fdm for elliptic equations with bitsadze-samarskii-dirichlet...
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 454831, 22 pagesdoi:10.1155/2012/454831
Research ArticleFDM for Elliptic Equations withBitsadze-Samarskii-Dirichlet Conditions
Allaberen Ashyralyev1, 2 and Fatma Songul Ozesenli Tetikoglu1
1 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey2 Department of Mathematics, ITTU, 74400 Ashgabat, Turkmenistan
Correspondence should be addressed to Fatma Songul Ozesenli Tetikoglu, [email protected]
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 articledistributed 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 themultidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichletcondition. The first and second-orders of accuracy stable difference schemes for the approximatesolution of this nonlocal boundary value problem are presented. The stability estimates, coercivity,and almost coercivity inequalities for solution of these schemes are established. The theoreticalstatements for the solutions of these nonlocal elliptic problems are supported by results ofnumerical 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 roleplayed by coercive inequalities in the study of local boundary-value problems for elliptic andparabolic differential equations is well known (see, e.g., [1, 2]).
In the present paper, we consider the Bitsadze-Samarskii type nonlocal boundaryvalue problem
−d2u(t)dt2
+Au(t) = f(t), (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 definiteoperator A. It is known (see, e.g., [3–11]) that various nonlocal boundary value problems forelliptic equations can be reduced to the boundary value problem (1.1). The simply nonlocalboundary value problem was presented and investigated for the first time by Bitsadzeand Samarskii [12]. Further, methods of solutions of Bitsadze-Samarskii nonlocal boundaryvalue problems for elliptic differential equations have been studied extensively by manyresearchers (see [13–21] and the references given therein).
A function u(t) is called a solution of problem (1.1) if the following conditions aresatisfied.
(i) u(t) is twice continuously differentiable on the segment [0, 1]. Derivatives at theendpoints of the segment are understood as the appropriate unilaterial derivatives.
(ii) The element u(t) belongs to D(A) for all t ∈ [0, 1], and the function Au(t) iscontinuous on [0, 1].
(iii) u(t) 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) withboundary S, Ω = Ω ∪ S. In present paper, we are interested in studying the stable differenceschemes for the numerical solution of the following nonlocal boundary value problem for themultidimensional elliptic equation
−utt −n∑
r=1
(ar(x)uxr )xr + δu = f(t, x), 0 < t < 1,
x = (x1, . . . , xn) ∈ Ω,
ut(0, x) = ϕ(x),
ut(1, x) = βut(λ, x) + ψ(x), x ∈ Ω,∣∣β∣∣ ≤ 1, 0 ≤ λ < 1,
u(t, x) = 0, 0 ≤ t ≤ 1, x ∈ S, S = ∂Ω.
(1.2)
Here ψ(x), ϕ(x) (x ∈ Ω), and f(t, x) (t ∈ (0, 1), x ∈ Ω) are given smooth functions, δ is alarge positive constant and ar(x) ≥ a > 0.
In the present paper, the first and second-orders of accuracy difference schemes arepresented for the approximate solution of problem (1.2). The stability and coercive stabilityestimates for the solution of these difference schemes are obtained. A numerical method isproposed for solving nonlocal boundary value problem for the multidimensional ellipticpartial differential equation with the Bitsadze-Samarskii-Dirichlet condition. A procedure ofmodified Gauss elimination method is used for solving these difference schemes in the caseof 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 definethe grid sets as follows:
Ωh = {x : x = xm = (h1m1, . . . , hnmn), m = (m1, . . . , mn),
0 ≤ mr ≤Nr, hrNr = L, r = 1, . . . , n}
Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.
(2.1)
We introduce the Hilbert space L2h = L2(Ωh) of the grid functions ϕh(x) = {ϕ(h1m1, . . . ,
hnmn)} defined on Ωh, equipped with the norm
∥∥∥ϕh∥∥∥L2(Ωh)
=
⎛
⎝∑
x∈Ωh
∣∣∣ϕh(x)
∣∣∣
2h1 · · ·hn
⎞
⎠
1/2
(2.2)
and the Hilbert spaceW22 (Ωh) defined on Ωh, equipped with the norm
∥∥∥ϕh∥∥∥W2
2 (Ωh)=
⎛
⎝∑
x∈Ωh
∣∣∣ϕh(x)
∣∣∣
2h1 · · ·hn
⎞
⎠
1/2
+
⎛
⎝∑
x∈Ωh
n∑
r=1
∣∣∣ϕhxr ,mr
∣∣∣
2h1 · · ·hn
⎞
⎠
1/2
+
⎛
⎝∑
x∈Ωh
n∑
r=1
∣∣∣ϕhxrxr ,mr
∣∣∣
2h1 · · ·hn
⎞
⎠
1/2
.
(2.3)
Finally, we introduce the Banach spaces C([0, 1]τ , L2h) and Cα([0, 1]τ , L2h) of grid abstractfunction {ϕh
k(x)}N−1
1 defined on [0, 1]τ with values L2h, equipped with the following norms:
∥∥∥∥
{
ϕhk
}N−1
1
∥∥∥∥C([0,1]τ ,L2h)
= max1≤k≤N−1
∥∥∥ϕhk
∥∥∥L2h,
∥∥∥∥
{
ϕhk
}N−1
1
∥∥∥∥Cα([0,1]τ ,L2h)
= max1≤k≤N−1
∥∥∥ϕhk
∥∥∥L2h
+ sup1≤k<k+r≤N−1
∥∥ϕk+r − ϕk
∥∥L2h
(rτ)α.
(2.4)
To the differential operator A generated by the problem (1.2) we assign the differenceoperator Ax
h by the formula
Axhu
h = −n∑
r=1
(
ar(x)uhxr)
xr ,mr
+ δuhxr (2.5)
acting in the space of grid functions uh(x), satisfying the condition uh(x) = 0 for all x ∈ Sh.It is known that Ax
his a self-adjoint positive definite operator in L2(Ωh). With the help of
4 Abstract and Applied Analysis
Axh, we arrive at the nonlocal boundary value problem for an infinite system of the following
ordinary differential equations:
−d2uh(t, x)dt2
+Axhu
h(t, x) = fh(t, x), 0 < t < 1, x ∈ Ωh,
uht (0, x) = ϕh(x), uht (1, x) = βu
ht (λ, x) + ψ
h(x), x ∈ Ωh.
(2.6)
In the second step, we replaced problem (2.6) by the first-order of accuracy difference schemeas follows:
−uhk+1(x) − 2uhk(x) + u
hk−1(x)
τ2+Ax
huhk(x) = f
hk (x),
fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,
uh1(x) − uh0(x)τ
= ϕh(x), x ∈ Ωh,
uhN(x) − uhN−1(x)τ
= βuh[λ/τ]+1(x) − uh[λ/τ](x)
τ+ ψh(x), x ∈ Ωh
(2.7)
and the second order of accuracy difference scheme as follows:
−uhk+1(x) − 2uh
k(x) + uhk−1(x)
τ2+Ax
huhk(x) = f
hk (x),
fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,
−3uh0(x) + 4uh1(x) − uh2(x)2τ
= ϕh(x), x ∈ Ωh,
uhN−2(x) − 4uhN−1(x) + 3uhN(x)2τ
= β
⎡
⎣
uh[λ/τ]−1(x) − 4uh[λ/τ](x) + 3uh[λ/τ]+1(x)
2τ
+uh[λ/τ]+1(x) − 2uh[λ/τ](x) + u
h[λ/τ]−1(x)
τ
(λ
τ−[λ
τ
])⎤
⎦
+ ψh(x), x ∈ Ωh.
(2.8)
Now, we will study the well-posedness of (2.7) and (2.8). We have the following theorem onstability 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 differenceschemes (2.7) and (2.8) satisfy the following stability estimate:
max1≤k≤N
∥∥∥uhk
∥∥∥L2h
≤M1
[∥∥∥ϕh∥∥∥L2h
+∥∥∥ψh∥∥∥L2h
+ max1≤k≤N−1
∥∥∥fhk
∥∥∥L2h
]
, (2.9)
whereM1 does not depend on τ, h, ψh(x), ϕh(x) and fhk(x), 1 ≤ k ≤N − 1.
Proof. The proof of (2.9) is based on the following formula:
uhk(x) =(
I − R2N)−1{(
Rk − R2N−k)
uh0(x) +(
RN−k − RN+k)
uhN(x)
−(
RN−k − RN+k)(
2I + τBxh)−1(
Bxh)−1N−1∑
i=1
(
RN−1−i − RN−1+i)
fhi (x)τ
}
+(
2I + τBxh)−1(
Bxh)−1N−1∑
i=1
(
R|k−i|−1 − Rk+i−1)
fhi (x)τ for k = 1, . . . ,N − 1,
(2.10)
where
uh0(x) = Pτ(
I + τBxh)(
2I + τBxh)−1(
Bxh)−1
×[
(I + R)RN−2N−1∑
i=1
(
RN−i − RN+i)
fhi (x)τ
+ (I + R)[
I + R2N−2 − β(
RN−[λ/τ]−1 + RN+[λ/τ])]
×N−1∑
i=1
Ri−1fhi (x)τ − β(
RN + RN−1)[λ/τ]−1∑
i=1
R[λ/τ]−ifhi (x)τ
+ β(
RN−2 + RN−1) N−1∑
i=[λ/τ]+1
Ri−[λ/τ]fhi (x)τ
+β(
RN + RN−1)N−1∑
i=1
R[λ/τ]+ifhi (x)τ − β(
RN−1 + RN)
fh[λ/τ](x)τ
]
− Pτ(I − R)−1[
I + R2N−1 − β(
RN−[λ/τ]−1 + RN+[λ/τ])]
ϕh(x)τ
+ Pτ(I − R)−1(
RN−1 + RN)
ψh(x)τ,
6 Abstract and Applied Analysis
uhN(x) = Pτ(
I + τBxh)(
2I + τBxh)−1(
Bxh)−1
×[[
R−1(I + R) − β(
RN−[λ/τ]−1 + RN+[λ/τ]−1)]N−1∑
i=1
(
RN−i − RN+i)
fhi (x)τ
− β(
I + R2N−1)[λ/τ]−1∑
i=1
R[λ/τ]−ifhi (x)τ + β(
I + R2N−1)
R−1N−1∑
i=[λ/τ]+1
Ri−[λ/τ]fhi (x)τ
+ β(
I + R2N−1)N−1∑
i=1
R[λ/τ]+ifhi (x)τ − β(
I + R2N−1)
fh[λ/τ](x)τ
+(I + R)[
RN + RN−1 − β(
R[λ/τ] + R2N−[λ/τ]−1)]N−1∑
i=1
Ri−1fhi (x)τ
]
− Pτ(I − R)−1[
RN + R2N−1 − β(
R[λ/τ] + R2N−[λ/τ]−1)]
ϕh(x)τ
+ Pτ(I − R)−1(
I + R2N−1)
ψh(x)τ,
Pτ =[
I − R2N−2 − β(
RN−[λ/τ]−1 + RN+[λ/τ]−1)]−1
,
(2.11)
for (2.7), and
uh0(x) = Dτ
(
I + τBxh)(
2I + τBxh)−1(
Bxh)−1
×{
(I + R)(
4R − I − R2)
(I − 3R)RN−4(
I − R2N)N−2∑
i=1
(
RN−i − RN+i)
fhi (x)τ
− (I + R)(
4R − I − R2)(
I − R2N)
×[
3I − R − R2N−2(I − 3R)
− β[
RN−[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−RN+[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]]
×N−1∑
i=2
Ri−2fhi (x)τ − β(I + R)RN−2(
4R − I − R2)(
I − R2N)
×[(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))[λ/τ]−1∑
i=1
R[λ/τ]−i−1fhi (x)τ
+(
3I − R + 2(λ
τ−[λ
τ
])
(I − R)) N−1∑
i=[λ/τ]+2
Ri−[λ/τ]−1fhi (x)τ
−(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))N−1∑
i=1
R[λ/τ]+i−1fhi (x)τ
]
Abstract and Applied Analysis 7
− β(
I − R2N)
(I + R)RN−2(
4R − R2 − I)(
4 + 4(λ
τ−[λ
τ
]))
×(
fh[λ/τ]+1(x)τ − fh[λ/τ](x)τ)
− (I + R)(4I − R)(
I − R2N)
×[
3I − R − R2N−2(I − 3R)
− β(
RN−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
−RN+[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R)))]
× fh1 (x)τ − (I + R)RN−1(
4R − R2 − I)(
I − R2N)(
4I + R2N−3(I − 3R))
×fhN−1(x)τ
}
−Dτ
(
I − R2N)
(I − R)−1(I + R)RN−2(
4R − I − R2)
2τψh(x)
+Dτ
(
I − R2N)
(I − R)−1
×[
3I − R − R2N−2(I − 3R)
− β[
RN−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
−RN+[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))]]
2τϕh(x),
(2.12)
uhN(x) = Dτ
(
I + τBxh)(
2I + τBxh)−1(
Bxh)−1
×{(
I − R2N)[
(I + R)(
4R − I − R2)
R−2(R − 3I)
+ β[
(R − 3I)(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
RN−[λ/τ]−1
−(3R − I)(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
RN+[λ/τ]−1]]
×N−2∑
i=1
(
RN−i − RN+i)
fhi (x)τ + (I + R)(
I − R2N)(
4R − I − R2)
×[
(I + R)RN−2(
4R − I − R2)
− β[
R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−R2N−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]]N−1∑
i=2
Ri−2fhi (x)τ
8 Abstract and Applied Analysis
+ β(
I − R2N)(
R − 3I + R2N−2(I − 3R))
×[(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))[λ/τ]−1∑
i=1
R[λ/τ]−i−1fhi (x)τ
+(
3I − R + 2(λ
τ−[λ
τ
])
(I − R)) N−1∑
i=[λ/τ]+2
Ri−[λ/τ]−1fhi (x)τ
−(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))N−1∑
i=1
R[λ/τ]+i−1fhi (x)τ
]
+ β(
R − 3I + R2N−2(3I − R))(
I − R2N)
×(
4 + 4(λ
τ−[λ
τ
])
(I − R))(
fh[λ/τ]+1(x)τ − fh[λ/τ](x)τ)
+ (I + R)(
I − R2N)
×[
(I + R)RN−2(
I − 4R + R2)
− β[
R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−R2N−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]]
fh1 (x)τ
+ R(
I − R2N−1)[
R − 4I + R2N−4(
I + R2 − 3R)(
3I − R − R2(4I − R))
− β[
RN−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
×(
3R − I + R2N(3I − R))
− RN+[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
×(
3I − R + R2N(I − 3R))]]
fhN−1(x)τ}
+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)
− β[
R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−R2N−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]]
2τψh(x),
Abstract and Applied Analysis 9
Dτ =(
I − R2N)−1
×{[
−(3I − R)2 + (I − 3R)2]
− β[
−RN+[λ/τ]−3(I − 3R)(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−RN−[λ/τ]−1(3I − R)(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]}−1
,
R =(
I + τBxh)−1
,
Bxh =12
(
τAxh +√
4Axh + τ
2(
Axh
)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 C([0, 1]τ , L2h). We have thefollowing theorem on almost coercive stability.
Theorem 2.2. Let τ and |h| be sufficiently small positive numbers. Then the solutions of differenceschemes (2.7) and (2.8) satisfy the following almost coercive stability estimate:
max1≤k≤N−1
∥∥∥∥∥
uhk+1 − 2uh
k+ uh
k−1τ2
∥∥∥∥∥L2h
+ max1≤k≤N−1
∥∥∥uhk
∥∥∥W2
2h
≤M2
[∥∥∥ϕh∥∥∥W2
2h
+∥∥∥ψh∥∥∥W2
2h
+ ln1
τ + |h| max1≤k≤N−1
∥∥∥fhk
∥∥∥L2h
]
,
(2.14)
whereM2 does not depend on τ, h, ψh(x), ϕh(x), and fhk (x), 1 ≤ k ≤N − 1.
Proof. The proof of (2.14) is based on the formulas (2.11), (2.12), (2.13), the symmetryproperties of the difference operator Ax
hdefined 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
Axhu
h(x) = wh(x), x ∈ Ωh,
uh(x) = 0, x ∈ Sh(2.15)
the following coercivity inequality holds (see [21]):
n∑
r=1
∥∥∥∥
(
uh)
xrxr , mr
∥∥∥∥L2h
≤M∥∥∥wh
∥∥∥L2h, (2.16)
where M does not depend on h and wh(x).
10 Abstract and Applied Analysis
Theorem 2.4. Let ϕh(x) = ψh(x) = 0. Then the difference problems (2.7) and (2.8) are well-posed inHolder spaces Cα([0, 1]τ , L2h) and the following coercivity inequality holds:
max1≤k≤N−1
∥∥∥∥∥∥
{
uhk+1 − 2uhk + uhk−1
τ2
}N−1
1
∥∥∥∥∥∥Cα([0,1]τ , L2h)
+∥∥∥∥
{
uhk
}N−1
1
∥∥∥∥Cα([0,1]τ , W
22h)
≤ M3
α(1 − α) max1≤k≤N−1
∥∥∥∥
{
fhk
}N−1
1
∥∥∥∥Cα([0,1]τ ,L2h)
.
(2.17)
HereM3 does not depend on τ, h, and fhk (x), 1 ≤ k ≤N − 1.
Proof. The proof of (2.17) is based on the formula
Axhu
h0(x) − fh1 (x) = Pτ
(
I + τBxh)(
2I + τBxh)−1(I + R)
×[
RN−1N−1∑
i=1
τBxhRN−i(
fhi − fhN−1)
− RN−1N−1∑
i=1
τBxhRN+i(
fhi (x) − fh1 (x))
−βRN[λ/τ]−1∑
i=1τBx
hR[λ/τ]−i
(
fhi (x) − fh[λ/τ](x))
+[
I + R2N−2 − β(
RN−[λ/τ]−1 + RN+[λ/τ])]
×N−1∑
i=1
τBxhRi(
fhi (x) − fh1 (x))
+ βRNN−1∑
i=[λ/τ]+1
τBxhRi−[λ/τ]
(
fhi (x) − fh[λ/τ](x))
+ βRNN−1∑
i=1
τBxhR[λ/τ]+i
(
fhi (x) − fh1 (x))
+ RN−1(
I − RN−1)
fhN−1(x)
+ β[
RN+[λ/τ]−1 − R2N−[λ/τ]−1 − τBxhRN]
fh[λ/τ](x)
+
[
R2N−2 − R2N−1 + R2N−2 − R3N−3 + R3N−2 − RN−1
+ β(
R2N−[λ/τ]−2 + R2N+[λ/τ]−1
+R2N+[λ/τ] − RN+[λ/τ]−1)]
fh1 (x)
]
Abstract and Applied Analysis 11
− Pτ(I − R)−1τ[
I + R2N−1 − β(
RN−[λ/τ]−1 + RN+[λ/τ])]
Axhϕ
h
+ Pτ(I − R)−1τ(
RN−1 + RN)
Axhψ
h,
Ahxu
hN(x) − fhN−1(x) = Pτ
(
I + τBxh)(
2I + τBxh)−1
×{[
R(I + R) − β(
RN−[λ/τ]+1 + RN+[λ/τ]+1)]
×N−1∑
i=1
τBxhRN−i(
fhi (x) − fhN−1(x))
−[
R(I + R) − β(
RN−[λ/τ]+1 + RN+[λ/τ]+1)]
×N−1∑
i=1
τBxhRN+i(
fhi (x) − fh1 (x))
− βR2(
I + R2N−1)[λ/τ]−1∑
i=1
τBxhR[λ/τ]−i
(
fhi (x) − fh[λ/τ](x))
+ βR(
I + R2N−1) N−1∑
i=[λ/τ]+1
τBxhRi−[λ/τ]
(
fhi (x) − fh[λ/τ](x))
+ βR2(
I + R2N−1)N−1∑
i=1
τBxhR[λ/τ]+i
(
fhi (x) − fh1 (x))
− β(
I + R2N−1)
×[
R(
I − R[λ/τ]−1)
−(
I − RN−[λ/τ]−1)
− τBxhR]
× fh[λ/τ](x)
+[
(I + R)(
R2N−2 − RN − I + R)
− β(
RN−[λ/τ]+1
+ RN+[λ/τ]+1 − R2N−[λ/τ]
− R2N+[λ/τ] − RN−[λ/τ]−1
+RN+[λ/τ]−1 − RN−[λ/τ] + RN+[λ/τ])]
× fhN−1(x) +[
(I + R)(
RN − R2N−1)
+ β(
RN+[λ/τ] + RN+[λ/τ]+1 − RN+[λ/τ]+2
+ R2N+[λ/τ]+1 + R2N+[λ/τ]+1 − R3N+[λ/τ]
− R3N+[λ/τ]+1 + R[λ/τ]+1 − R2N−[λ/τ]
+R3N−[λ/τ]−1)]
fh1 (x)
}
12 Abstract and Applied Analysis
+ Pτ(I − R)−1Rτ(
I + R2N−1)
Ahxψ
− Pτ(I − R)−1Rτ[
RN−1 + RN − β(
R[λ/τ] + R2N−[λ/τ]−1)]
Ahxϕ
(2.18)
for (2.7) and
Axhu
h0(x) − fh1 (x) = Dτ
(
I + τBxh)(
2I + τBxh)−1
×{
(I + R)(
4R − I − R2)
(I − 3R)RN−3(
I − R2N)
×[N−2∑
i=1
BxhτRN−i(
fhi (x)−fhN−1(x))
−N−2∑
i=1
BxhτRN+i(
fhi (x)−fh1 (x))]
− (I + R)(
4R − I − R2)(
I − R2N)
×[
3I − R − R2N−2(I − 3R)
− β[
RN−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
−RN+[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))]]
×N−1∑
i=2
BxhτRi−1(
fhi (x) − fh1 (x))
− β(I + R)RN−2(
4R − I − R2)(
I − R2N)
×[(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
×[λ/τ]−1∑
i=1
BxhτR[λ/τ]−i
(
fhi (x) − fh[λ/τ](x))
+(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
×N−1∑
i=[λ/τ]+2
BxhτRi−[λ/τ]
(
fhi (x) − fh[λ/τ](x))
Abstract and Applied Analysis 13
−(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
×N−1∑
i=1
BxhτR[λ/τ]+i
(
fhi (x) − fh1 (x))]
− β(
I − R2N)
(I + R)RN−1(
4R − R2 − I)
×(
4 + 4(λ
τ−[λ
τ
]))
τBxhfh[λ/τ]+1(x)
+ β(
I − R2N)
(I + R)RN−2(
4R − R2 − I)
×[
2R(
R + 2(λ
τ−[λ
τ
]))
+ R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
+ RN−[λ/τ]−2
×(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]
fh[λ/τ](x)
×[
(I + R)(
I − R2N)
×[
(I − 3R)(
3R − I − R2N−5(I − R)
×(
R2 − R2N(I + R)(
R2 − 4R + I)))
− (3I − R)RN−1(
R2 − 4R + I)
− β(
4R − I − R2)
×((
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
RN−[λ/τ]−3
−(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
×RN+[λ/τ]−3)]]
fh1 (x)
+ (I + R)(
4R − I − R2)(
I − R2N)
RN−2
×(
R − 3I − RN−2(I − 3R)(
I + RN(I − R)))
fhN−1(x)
}
−Dτ
(
I − R2N)
(I − R)−1(I + R)RN−2(
4R − I − R2)
2τAxhψ
h(x)
+Dτ
(
I − R2N)
(I − R)−1
14 Abstract and Applied Analysis
×[
3I − R − R2N−2(I − 3R)
− β[
RN−[λ/τ](
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
−RN+[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))]]
2τAxhϕ
h(x),
AxhRu
hN(x)− fhN−1(x)
= Dτ
(
I + τBxh)(
2I + τBxh)−1
×{(
I − R2N)
×[
(I + R)(
4R − I − R2)
(R − 3I)
+ β[
(R − 3I)(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
× RN−[λ/τ]+1 − (3R − I)
×(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
RN+[λ/τ]+1]]
×(
N−2∑
i=1
BxhτRN−i(
fhi (x) − fhN−1(x))
−N−2∑
i=1
BxhτRN+i(
fhi (x) − fh1 (x)))
+ (I + R)(
I − R2N)(
4R − I − R2)
×[
(I + R)RN−2(
−4R + I + R2)
− β[
R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−R2N−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]]
×N−1∑
i=2
BxhτRi(
fhi (x) − fh1 (x))
+ βR(
R − 3I + R2N−2(I − 3R))(
I − R2N)
×[(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
×[λ/τ]−1∑
i=1
BxhτR[λ/τ]−i
(
fhi (x) − fh[λ/τ](x))
Abstract and Applied Analysis 15
+(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
×N−1∑
i=[λ/τ]+2
BxhτR[λ/τ]+i
(
fhi (x) − fh1 (x))
+(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
×N−1∑
i=1
BxhτR[λ/τ]+i
(
fhi (x) − fh1 (x))]
+ βR2(
R − 3I + R2N−2(I − 3R))(
I − R2N)(
4 + 4(λ
τ−[λ
τ
]))
× fh[λ/τ]−1(x) − βR(
I − R2N)
×[(
I − R + RN−[λ/τ])(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
+R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))]
× fh[λ/τ](x) +(
I − R2N)
×[
(I + R)(
4R − R2 − I)
RN+1
× (R − 3I)(
I − RN−2)
+ β[(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
× R2N−[λ/τ](
R2(3I − R) + RN−2(
R3 − 4R2 + I))
−(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
R[λ/τ]
×(
R2(3I − R) + (I − 3R)RN−3(
R + RN(I − R)))]]
× fh1 (x) +(
I − R2N)
×[
(I − 3R)(
4R − R2 − I)(
3I − R + R2N−2)
− β[(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))
RN−[λ/τ]−1(3I − R)
×(
R + I + RN(
I − R + RN−2(I − R) + R3N−2(2I − R)))
−(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
RN+[λ/τ]−1(3R − I)
16 Abstract and Applied Analysis
×(
I − R − R2N−3
×(
1 + R − RN(
3I − R2)
+ R2N(
R3 − 4R2 + I)))]]
×fhN−1(x)}
+Dτ
(
I − R2N)
(I − R)−1R(
R − 3I + R2N−2(I − 3R))
2τAxhϕ
h(x)
−Dτ
(
I − R2N)
(I − R)−1R
×[
RN−2(I + R)(
R2 − 4R + I)
− β[
R[λ/τ]−1(
I − 3R + 2(λ
τ−[λ
τ
])
(I − R))
−R2N−[λ/τ]−1(
3I − R + 2(λ
τ−[λ
τ
])
(I − R))]]
2τAxhψ
h(x)
(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 stabilityinequality. Therefore, we will give the following results of numerical experiments of theBitsadze-Samarskii-Dirichlet problem:
−∂2u(t, x)∂t2
− ∂2u(t, x)∂x2
+ u = f(t, x), 0 < t < 1, 0 < x < 1,
f(t, x) = exp(−πt) sin(πx), 0 < t < 1, 0 < x < 1,
ut(0, x) = −π sin(πx), 0 ≤ x ≤ 1,
ut(1, x) = ut(12, x
)
+ π sin(πx)(
exp(
−π2
)
− exp(−π))
, 0 ≤ x ≤ 1,
u(t, 0) = u(t, 1) = 0, 0 ≤ t ≤ 1
(3.1)
for the two-dimensional elliptic equation.
Abstract and Applied Analysis 17
The exact solution of this problem is
u(t, x) = exp(−πt) sin(πx). (3.2)
For the approximate solution of problem (3.1), we consider the set [0, 1]τ × [0, 1]h of a familyof grid points depending on small parameters τ and h as follows:
[0, 1]τ × [0, 1]h = {(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 forthe approximate solution of problem (3.1):
−unk+1 − 2unk + u
nk−1
τ2− un+1
k− 2un
k+ un−1
k
h2+ unk = exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,
1 ≤ n ≤M − 1,
un1 − un0τ
= −π sin(πxn), 0 ≤ n ≤M,
unN − unN−1τ
= βunN/2 − unN/2−1
τ+ π sin(πxn)
×(
exp(
−π2
)
− exp(−π))
, 0 ≤ n ≤M,
u0k = uMk = 0, 0 ≤ k ≤N.
(3.4)
We have (N + 1) × (M + 1) system of linear equations in (3.4) and we will write themin the following matrix form:
Aun+1 + Bun + Cun−1 = Dϕn, 1 ≤ n ≤M − 1,
u0 = uM = 0.(3.5)
18 Abstract and Applied Analysis
Here,
A =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 0 0 · 0 0 · 0 0 00 a 0 · 0 0 · 0 0 00 0 a · 0 0 · 0 0 0· · · · · · · · · ·0 0 0 · 0 0 · a 0 00 0 0 · 0 0 · 0 a 00 0 0 · 0 0 · 0 0 0
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(N+1)×(N+1)
,
B =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
−1 1 0 · 0 0 · 0 0 0c b c · 0 0 · 0 0 00 c b · 0 0 · 0 0 0· · · · · · · · · ·0 0 0 · 0 0 · b c 00 0 0 · 0 0 · c b c0 0 0 · 1 −1 · 0 −1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(N+1)×(N+1)
,
C = A, D = [I](N+1)×(N+1), us =
⎡
⎢⎢⎢⎢⎢⎣
u0su1s·
uN−1s
uNs
⎤
⎥⎥⎥⎥⎥⎦
(N+1)×1
,
(3.6)
where s = n − 1, n, n + 1 and
ϕn =
⎡
⎢⎢⎢⎢⎢⎣
ϕ0n
ϕ1n
.ϕN−1n
ϕNn
⎤
⎥⎥⎥⎥⎥⎦
(N+1)×1
,
a =1h2, b = − 2
τ2− 2h2
− 1, c =1τ2,
ϕkn =
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
−τπ sin(πxn), k = 0,
− exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,
τπ sin(πxn)(
exp(
−π2
)
− exp(−π))
, k =N.
(3.7)
We seek a solution of the matrix equation in following form:
un = αn+1un+1 + βn+1, 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:
αn+1 = − (B + Cαn)−1A,
βn+1 = (B + Cαn)−1(
Dϕn − Cβn)
, n = 1, . . . ,M − 1,(3.9)
where
α1 = [0](N+1)×(N+1), β1 = [0](N+1)×1. (3.10)
Now, applying (2.8) for N even number, we can present the following second-order ofaccuracy difference scheme:
−unk+1 − 2un
k+ un
k−1τ2
− un+1k − 2unk + un−1k
h2+ unk = − exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,
1 ≤ h ≤M − 1,
−3un0 + 4un1 − un22τ
= −π sin(πxn), 0 ≤ n ≤M,
unN−2(x) − 4unN−1(x) + 3unN(x)2τ
=unN/2−2 − 4un
N/2−1 + 3unN/2
2τ
+ π sin(πxn)(
exp(
−π2
)
− exp(−π))
, 0 ≤ n ≤M,
u0k = uMk = 0, 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:
Aun+1 + Bun + Cun−1 = Dϕn, 1 ≤ n ≤M − 1,
u0 = uM = 0,(3.12)
20 Abstract and Applied Analysis
where
A =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 0 0 · 0 0 0 · 0 0 00 a 0 · 0 0 0 · 0 0 00 0 a · 0 0 0 · 0 0 0· · · · · · · · · · ·0 0 0 · 0 0 0 · a 0 00 0 0 · 0 0 0 · 0 a 00 0 0 · 0 0 0 · 0 0 0
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(N+1)×(N+1)
,
B =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
−3 4 −1 · 0 0 0 · 0 0 0b c b · 0 0 0 · 0 0 00 b c · 0 0 0 · 0 0 0· · · · · · · · · · ·0 0 0 · 0 0 0 · c b 00 0 0 · 0 0 0 · b c b0 0 0 · −1 4 −3 · 1 −4 3
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(N+1)×(N+1)
,
C = A, D = [I](N+1)×(N+1),
us =
⎡
⎢⎢⎢⎢⎢⎣
u0su1s·
uN−1s
uNs
⎤
⎥⎥⎥⎥⎥⎦
(N+1)×1
,
(3.13)
where s = n − 1, n, n + 1 and
ϕkn =
⎡
⎢⎢⎢⎢⎢⎣
ϕ0n
ϕ1n
·ϕN−1n
ϕNn
⎤
⎥⎥⎥⎥⎥⎦
(N+1)×1
. (3.14)
Here,
a =1h2, b =
1τ2, c = − 2
h2− 2τ2
− 1, (3.15)
ϕkn =
⎧
⎪⎨
⎪⎩
−2τπ sin(πxn), k = 0,− exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,
2τπ sin(πxn)(
exp(
−π2
)
− exp(−π))
, k =N.(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 N =M = 20 N =M = 40 N =M = 60First-order difference scheme (2.7) 0.05384197882300 0.02631633782987 0.01740639813932Second-order difference scheme (2.8) 0.00631619894867 0.00164455475890 7.4144589892985e -004
The errors in numerical solutions are computed by
ENM = max1≤k≤N−1
(M−1∑
n=1
∣∣∣u(tk, xn) − ukn
∣∣∣
2h
)1/2
(3.17)
for different values of M and N, where u(tk, xn) represents the exact solution and uknrepresents the numerical solution at (tk, xn). The results are shown in Table 1 forN =M = 20,N =M = 40, andN =M = 60.
Thus, second-order of accuracy difference scheme is more accurate compared with thefirst-order of accuracy difference scheme.
4. Conclusion
The first and second-orders of accuracy difference schemes for approximate solutions ofthe Bitsadze-Samarskii-Dirichlet type nonlocal boundary value problem for the multidimen-sional elliptic partial differential equation are presented. Theorems on the stability, almostcoercive stability, and coercive stability estimates for the solution of these difference schemesare established. Numerical experiments are given.
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