤ N − 1.
Proof. The proof of 2.17 is based on the formula
−1
Axh uh0 x − f1h x Pτ I τBhx 2I τBhx I R
N−1
h
× RN−1
τBhx RN−i fih − fN−1
i1
− RN−1
τBhx RNi fih x − f1h x
N−1
i1
−βRN
h
τBhx Rλ/τ−i fih x − fλ/τ
x
λ/τ−1
!
i1
I R2N−2 − β RN−λ/τ−1 RNλ/τ
×
τBhx Ri fih x − f1h x
N−1
i1
h
τBhx Ri−λ/τ fih x − fλ/τ
x
N−1
βRN
iλ/τ1
βRN
τBhx Rλ/τi fih x − f1h x
N−1
i1
h
RN−1 I − RN−1 fN−1
x
h
β RNλ/τ−1 − R2N−λ/τ−1 − τBhx RN fλ/τ
x
R2N−2 − R2N−1 R2N−2 − R3N−3 R3N−2 − RN−1
β R2N−λ/τ−2 R2Nλ/τ−1
R
2Nλ/τ
−R
Nλ/τ−1
f1h x
2.17
Abstract and Applied Analysis
11
− Pτ I − R−1 τ I R2N−1 − β RN−λ/τ−1 RNλ/τ Axh ϕh
Pτ I − R−1 τ RN−1 RN Axh ψ h ,
−1
h
Ahx uhN x − fN−1
x Pτ I τBhx 2I τBhx
× RI R − β RN−λ/τ1 RNλ/τ1
×
h
τBhx RN−i fih x − fN−1
x
N−1
i1
− RI R − β RN−λ/τ1 RNλ/τ1
×
τBhx RNi fih x − f1h x
N−1
i1
λ/τ−1
h
− βR2 I R2N−1
τBhx Rλ/τ−i fih x − fλ/τ
x
i1
N−1
h
βR I R2N−1
τBhx Ri−λ/τ fih x − fλ/τ
x
iλ/τ1
N−1
βR2 I R2N−1
τBhx Rλ/τi fih x − f1h x
− β I R2N−1
i1
× R I − Rλ/τ−1 − I − RN−λ/τ−1 − τBhx R
h
× fλ/τ
x
I R R2N−2 − RN − I R
− β RN−λ/τ1
RNλ/τ1 − R2N−λ/τ
− R2Nλ/τ − RN−λ/τ−1
RNλ/τ−1 − RN−λ/τ RNλ/τ
h
× fN−1
x I R RN − R2N−1
β RNλ/τ RNλ/τ1 − RNλ/τ2
R2Nλ/τ1 R2Nλ/τ1 − R3Nλ/τ
− R3Nλ/τ1 Rλ/τ1 − R2N−λ/τ
3N−λ/τ−1
h
f1 x
R
12
Abstract and Applied Analysis
Pτ I − R−1 Rτ I R2N−1 Ahx ψ
− Pτ I − R−1 Rτ RN−1 RN − β Rλ/τ R2N−λ/τ−1 Ahx ϕ
2.18
for 2.7 and
−1
Axh uh0 x − f1h x Dτ I τBhx 2I τBhx
× I R 4R − I − R2 I − 3RRN−3 I − R2N
×
h
Bhx τRN−i fih x−fN−1
x
N−2
i1
N−2
−
i1
Bhx τRNi
fih x−f1h x
− I R 4R − I − R2 I − R2N
× 3I − R − R2N−2 I − 3R
λ
λ
−
− β RN−λ/τ−1 3I − R 2
I − R
τ
τ
λ
λ
−RNλ/τ−1 I − 3R 2
−
I − R
τ
τ
×
Bhx τRi−1 fih x − f1h x
N−1
i2
− βI RRN−2 4R − I − R2 I − R2N
×
λ
λ
−
I − 3R 2
I − R
τ
τ
×
h
Bhx τRλ/τ−i fih x − fλ/τ
x
λ/τ−1
i1
λ
λ
−
3I − R 2
I − R
τ
τ
×
N−1
h
Bhx τRi−λ/τ fih x − fλ/τ
x
iλ/τ2
Abstract and Applied Analysis
13
λ
λ
−
− I − 3R 2
I − R
τ
τ
N−1
x
λ/τi
h
h
fi x − f1 x
× Bh τR
i1
− β I − R2N I RRN−1 4R − R2 − I
λ
λ
h
−
× 44
τBhx fλ/τ1
x
τ
τ
β I − R2N I RRN−2 4R − R2 − I
λ
λ
−
× 2R R 2
τ
τ
λ
λ
Rλ/τ−1 I − 3R 2
−
I − R RN−λ/τ−2
τ
τ
λ
λ
h
−
× 3I − R 2
I − R fλ/τ
x
τ
τ
× I R I − R2N
× I − 3R 3R − I − R2N−5 I − R
× R2 − R2N I R R2 − 4R I
− 3I − RRN−1 R2 − 4R I − β 4R − I − R2
λ
λ
−
3I − R 2
I − R RN−λ/τ−3
τ
τ
λ
λ
−
− I − 3R 2
I − R
τ
τ
×RNλ/τ−3
f1h x
×
I R 4R − I − R2 I − R2N RN−2
× R − 3I − R
N−2
I − 3R I R I − R
N
h
fN−1
x
− Dτ I − R2N I − R−1 I RRN−2 4R − I − R2 2τAxh ψ h x
Dτ I − R2N I − R−1
14
Abstract and Applied Analysis
× 3I − R − R2N−2 I − 3R
λ
λ
−
− β RN−λ/τ 3I − R 2
I − R
τ
τ
λ
λ
−RNλ/τ−1 I − 3R 2
−
2τAxh ϕh x,
I − R
τ
τ
h
Axh RuhN x− fN−1
x
−1
Dτ I τBhx 2I τBhx
I − R2N
×
× I R 4R − I − R2 R − 3I
λ
λ
−
β R − 3I 3I − R 2
I − R
τ
τ
× RN−λ/τ1 − 3R − I
λ
λ
−
× I − 3R 2
I − R RNλ/τ1
τ
τ
"
×
N−2
i1
Bhx τRN−i
fih x
−
h
fN−1
x
I R I − R2N 4R − I − R2
−
N−2
i1
Bhx τRNi
fih x
−
× I RRN−2 −4R I R2
λ
λ
−
− β Rλ/τ−1 I − 3R 2
I − R
τ
τ
λ
λ
−R2N−λ/τ−1 3I − R 2
−
I − R
τ
τ
×
Bhx τRi fih x − f1h x
N−1
i2
βR R − 3I R2N−2 I − 3R I − R2N
×
λ
λ
−
I − 3R 2
I − R
τ
τ
×
h
Bhx τRλ/τ−i fih x − fλ/τ
x
λ/τ−1
i1
f1h x
#
Abstract and Applied Analysis
15
λ
λ
−
3I − R 2
I − R
τ
τ
N−1
×
Bhx τRλ/τi fih x − f1h x
iλ/τ2
λ
λ
−
I − 3R 2
I − R
τ
τ
N−1
x
λ/τi
h
h
fi x − f1 x
× Bh τR
i1
λ
λ
−
βR2 R − 3I R2N−2 I − 3R I − R2N 4 4
τ
τ
h
× fλ/τ−1
x − βR I − R2N
λ
λ
−
3I − R 2
× I −RR
I − R
τ
τ
λ
λ
λ/τ−1
−
I − 3R 2
R
I − R
τ
τ
h
× fλ/τ
x I − R2N
N−λ/τ
× I R 4R − R2 − I RN1
× R − 3I I − RN−2
λ
λ
−
β 3I − R 2
I − R
τ
τ
× R2N−λ/τ R2 3I − R RN−2 R3 − 4R2 I
λ
λ
−
− I − 3R 2
I − R Rλ/τ
τ
τ
× R2 3I − R I − 3RRN−3 R RN I − R
× f1h x I − R2N
× I − 3R 4R − R2 − I 3I − R R2N−2
λ
λ
−
3I − R 2
I − R RN−λ/τ−1 3I − R
τ
τ
× R I RN I − R RN−2 I − R R3N−2 2I − R
−β
λ
λ
− I − 3R 2
−
I − R RNλ/τ−1 3R − I
τ
τ
16
Abstract and Applied Analysis
× I − R − R2N−3
× 1 R − RN 3I − R2 R2N R3 − 4R2 I
×f hN−1 x
Dτ I − R2N I − R−1 R R − 3I R2N−2 I − 3R 2τAxh ϕh x
− Dτ I − R2N I − R−1 R
× RN−2 I R R2 − 4R I
λ
λ
−
− β Rλ/τ−1 I − 3R 2
I − R
τ
τ
λ
λ
−R2N−λ/τ−1 3I − R 2
−
2τAxh ψ h x
I − R
τ
τ
2.19
for 2.8, the symmetry properties of the difference operator Axh defined by the formula 2.5
and on Theorem 2.3 on well-posedness of the elliptic difference problem.
3. Numerical Analysis
We have not been able to obtain a sharp estimate for the constants figuring in the stability
inequality. Therefore, we will give the following results of numerical experiments of the
Bitsadze-Samarskii-Dirichlet problem:
−
∂2 ut, x ∂2 ut, x
−
u ft, x,
∂t2
∂x2
ft, x exp−πt sinπx,
0 < t < 1, 0 < x < 1,
0 < t < 1, 0 < x < 1,
ut 0, x −π sinπx, 0 ≤ x ≤ 1,
π
1
, x π sinπx exp −
− exp−π ,
ut 1, x ut
2
2
ut, 0 ut, 1 0,
0 ≤ x ≤ 1,
0≤t≤1
3.1
for the two-dimensional elliptic equation.
Abstract and Applied Analysis
17
The exact solution of this problem is
ut, x exp−πt sinπx.
3.2
For the approximate solution of problem 3.1, we consider the set 0, 1τ × 0, 1h of a family
of grid points depending on small parameters τ and h as follows:
0, 1τ × 0, 1h {tk , xn : tk kτ, 0 ≤ k ≤ N, Nτ 1,
xn nh, 0 ≤ n ≤ M, Mh 1}.
3.3
Applying 2.7, we present the following first-order of accuracy difference scheme for
the approximate solution of problem 3.1:
−
unk1 − 2unk unk−1
τ2
−
− 2unk un−1
un1
k
k
h2
unk exp−πtk sinπxn ,
1 ≤ k ≤ N − 1,
1 ≤ n ≤ M − 1,
un1 − un0
−π sinπxn ,
τ
0 ≤ n ≤ M,
unN/2 − unN/2−1
unN − unN−1
β
π sinπxn τ
τ
π
× exp −
− exp−π , 0 ≤ n ≤ M,
2
u0k uM
k 0,
0 ≤ k ≤ N.
3.4
We have N 1 × M 1 system of linear equations in 3.4 and we will write them
in the following matrix form:
Aun1 Bun Cun−1 Dϕn ,
u0 uM 0.
1 ≤ n ≤ M − 1,
3.5
18
Abstract and Applied Analysis
Here,
⎡
0
⎢0
⎢
⎢0
⎢
⎢
A ⎢·
⎢
⎢0
⎢
⎣0
0
0
a
0
·
0
0
0
⎡
−1
⎢c
⎢
⎢0
⎢
⎢
B⎢·
⎢
⎢0
⎢
⎣0
0
0
0
a
·
0
0
0
1
b
c
·
0
0
0
·
·
·
·
·
·
·
0
c
b
·
0
0
0
0
0
0
·
0
0
0
·
·
·
·
·
·
·
0
0
0
·
0
0
0
0
0
0
·
0
0
1
·
·
·
·
·
·
·
0
0
0
·
a
0
0
0
0
0
·
0
a
0
⎤
0
0⎥
⎥
0⎥
⎥
⎥
·⎥
,
⎥
0⎥
⎥
0⎦
0 N1×N1
0
0
0
·
0
0
−1
·
·
·
·
·
·
·
0
0
0
·
b
c
0
0
0
0
·
c
b
−1
⎤
0
0⎥
⎥
0⎥
⎥
⎥
·⎥
,
⎥
0⎥
⎥
c⎦
1 N1×N1
3.6
⎡
C A,
⎤
u0s
⎢ u1 ⎥
⎢ s ⎥
⎢
⎥
us ⎢ · ⎥
,
⎢ N−1 ⎥
⎣us ⎦
uN
s
N1×1
D IN1×N1 ,
where s n − 1, n, n 1 and
⎡
⎤
ϕ0n
⎢ ϕ1 ⎥
⎢ n ⎥
⎢
⎥
ϕn ⎢ . ⎥
,
⎢ N−1 ⎥
⎣ϕn ⎦
ϕN
n
N1×1
a
1
,
h2
b−
2
2
− 2 − 1,
2
τ
h
c
⎧
−τπ sinπxn ,
⎪
⎪
⎪
⎪
⎨
− exp−πtk sinπxn ,
ϕkn ⎪
⎪
⎪
⎪
⎩ τπ sinπxn exp − π − exp−π ,
2
1
,
τ2
3.7
k 0,
1 ≤ k ≤ N − 1,
k N.
We seek a solution of the matrix equation in following form:
un αn1 un1 βn1 ,
n M − 1, . . . , 1,
uM 0,
3.8
Abstract and Applied Analysis
19
where αn n 1, . . . , M are N 1 × N 1 square matrices and βn n 1, . . . , M are
N 1 × 1 column matrices defined by see, 17 as follows:
αn1 − B Cαn −1 A,
βn1 B Cαn −1 Dϕn − Cβn ,
3.9
n 1, . . . , M − 1,
where
α1 0N1×N1 ,
β1 0N1×1 .
3.10
Now, applying 2.8 for N even number, we can present the following second-order of
accuracy difference scheme:
−
unk1 − 2unk unk−1
τ2
−
un1
− 2unk un−1
k
k
h2
unk − exp−πtk sinπxn ,
1 ≤ k ≤ N − 1,
1 ≤ h ≤ M − 1,
−3un0 4un1 − un2
−π sinπxn ,
2τ
0 ≤ n ≤ M,
unN−2 x − 4unN−1 x 3unN x unN/2−2 − 4unN/2−1 3unN/2
2τ
2τ
π
− exp−π ,
π sinπxn exp −
2
u0k uM
k 0,
0 ≤ n ≤ M,
0≤k≤N
3.11
for the approximate solution of problem 3.1.
So, again we have N 1 × M 1 system of linear equations in 3.11 and we will
write them in the following matrix form:
Aun1 Bun Cun−1 Dϕn ,
u0 uM 0,
1 ≤ n ≤ M − 1,
3.12
20
Abstract and Applied Analysis
where
⎡
0
⎢0
⎢
⎢0
⎢
⎢
A ⎢·
⎢
⎢0
⎢
⎣0
0
⎡
−3
⎢b
⎢
⎢0
⎢
⎢
B⎢·
⎢
⎢0
⎢
⎣0
0
0
a
0
·
0
0
0
4
c
b
·
0
0
0
0
0
a
·
0
0
0
−1
b
c
·
0
0
0
·
·
·
·
·
·
·
·
·
·
·
·
·
·
0
0
0
·
0
0
0
0
0
0
·
0
0
0
0
0
0
·
0
0
0
· 0
· 0
· 0
· ·
· 0
· 0
· −1
0
0
0
·
0
0
4
0
0
0
·
0
0
−3
0
0
0
·
a
0
0
·
·
·
·
·
·
·
⎤
0
0⎥
⎥
0⎥
⎥
⎥
·⎥
,
⎥
0⎥
⎥
0⎦
0 N1×N1
0
0
0
·
0
a
0
0
0
0
·
c
b
1
0
0
0
·
b
c
−4
⎤
0
0⎥
⎥
0⎥
⎥
⎥
·⎥
,
⎥
⎥
0⎥
b⎦
3 N1×N1
3.13
C A,
D IN1×N1 ,
⎤
u0s
⎢ u1 ⎥
⎢ s ⎥
⎢
⎥
us ⎢ · ⎥
,
⎢ N−1 ⎥
⎣us ⎦
uN
s
N1×1
⎡
where s n − 1, n, n 1 and
⎡
⎤
ϕ0n
⎢ ϕ1 ⎥
⎢ n ⎥
⎢
⎥
k
.
ϕn ⎢ · ⎥
⎢ N−1 ⎥
⎣ϕn ⎦
ϕN
n
N1×1
3.14
Here,
a
1
,
h2
b
1
,
τ2
c−
2
2
− 2 − 1,
2
h
τ
⎧
⎪
⎨ −2τπ sinπxn ,
k
− exp−πtk sinπx
ϕn n ,π ⎪
⎩
− exp−π ,
2τπ sinπxn exp −
2
k 0,
1 ≤ k ≤ N − 1,
k N.
3.15
3.16
So, we have the second-order difference equation with respect to n with matrix coefficients.
To solve this difference equation, we use the same algorithm 3.8 and 3.9.
Now, we will give the results of the numerical experiments.
Abstract and Applied Analysis
21
Table 1: Comparison of the errors of difference schemes.
Difference schemes
First-order difference scheme 2.7
Second-order difference scheme 2.8
N M 20
0.05384197882300
0.00631619894867
N M 40
0.02631633782987
0.00164455475890
N M 60
0.01740639813932
7.4144589892985e -004
The errors in numerical solutions are computed by
"
N
EM
max
1≤k≤N−1
#1/2
2
k
utk , xn − un h
M−1
3.17
n1
for different values of M and N, where utk , xn represents the exact solution and ukn
represents the numerical solution at tk , xn . The results are shown in Table 1 for N M 20,
N M 40, and N M 60.
Thus, second-order of accuracy difference scheme is more accurate compared with the
first-order of accuracy difference scheme.
4. Conclusion
The first and second-orders of accuracy difference schemes for approximate solutions of
the Bitsadze-Samarskii-Dirichlet type nonlocal boundary value problem for the multidimensional elliptic partial differential equation are presented. Theorems on the stability, almost
coercive stability, and coercive stability estimates for the solution of these difference schemes
are established. Numerical experiments are given.
References
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