Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 846582, 29 pages
doi:10.1155/2012/846582
Research Article
A Note on the Second Order of Accuracy
Stable Difference Schemes for the Nonlocal
Boundary Value Hyperbolic Problem
Allaberen Ashyralyev1 and Ozgur Yildirim2
1
2
Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey
Department of Mathematics, Yildiz Technical University, Esenler 34210, Istanbul, Turkey
Correspondence should be addressed to Ozgur Yildirim, ozgury@yildiz.edu.tr
Received 23 March 2012; Accepted 11 June 2012
Academic Editor: Sergey Piskarev
Copyright q 2012 A. Ashyralyev and O. Yildirim. 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.
The second order of accuracy absolutely stable difference schemes are presented for the nonlocal
boundary value hyperbolic problem for the differential equations in a Hilbert space H with the
self-adjoint positive definite operator A. The stability estimates for the solutions of these difference
schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary
conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered.
The stability estimates for the solutions of these difference schemes for the nonlocal boundary
value hyperbolic problem are established. Finally, a numerical method proposed and numerical
experiments, analysis of the errors, and related execution times are presented in order to verify
theoretical statements.
1. Introduction
Hyperbolic partial differential equations play an important role in many branches of science
and engineering and can be used to describe a wide variety of phenomena such as acoustics,
electromagnetics, hydrodynamics, elasticity, fluid mechanics, and other areas of physics see
1–5 and the references given therein.
While applying mathematical modelling to several phenomena of physics, biology,
and ecology, there often arise problems with nonclassical boundary conditions, which the
values of unknown function on the boundary are connected with inside of the given domain.
Such type of boundary conditions are called nonlocal boundary conditions. Over the last
decades, boundary value problems with nonlocal boundary conditions have become a
rapidly growing area of research see, e.g., 6–16 and the references given therein.
2
Abstract and Applied Analysis
In the present work, we consider the nonlocal boundary value problem
d2 ut
Aut ft 0 ≤ t ≤ 1,
dt2
n
n
u0 αj u λj ϕ,
βj ut λj ψ,
ut 0 j1
1.1
j1
0 < λ1 < λ2 < · · · < λn ≤ 1,
where A is a self-adjoint positive definite operator in a Hilbert space H.
A function ut is called a solution of the problem 1.1, if the following conditions are
satisfied:
i ut is twice continuously differentiable on the segment 0, 1. The derivatives at the
endpoints of the segment are understood as the appropriate unilateral derivatives.
ii The element ut belongs to DA, independent of t, and dense in H for all t ∈ 0, 1
and the function Aut is continuous on the segment 0, 1.
iii ut satisfies the equation and nonlocal boundary conditions 1.1.
In the paper of 8, the following theorem on the stability estimates for the solution of
the nonlocal boundary value problem 1.1 was proved.
Theorem 1.1. Suppose that ϕ ∈ DA, ψ ∈ DA1/2 , and ft is a continuously differentiable
function on 0, 1 and the assumption
n n
n n
αk βk βk < 1 αk βk |αm |
m1
k1
k1
k1
1.2
k/
m
holds. Then, there is a unique solution of problem 1.1 and the stability inequalities
maxutH ≤ M ϕH A−1/2 ψ maxA−1/2 ft ,
H
0≤t≤1
maxA1/2 ut
H
0≤t≤1
0≤t≤1
≤ M A1/2 ϕ ψ H maxftH ,
H
H
0≤t≤1
1
d2 ut 1/2 f t H dt
max
maxAutH ≤ M Aϕ H A ψ f0 H H
0≤t≤1 dt2 0≤t≤1
0
H
1.3
hold, where M does not depend on ϕ, ψ, and ft, t ∈ 0, 1.
Moreover, the first order of accuracy difference scheme
τ −2 uk1 − 2uk uk−1 Auk1 fk ,
tk kτ,
1 ≤ k ≤ N − 1, Nτ 1,
fk ftk ,
Abstract and Applied Analysis
u0 n
3
αr uλr /τ ϕ,
r1
τ −1 u1 − u0 n
1
ψ,
βr uλr /τ1 − uλr /τ
τ
r1
1.4
for the approximate solution of problem 1.1 was presented. The stability estimates for the
solution of this difference scheme, under the assumption
n
|αk | k1
n n
n βk |αk | βk < 1,
k1
k1
1.5
k1
were established.
In the development of numerical techniques for solving PDEs, the stability has been
an important research topic see 6–31. A large cycle of works on difference schemes for
hyperbolic partial differential equations, in which stability was established under the
assumption that the magnitude of the grid steps τ and h with respect to the time and space
variables, are connected. In abstract terms, this particularly means that τAh → 0 when
τ → 0.
We are interested in studying the high order of accuracy difference schemes for
hyperbolic PDEs, in which stability is established without any assumption with respect to the
grid steps τ and h. Particularly, a convenient model for analyzing the stability is provided by
a proper unconditionally absolutely stable difference scheme with an unbounded operator.
In the present paper, the second order of accuracy unconditionally stable difference
schemes for approximately solving boundary value problem 1.1 is presented. The stability
estimates for the solutions of these difference schemes and their first and second order
difference derivatives are established. This operator approach permits one to obtain the stability estimates for the solutions of difference schemes of nonlocal boundary value problems,
for one-dimensional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional hyperbolic equation with Dirichlet condition in space variables.
Some results of this paper without proof were presented in 7.
Note that nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists see,
e.g., 11–16, 20–24, 32–38 and the references therein.
2. The Second Order of Accuracy Difference Scheme Generated by A2
Throughout this paper for simplicity λ1 > 2τ and λn < 1 will be considered. Let us associate
boundary value problem 1.1 with the second order of accuracy difference scheme
τ2 2
A uk1 fk ,
4
tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
τ −2 uk1 − 2uk uk−1 Auk fk ftk ,
4
Abstract and Applied Analysis
λm
αm uλm /τ τ −1 uλm /τ − uλm /τ−1 × λm −
τ
ϕ,
τ
m1
τ 2A
τ
τ −1 u1 − u0 −
I
f0 − Au0
2
2
n
λk
τ
λk −
τ
× fλk /τ − Auλk /τ
βk τ −1 uλk /τ − uλk /τ−1 ψ,
2
τ
k1
u0 n
f0 f0.
2.1
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 operator Ah that act in the Hilbert space and
are uniformly self-adjoint positive definite in h for 0 < h ≤ h0 .
In general, we have not been able to obtain the stability estimates for the solution
of difference scheme 2.1 under assumption 1.5. Note that the stability of solution of
difference scheme 2.1 will be obtained under the strong assumption
n λk λk τ βk 1 λk −
τ
|αk | 1 λk −
τ
τ
k1
k1
n
n λk |αk | βk 1 λk −
τ
τ
k1
k1
n
λk n λk |αk |λk −
βk λk −
τ
τ < 1.
τ
τ
k1
k1
n
2.2
Now, let us give some lemmas that will be needed in the sequel.
Lemma 2.1. The following estimates hold:
≤ 1,
RH→H ≤ 1,
R
H→H
−1 −1 ≤ 1,
≤ 1,
R R
R R
H→H
H→H
1/2 1/2 ≤ 1,
≤ 1.
τA R
τA R
H→H
2.3
H→H
−1
−1
I − iτA1/2 − τ 2 /2A .
Here, H is the Hilbert space, R I iτA1/2 − τ 2 /2A , and R
Lemma 2.2. Suppose that assumption 2.2 holds. Denote
−1 λ /τ−1 λm /τ−1 I iτA1/2
R m
I − iτA1/2 R
τ 2A
Bτ αm I 2
2
m1
−1 n
λk /τ−1 R−1 Rλk /τ−1 R
−1
τ 2A
R
βk I 2
2
k1
n
Abstract and Applied Analysis
5
−1
n
n τ 2A
1
λm /τ−1 Rλk /τ−1
λk /τ−1 R
Rλm /τ−1 R
−
αm βk I 2 m1 k1
2
−1
n
λm
τ 2A
1/2
αm λm −
τ iA
I
τ
2
m1
λ /τ−1 m
I − iτ/2A1/2 I iτA1/2 − Rλm /τ−1 I iτ/2A1/2 I − iτA1/2
R
×
2
−1 n
λk /τ−1 R−1 − Rλk /τ−1 R
−1
R
τ 2A
λk
1/2
βk λk −
I
τ iA
τ
2
2
k1
−2
n
n τ 2A
1
λm
τ
I
αm βk λm −
4 m1 k1
τ
2
iτ 1/2
iτ 1/2
1/2
λm /τ−1 λk /τ−1
λm /τ−1 λk /τ−1
× iA
R
R
I A
−R
I− A
R
2
2
−1
n
n 1
λk
τ 2A
−
τ iA1/2
αm βk λk −
I
2 m1 k1
τ
2
λm /τ−1 Rλk /τ−1
λk /τ−1 − R
× Rλm /τ−1 R
−1
n
n τ 2A
λm
λk
1
τ
λk −
τ A I
αm βk λm −
−
2 m1 k1
τ
τ
2
iτ 1/2
iτ 1/2
λm /τ−1 λk /τ−1
λm /τ−1 λk /τ−1
I− A
R
I A
.
× R
R
R
2
2
2.4
Then, the operator I − Bτ has an inverse
Tτ I − Bτ −1 ,
2.5
Tτ H→H ≤ M.
2.6
and the following estimate holds:
Proof. The proof of estimate 2.6 is based on the estimate
n
n λk λk τ − βk 1 λk −
τ
I − Bτ H→H ≥ 1 − |αk | 1 λk −
τ
τ
k1
k1
n λk − |αk | βk 1 λk −
τ
τ
k1
k1
n
λk n λk − |αk |λk −
βk λk −
τ
τ .
τ
τ
k1
k1
n
2.7
Estimate 2.7 follows from the triangle inequality and estimate 2.3. Lemma 2.2 is proved.
6
Abstract and Applied Analysis
Now, we will obtain the formula for the solution of problem 2.1. It is easy to show
that see, e.g., 18 there is unique solution of the problem
τ2 2
A uk1 fk ,
4
tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
τ −2 uk1 − 2uk uk−1 Auk fk ftk ,
τ 2A
τ
τ −1 u1 − u0 −
I
f0 − Au0 ω,
2
2
2.8
f0 f0,
u0 μ,
and the following formula holds:
−1 τ2
τ 2A
μ τω f0 ,
u1 I u0 μ,
2
2
−1 k−1 k−1 I iτA1/2
R
I − iτA1/2 R
τ 2A
uk I μ
2
2
−1
−1 R
−1 k−1 R−1 − Rk−1 R
τ 2A
τ 1/2
I
ω f0
iA
2
2
2
−
k−1
τ
s1
2i
k−s fs ,
A−1/2 Rk−s − R
2.9
2 ≤ k ≤ N.
Applying formula 2.9 and the nonlocal boundary conditions in problem 2.1, we get
μ Tτ
n
τ
αm f0
2
m1
⎛
τ 2A
×⎝ I 2
n
−1
αm
m1
×
−
n
iA1/2
−1
×
λm /τ−1 R−1 − Rλm /τ−1 R
−1
R
2
−1
τ 2A
λm
I
λm −
τ
τ
2
⎞
−1 I iτ/2A1/2
λm /τ−1 R−1 I − iτ/2A1/2 Rλm /τ−1 R
R
⎠
2
αm
m1
/τ−1
λm
s1
n
τ −1/2 λm /τ−s λm /τ−s λm
A
R
fs −
−R
αm λm −
τ
2i
τ
m1
τ 1/2 −1/2 λm /τ−s
iτ 1/2
iτ 1/2
λm /τ−s
iA
R
A
I− A
R
I− A
fs
×
2i
2
2
s1
n
λm
αm τ λm −
τ RRfλm /τ−1 − ϕ
τ
m1
λm
/τ−2
Abstract and Applied Analysis
⎡
−1
n
−1
2
τ
A
λ
τ
k
iA1/2
λk −
τ
A I
× ⎣I βk
2
τ
2
k1
−1
n
λk /τ−1 R−1 − Rλk /τ−1 R
−1
τ 2A
R
λm
τ
I
×
αm λm −
−
2
τ
2
m1
λ /τ−1 −1 −1 I iτ/2A1/2
m
R I − iτ/2A1/2 Rλm /τ−1 R
R
×
2
⎡
−1
n
−1 R
λm /τ−1 R−1 − Rλm /τ−1 R
−1
τ 2A
iA1/2
⎣ αm I 2
2
m1
−1
τ 2A
λm
I
τ
αm λm −
τ
2
m1
λ /τ−1 −1 −1 I iτ/2A1/2
m
R I − iτ/2A1/2 Rλm /τ−1 R
R
×
2
⎛
⎡
−1
n
τ 2A
τ
× ⎣ f0 ⎝ βk I 2
2
k1
n
−1 I iτ/2A1/2
λk /τ−1 R−1 I − iτ/2A1/2
Rλk /τ−1 R
R
×
2
2
−1
n
−1
τ 2A
λk
τ
− βk
iA1/2
λk −
τ
A× I 2
τ
2
k1
⎞
/τ−2
λk
n
λk /τ−1 R−1 − Rλk /τ−1 R
−1
τ
R
⎠ βk
×
2
2i
s1
k1
iτ
λk /τ−s I − iτ A1/2
× A−1/2 iA1/2 Rλk /τ−s I A1/2 R
fs
2
2
n
λk /τ−1 βk τRRf
k1
λk /τ−s
× R
λk /τ−s
−R
⎧⎡
−1
n
⎨
2
τ
A
αm I ω Tτ ⎣I −
⎩
2
m1
×
−
λk
/τ−1
n
τ −1/2
λk
τ
λk −
τ
A
A
βk
2
τ
2i
s1
k1
n
λk
τ
λk −
fs βk
τ
fλk /τ−1 ψ ,
2
τ
k1
λm /τ−1 I iτA1/2
R
Rλm /τ−1 I − iτA1/2
2
2
n
αm
m1
−1
τ 2A
λm
I
τ
iA1/2
λm −
τ
2
7
8
Abstract and Applied Analysis
λm /τ−1 I − iτ/2A1/2 I iτA1/2 − Rλm /τ−1 I iτ/2A1/2 I − iτA1/2
R
×
2
⎛
⎡
−1
n
τ 2A
τ
× ⎣ f0 ⎝ βk I 2
2
k1
−1 I iτ/2A1/2
λk /τ−1 R−1 I − iτ/2A1/2
Rλk /τ−1 R
R
×
2
2
−1
n
−1
τ 2A
λk
τ
λk −
τ
A× I iA1/2
− βk
2
τ
2
k1
⎞
/τ−2
λk
n
λk /τ−1 R−1 − Rλk /τ−1 R
−1
τ
R
⎠ βk
×
2
2i
s1
k1
iτ
λk /τ−s I − iτ A1/2
fs
× A−1/2 iA1/2 Rλk /τ−s I A1/2 R
2
2
λk
/τ−1
n
n
τ
λk /τ−1 βk τ λk − λk τ
βk RRf
A
2
τ
2i
s1
k1
k1
⎤
τ
λ
k
λk /τ−s fs λk −
τ
fλk /τ−1 ψ ⎦
×A−1/2 Rλk /τ−s − R
2
τ
⎡
−1 λ /τ−1 n
λk /τ−1 I iτA1/2
R k
I − iτA1/2 R
τ 2A
τ
⎣
−
βk A I 2
2
2
k1
−1 n
τ 2A
iτ 1/2 1
λk /τ−1
βk I R
I− A
I iτA1/2
−
2 k1
2
2
−Rλk /τ−1
⎤
iτ 1/2 I A
I − iτA1/2 ⎦
2
⎡
⎛
−1
n
−1 R
λm /τ−1 R−1 − Rλm /τ−1 R
−1
τ 2A
τ
⎝
⎣
×
iA1/2
f0
αm I 2
2
2
m1
n
αm
m1
×
−
n
αm
×
⎞
−1 I iτ/2A1/2
λm /τ−1 R−1 I − iτ/2A1/2 Rλm /τ−1 R
R
⎠
2
/τ−1
λm
m1
s1
λm
/τ−2
s1
−1
τ 2A
λm
I
λm −
τ
τ
2
n
τ −1/2 λm /τ−s λm /τ−s λm
A
τ
R
fs −R
αm λm −
2i
τ
m1
τ −1/2 1/2 λm /τ−s
iτ 1/2
iτ 1/2
λm /τ−s
iA
R
A
I A
R
I− A
fs
2i
2
2
Abstract and Applied Analysis
9
λm
τ RRfλm /τ−1 − ϕ
.
αm τ λm −
τ
m1
n
2.10
Thus, formulas 2.9 and 2.10 give a solution of problem 2.1.
Theorem 2.3. Suppose that assumption 2.2 holds and ϕ ∈ DA, ψ ∈ DA1/2 . Then, for the
solution of difference scheme 2.1 the stability inequalities
max uk H ≤ M
0≤k≤N
k0
max A1/2 uk H
0≤k≤N
−1/2 −1/2 fk τ A
ψ ϕ H ,
A
N−1
H
H
1/2 fk τ A ϕ ψ H ,
H
N−1
≤M
k0
H
τ2 2
−2
max τ uk1 − 2uk uk−1 max Auk A uk1 H
4
1≤k≤N−1
0≤k≤N−1
H
N−1
fk − fk−1 f0 ≤M
A1/2 ψ AϕH
H
H
2.11
2.12
2.13
H
k1
hold, where M does not depend on τ, ϕ, ψ, and fk , 0 ≤ k ≤ N − 1.
Proof. By 18, the following estimates
max uk H ≤ M
0≤k≤N
k0
max A1/2 uk H
0≤k≤N
−1/2 −1/2 fk τ A
ω μH ,
A
N−1
H
≤M
H
1/2 fk τ A μ ωH ,
H
N−1
k0
H
τ2 2
−2
max τ uk1 − 2uk uk−1 max Auk A uk1 H
4
1≤k≤N−1
0≤k≤N−1
H
N−1
fk − fk−1 f0 ≤M
A1/2 ω Aμ
H
k1
H
2.14
2.15
2.16
H
H
hold for the solution of 2.8. Using formulas of μ, ω, and 2.3 and 2.6 the following estimates obtained
N−1
−1/2
−1/2
μ ≤ M
fs τ A
ψ ϕH ,
A
H
H
s0
−1/2 ω
A
H
≤M
H
−1/2 −1/2 fs τ A
ψ ϕ H .
A
N−1
s0
H
H
2.17
2.18
10
Abstract and Applied Analysis
Estimate 2.11 follows from 2.14, 2.17, and 2.18. In a similar manner, we obtain
max A1/2 uk H
0≤k≤N
≤M
1/2
fk τ A ϕ ψ .
H
H
N−1
k0
H
2.19
Now, we obtain the estimates for AμH and A1/2 ωH . First, applying A to the formula of μ
and using Abel’s formula, we can write
⎧⎡
⎛
−1
n
⎨ τ
λm /τ−1 R−1 − Rλm /τ−1 R
−1
τ 2A
R
⎝
⎣
f0
αm I A1/2
Aμ Tτ
⎩ 2
2
2i
m1
n
αm
m1
−1
λm
τ 2A
λm −
τ A I
τ
2
⎞
−1 I iτ/2A1/2
λm /τ−1 R−1 I − iτ/2A1/2 Rλm /τ−1 R
R
⎠
×
2
−
/τ−1
λm
n
1
iτ 1/2 −1 λm /τ−s
iτ 1/2 −1
λm /τ−s
R
I A
I− A
αm
R
2 m1
2
2
s2
n
1
iτ 1/2 −1 λm /τ−1
iτ 1/2 −1
λm /τ−1
× fs −fs−1 αm
R
I A
I− A
f1
R
2
2
2
m1
−
n
αm
m1
λ
×
m /τ−2
s2
τ 2A
I
4
−1
fλm /τ−1 λm
τ
αm λm −
τ
m1
n
1 1/2 λm /τ−s λm /τ−s A × R
R
fs − fs−1
2i
1
λm /τ−1 f1 iA1/2 fλm /τ−2
A1/2 Rλm /τ−1 R
2i
⎤
n
λm
λm /τ−1 − Aϕ ⎦
τ ARRf
αm τ λm −
τ
m1
⎡
−1
n
−1
2
A
λ
τ
τ
k
λk −
iA1/2
× ⎣I βk
τ
A I
2
τ
2
k1
−1
τ 2A
λm
I
−
τ
×
αm λm −
τ
2
m1
⎤
−1 I iτ/2A1/2
λm /τ−1 R−1 I − iτ/2A1/2 Rλm /τ−1 R
R
⎦
×
2
−1
λk /τ−1 R−1 − Rλk /τ−1 R
R
2
n
Abstract and Applied Analysis
⎡
−1
n
n
2
λm /τ−1 R−1 − Rλm /τ−1 R
−1 τ
A
λm
R
τ
⎣ αm I αm λm −
2
2i
τ
m1
m1
τ 2A
I
2
×A
1/2
11
−1
⎤
−1 I iτ/2A1/2
λm /τ−1 R−1 I − iτ/2A1/2 Rλm /τ−1 R
R
⎦
×
2
⎛
−1
n
2
τ
A
τ
1/2
I
× ⎣ f0 ⎝ βk A
2
2
k1
⎡
×
−1 I iτ/2A1/2
λk /τ−1 R−1 I − iτ/2A1/2
Rλk /τ−1 R
R
2
2
−1
τ 2A
τ
λk
λk −
τ
A× I βk i
2
τ
2
k1
n
⎞
λk /τ−1 −1
λk /τ−1 −1
R −R
R ⎠
R
×
2
−
n
βk
s2
k1
−
n
βk
k1
n
1 λk /τ−s λk /τ−s R
R
fs − fs−1
2i
/τ−2
λk
n
1 λk /τ−1 λk /τ−1 f1 − βk ifλm /τ−2
R
R
2i
k1
λk /τ−1 βk A1/2 τRRf
k1
n
λk
τ
λk −
βk
τ
2
τ
k1
λ /τ−1
iτ 1/2 −1 λk /τ−s
iτ 1/2 −1
1 −1/2 k
λk /τ−s
R
A
R
I A
I− A
×
2
2
2
s2
1 −1/2 λk /τ−1
iτ 1/2 −1 λk /τ−1
iτ 1/2 −1
R
f1
× fs −fs−1 A
R
I A
I− A
2
2
2
−A−1/2
τ 2A
I
4
−1
⎤
fλk /τ−1 ⎦
⎤⎫
n
⎬
τ
λk
λk −
τ
A1/2 fλk /τ−1 A1/2 ψ ⎦ .
βk
⎭
2
τ
k1
2.20
12
Abstract and Applied Analysis
Second, applying A1/2 to the formula of ω and using Abel’s formula, we can write
⎧⎡
−1
n
⎨
τ 2A
1/2
⎣
I−
αm I A ω Tτ
⎩
2
m1
λm /τ−1 I iτA1/2
R
Rλm /τ−1 I − iτA1/2
×
2
2
−1
n
τ 2A
λm
I
iA1/2
τ
αm λm −
−
τ
2
m1
⎤
λm /τ−1 I −iτ/2A1/2 I iτA1/2 −Rλm /τ−1 I iτ/2A1/2 I −iτA1/2
R
⎦
×
2
⎛
−1
n
2
A
τ
1
1/2
× ⎣ f0 ⎝ βk A τ I 2
2
k1
⎡
×
−1 I iτ/2A1/2
λk /τ−1 R−1 I − iτ/2A1/2
Rλk /τ−1 R
R
2
2
−1
n
τ 2A
λk
τ
I
λk −
τ
βk
2
τ
2
k1
⎞
λk /τ−1 R−1 − Rλk /τ−1 R
−1
1/2 R
⎠
×A i
2
−
n
βk
λk
/τ−2
s2
k1
1 λk /τ−s λk /τ−s R
R
2i
n
n
1 λk /τ−1 λk /τ−1 R
f1 − βk ifλk /τ−2
R
× fs − fs−1 − βk
2i
k1
k1
n
λk /τ−1 βk A1/2 τRRf
k1
⎡
n
λk
τ
λk −
τ
A−1/2
βk
2
τ
k1
−1
−1 λk
/τ−1
1
iτ
iτ
λ
/τ−s
1/2
λ
/τ−s
1/2
k
k
R
×⎣
R
I A
I− A
2 s2
2
2
1
× fs −fs−1 2
τ 2A
− I
4
−1
R
iτ
I A1/2
2
⎤
⎤
λk /τ−1
fλk /τ−1 ⎦ A1/2 ψ ⎦
−1
λk /τ−1
R
iτ
I − A1/2
2
−1 f1
Abstract and Applied Analysis
13
⎡
−1 λ /τ−1 n
2
λk /τ−1 I iτA1/2
I − iτA1/2 R
R k
τ
τ
A
1/2
I
− ⎣ βk A
×
2
2
2
k1
−
n
βk A
−1/2
k1
τ 2A
I
2
−1
λk /τ−1 I − iτ/2A1/2 I iτA1/2
R
×
2
Rλk /τ−1 I iτ/2A1/2 I − iτA1/2
−
2
⎛
⎡
−1
n
λm /τ−1 R−1 − Rλm /τ−1 R
−1
1
τ 2A
R
× ⎣ f0 ⎝ αm I × τA1/2
2
2
2i
m1
−1
λm
τ 2A
αm λm −
τ A I
τ
2
m1
λ /τ−1 −1 −1 I iτ/2A1/2
m
R I − iτ/2A1/2 Rλm /τ−1 R
R
×
2
n
−
n
1
αm A−1/2
2 m1
λm
/τ−1
iτ 1/2 −1 λm /τ−s
iτ 1/2 −1
λm /τ−s
×
R
I A
I− A
R
2
2
s2
n
1
αm A−1/2
× fs − fs−1 2 m1
iτ 1/2 −1 λm /τ−1
iτ 1/2 −1
λm /τ−1
f1
× R
R
I A
I− A
2
2
−1
n
n
τ 2A
λm
−1/2
I
−
αm A
fλm /τ−1 −
αm λm −
τ
4
τ
m1
m1
λ /τ−2
m
1 1/2 λm /τ−s λm /τ−s A × R
×
R
fs − fs−1
2i
s2
1 1/2 λm /τ−1 λm /τ−1 1/2
R
f1 iA fλm /τ−2
A
R
2i
⎤⎫
n
⎬
λm
λm /τ−1 − Aϕ ⎦ .
αm τ λm −
τ ARRf
⎭
τ
m1
2.21
The following estimates
14
Abstract and Applied Analysis
N−1
1/2 fs − fs−1 H f0 H A ψ Aϕ H ,
≤M
Aμ
H
H
s1
1/2 A ω
H
fs − fs−1 f0 A1/2 ψ AϕH
H
H
N−1
≤M
s1
2.22
H
are obtained by using formulas 2.20, 2.21, 2.3, and 2.6.
Estimate 2.13 follows from 2.16, and 2.22. Theorem 2.3 is proved.
Now, let us consider the applications of Theorem 2.3. First, the nonlocal the mixed
boundary value problem for hyperbolic equation
utt − axux x δu ft, x,
n
u0, x 0 < t < 1, 0 < x < 1,
αm uλm , x ϕx,
0 ≤ x ≤ 1,
m1
n
ut 0, x 2.23
βk ut λk , x ψx,
0 ≤ x ≤ 1,
k1
ut, 0 ut, 1,
ux t, 0 ux t, 1,
0 ≤ t ≤ 1,
under assumption 2.2, is considered. Here ar x, x ∈ 0, 1, ϕx, ψx x ∈ 0, 1 and
ft, x t ∈ 0, 1, x ∈ 0, 1 are given smooth functions and ar x ≥ a > 0, δ > 0. The
discretization of problem 2.23 is carried out in two steps.
In the first step, the grid space is defined as follows:
0, 1h {x : xr rh, 0 ≤ r ≤ K, Kh 1}.
2.24
1
1
2
2
We introduce the Hilbert space L2h L2 0, 1h , W2h
W2h
0, 1h and W2h
W2h
0, 1h K−1
h
of the grid functions ϕ x {ϕr }1 defined on 0, 1h , equipped with the norms
1/2
K−1
2
h
h
,
ϕ x h
ϕ L2h
h
ϕ 1
W2h
h
ϕ 2
W2h
ϕh ϕh L2h
L2h
r1
r1
ϕh
1/2
2
h
,
x,j 2.25
1/2 1/2
K−1
2
2
h
ϕ
h
ϕ
h
,
x,j xx,j K−1
r1
K−1
h
r1
respectively. To the differential operator A generated by problem 2.23, we assign the difference operator Axh by the formula
'
(K−1
2.26
Axh ϕh x −axϕx x,r δϕr 1 ,
acting in the space of grid functions ϕh x {ϕr }K
0 satisfying the conditions ϕ0 ϕK , ϕ1 −ϕ0 ϕK − ϕK−1 . With the help of Axh , we arrive at the nonlocal boundary value problem
Abstract and Applied Analysis
15
d2 vh t, x
Axh vh t, x f h t, x, 0 ≤ t ≤ 1, x ∈ 0, 1h ,
dt2
n
vh 0, x αj vh λj , x ϕh x, x ∈ 0, 1h ,
2.27
j1
vth 0, x n
βj vth λj , x ψ h x,
x ∈ 0, 1h ,
j1
for an infinite system of ordinary differential equations.
In the second step, we replace problem 2.27 by difference scheme 2.28
uhk1 x − 2uhk x uhk−1 x
τ2
Axh uhk x τ 2 x 2 h
Ah uk1 x fkh x,
4
x ∈ 0, 1h ,
tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
n
λj
h
h
−1
h
h
uλj /τ x−uλj /τ−1 x λj −
τ
ϕh x, x ∈ 0, 1h ,
u0 x αj u λ /τ xτ
j τ
j1
τ 2 Axh uh1 x − uh0 x τ h
−
f0 x − Axh uh0 x
I
2
τ
2
⎫
⎧ h
h
n
⎬
⎨ uλj /τ x − uλj /τ −1 x
λ
τ
j
h
λj −
τ
× fλ
βj
x − Axh uhλj /τ x
/τ
j
⎭
⎩
τ
2
τ
j1
fkh x f h tk , x,
ψ h x,
f0h x f h 0, x,
x ∈ 0, 1h .
2.28
Theorem 2.4. Let τ and h be sufficiently small positive numbers. Suppose that assumption 2.2
holds. Then, the solution of difference scheme 2.28 satisfies the following stability estimates:
max uhk max uhk L2h
x L2h
0≤k≤N
0≤k≤N
≤ M1 max fkh ψ h ϕhx ,
L2h
L2h
L2h
0≤k≤N−1
2.29
max τ −2 uhk1 − 2uhk uhk−1 max uhk
L2h
1≤k≤N−1
≤ M1 f0h L2h
0≤k≤N
h
max τ −1 fkh − fk−1
1≤k≤N−1
L2h
xx L2h
ψxh L2h
ϕhx x L2h
.
Here, M1 does not depend on τ, h, ϕh x, ψ h x and fkh , 0 ≤ k < N.
The proof of Theorem 2.4 is based on abstract Theorem 2.3 and symmetry properties
of the operator Axh defined by 2.26.
16
Abstract and Applied Analysis
Second, for the m-dimensional hyperbolic equation under assumption 2.2 is considered. Let Ω be the unit open cube in the m-dimensional Euclidean space Rm {x x1 , . . . , xm :
0 < xj < 1, 1 ≤ j ≤ m} with boundary S, Ω Ω ∪ S. In 0, 1 × Ω, the mixed boundary value
problem for the multidimensional hyperbolic equation
m
∂2 ut, x −
ar xuxr xr ft, x,
∂t2
r1
x x1 , . . . , xm ∈ Ω,
u0, x 0 < t < 1,
n
αj u λj , x ϕx,
x ∈ Ω;
j1
n
ut 0, x βk ut λk , x ψx,
x ∈ Ω;
k1
ut, x 0,
x∈S
2.30
is considered.
Here, ar x, x ∈ Ω, ϕx, ψx x ∈ Ω and ft, x t ∈ 0, 1, x ∈ Ω are given
smooth functions and ar x ≥ a > 0. The discretization of problem 2.30 is carried out in two
steps. In the first step, let us define the grid sets
'
(
h x xr h1 r1 , . . . , hm rm , r r1 , . . . , rm , 0 ≤ rj ≤ Nj , hj Nj 1, j 1, . . . , m ,
Ω
h ∩ Ω,
Ωh Ω
h ∩ S.
Sh Ω
2.31
h , W 1 W 1 Ω
h and W 2 W 2 Ω
h of the grid
We introduce the Banach space L2h L2 Ω
2h
2h
2h
2h
h
functions ϕ x {ϕh1 r1 , . . . , hm rm } defined on Ωh , equipped with the norms
h
L2 Ω
h
ϕ 1
W2h
h
ϕ x∈Ωh
⎛
ϕh 2
W2h
⎞1/2
2
⎝
ϕh x h1 · · · hm ⎠ ,
⎛
h
ϕ L2h
ϕh ⎞1/2
m h 2
ϕ
h1 · · · hm ⎠ ,
⎝
xr ,jr L2h
x∈Ωh r1
⎛
⎞1/2
m h 2
ϕ
h1 · · · hm ⎠
⎝
xr ⎛
x∈Ωh r1
⎞1/2
2
m h ϕ
h1 · · · hm ⎠ ,
⎝
xr xr ,jr x∈Ωh r1
2.32
Abstract and Applied Analysis
17
respectively. To the differential operator A generated by problem 2.27, we assign the difference operator Axh by the formula
Axh uh −
m r1
ar xuhxr
xr ,jr
2.33
,
acting in the space of grid functions uh x, satisfying the conditions uh x 0 for all x ∈ Sh .
h . With the help of Ax we
Note that Axh is a self-adjoint positive definite operator in L2 Ω
h
arrive at the nonlocal boundary value problem
d2 vh t, x
Axh vh t, x f h t, x,
dt2
vh 0, x n
αl vh λl , x ϕh x,
0 < t < 1, x ∈ Ωh ,
h,
x∈Ω
l1
n
dvh 0, x βl vth λl , x ψ h x,
dt
l1
x ∈ Ω,
2.34
for an infinite system of ordinary differential equations.
In the second step, we replace problem 2.34 by difference scheme 2.35
uhk1 x − 2uhk x uhk−1 x
τ2
Axh uhk x fkh x f h tk , x,
uh0 x τ 2 x 2 h
Ah uk1 x fkh x,
4
x ∈ Ωh ,
tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
n
*
)
αl uhλl /τ x τ −1 uhλl /τ x − uhλl /τ−1 x λl − λl /ττ ϕh x,
h,
x∈Ω
l1
I
τ 2 Axh
2
uh1 x − uh0 x τ h
−
f0 x − Axh uh0 x
τ
2
⎫
⎧
n
⎨ uh
⎬
x − uhλl /τ−1 x τ λl
λl /τ
h
λl −
βl
τ
× fλ
x − Axh uhλl /τ x
l /τ
⎭
⎩
τ
2
τ
l1
ψ h x,
f0h x f h 0, x,
h.
x∈Ω
2.35
18
Abstract and Applied Analysis
Theorem 2.5. Let τ and h be sufficiently small positive numbers. Suppose that assumption 2.2
holds. Then, the solution of difference scheme 2.35 satisfies the following stability estimates:
max uhk 0≤k≤N
L2h
≤ M1
m h u
k xr ,jr 0≤k≤N
max
L2h
r1
max fkh 0≤k≤N−1
L2h
ψ h L2h
max τ −2 uhk1 − 2uhk uhk−1 L2h
1≤k≤N−1
≤ M1 f0h L2h
m ϕhxr ,jr L2h
r1
,
m h max uk
xr xr ,jr 0≤k≤N
r1
h
max τ −1 fkh − fk−1
L2h
1≤k≤N−1
2.36
L2h
m ψxhr ,jr L2h
r1
m ϕhxr xr ,jr r1
L2h
.
Here, M1 does not depend on τ, h, ϕh x, ψ h x and fkh , 0 ≤ k < N.
The proof of Theorem 2.5 is based on abstract Theorem 2.3, symmetry properties of the
operator Axh defined by formula 2.33, and the following theorem on the coercivity inequality
for the solution of the elliptic difference problem in L2h .
Theorem 2.6. For the solutions of the elliptic difference problem
Axh uh x ωh x,
uh x 0,
x ∈ Ωh ,
x ∈ Sh ,
2.37
the following coercivity inequality holds [21]:
m h
uxr xr ,jr r1
L2h
≤ Mωh L2h
.
2.38
3. The Second Order of Accuracy Difference Scheme Generated by A
Note that the difference scheme 2.1 generated by A2 is based on the second order approximation formula for the differential equation u t Aut ft and second order of approximation formulas for nonlocal boundary conditions. Using the differential equation above,
Abstract and Applied Analysis
19
one can construct the second order of approximation formula for this differential equation.
Thus, we obtain the following difference scheme:
1
1
τ −2 uk1 − 2uk uk−1 Auk Auk1 uk−1 fk ,
2
4
fk ftk , tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
n
λm
−1
τ
ϕ,
u0 αm uλm /τ τ uλm /τ − uλm /τ−1 × λm −
τ
m1
τ 2A
τ 2A
τ
−1
I
I
τ u1 − u0 −
f0 − Au0
4
4
2
n
λk
τ
λk −
βk τ −1 uλk /τ − uλk /τ−1 ψ,
τ
× fλk /τ − Auλk /τ
2
τ
k1
f0 f0.
3.1
In exactly the same manner using the method of Section 2 one proves
Theorem 3.1. Suppose that assumption 2.2 holds and ϕ ∈ DA, ψ ∈ DA1/2 . Then, for the
solution of difference scheme 3.1 the following stability inequalities
max uk H ≤ M
0≤k≤N
k0
max A1/2 uk H
0≤k≤N
−1/2 −1/2 fk τ A
ψ ϕH ,
A
N−1
≤M
H
H
1/2 fk τ A ϕ ψ H ,
H
N−1
k0
H
1
1
Au
Au
max τ −2 uk1 − 2uk uk−1 max u
k
k1
k−1 H
4
1≤k≤N−1
1≤k≤N−1 2
H
N−1
fk − fk−1 f0 ≤M
A1/2 ψ Aϕ
k1
H
H
H
3.2
3.3
3.4
H
hold, where M does not depend on τ, ϕ, ψ and fk , 0 ≤ k ≤ N − 1.
Moreover, we can construct the difference schemes of a second order of accuracy with
respect to one variable for approximate solution of nonlocal boundary value problems 2.30
and 2.34. Therefore, abstract Theorem 3.1 permits us to obtain the stability estimates for the
solution of these difference schemes.
4. Numerical Analysis
In this section, several numerical examples are presented for the solution of problem 4.1,
demonstrating the accuracy of the difference schemes. In general, we have not been able to
20
Abstract and Applied Analysis
determine a sharp estimate for the constants figuring in the stability inequalities. However,
the numerical examples are presented for the following nonlocal boundary value problem
∂2 ut, x ∂2 ut, x
3
sin x, 0 < t < 1, 0 < x < π,
−
ut,
x
6t
2t
∂t2
∂x2
1
1
7
1
sin x, 0 ≤ x ≤ π,
u0, x u1, x − u , x −
4
4
2
32
9
1
1
1
,x −
sin x, 0 ≤ x ≤ π,
ut 0, x ut 1, x − ut
4
4
2
16
ut, 0 ut, π 0,
4.1
0 ≤ t ≤ 1.
The exact solution of this problem is ut, x t3 sin x.
We consider the set 0, 1τ × 0, πh of a family of grid points depending on the small
parameters τ and h: 0, 1τ × 0, πh {tk , xn : tk kτ, 0 ≤ k ≤ N, Nτ 1, xn nh, 0 ≤ n ≤ M,Mh π}. For the approximate solution of problem 4.1, we have applied
the first order and the two different types of second orders of accuracy difference schemes,
respectively. Solving these difference equations we have applied a procedure of modified
Gauss elimination method with respect to n with matrix coefficients.
First let us consider the first order of accuracy in time difference scheme 1.4 for the
approximate solution of the Cauchy problem 4.1. Using difference scheme 1.4, we obtain
k1
k1
k
k−1
uk1
uk1
n1 − 2un un−1
n − 2un un
−
uk1
ftk1 , xn , tk kτ,
n
τ2
h2
1 ≤ k ≤ N − 1, Nτ 1, xn nh, 1 ≤ n ≤ M − 1, Mh π,
1 N 1 N/21 7
u − un
sin x, 1 ≤ n ≤ M − 1,
−
4 n 4
32
1
1
9
N/21
N−1
τ −1 u1n − u0n τ −1 uN
− τ −1 un
− sin x,
−uN/2
n − un
n
4
4
16
u0n xn nh,
1 ≤ n ≤ M − 1,
uk0 ukM 0,
0 ≤ k ≤ N,
4.2
for the approximate solution of Cauchy problem 4.1. We have N 1 × N 1 system of
linear equation in 4.2 and we can write them in the matrix form as
AUn1 BUn CUn−1 Dϕn ,
U0 0,
1 ≤ n ≤ M − 1,
UM 0,
where
⎡
0
⎢0
⎢
⎢0
⎢
A⎢
⎢·
⎢
⎣0
0
0
0
0
·
0
0
0
a
0
·
0
0
0
0
a
·
0
0
·
·
·
·
·
·
0
0
0
·
0
0
0
0
0
·
a
0
⎤
0
0⎥
⎥
0⎥
⎥
,
⎥
·⎥
⎥
0⎦
0 N1×N1
4.3
Abstract and Applied Analysis
⎡
21
1 0 0 0 0 ·
⎢
⎢b
⎢
⎢
⎢0
⎢
⎢0
⎢
B⎢
⎢·
⎢0
⎢
⎢0
⎢
⎢
⎢0
⎣
−1
c
b
0
·
0
0
0
d
c
b
·
0
0
0
0
d
c
·
0
0
0
·
·
·
·
·
·
·
0
0
d
·
0
0
0
1 0 0 0 ·
1
4
0
0
0
·
0
0
0
1
−
4
·
·
·
·
·
·
·
0
0
0
·
0
0
0
1
4
⎡
1
⎢0
D⎢
⎣·
0
C A,
1⎤
4⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
,
· ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
⎥
d⎥
1⎦
−
4 N1×N1
0 · 0 0 0 −
0
0
0
·
0
d
c
1
· 0 0
4
·
·
·
·
0
1
·
0
⎤
ϕ0n
⎢ ϕ1n ⎥
⎢ ⎥
ϕkn ⎢ . ⎥
⎣ .. ⎦
0
0
0
·
0
b
0
0
0
0
·
0
c
b
⎤
0
0⎥
⎥
,
·⎦
1 N1×N1
⎡
ϕN
n
,
0 ≤ k ≤ N,
N1×1
7
9
sinxn ,
sinxn ,
ϕN
n −
32
16
⎧
⎨ 6t 2t 3 sinx ,
k1
k1
n
k
ϕn ⎩ftk1 , xn , 1 ≤ k ≤ N − 1,
ϕ0n −
⎤
u0s
1
⎢ us ⎥
⎢ ⎥
Usk ⎢ . ⎥
⎣ .. ⎦
⎡
uN
s
,
0 ≤ k ≤ N, s n ± 1, n.
N1×1
4.4
Here,
a−
1
,
h2
b
1
,
τ2
c
−2
,
τ2
d
1
2
2 1.
2
τ
h
4.5
For the solution of difference equation 4.2, we have applied the modified Gauss elimination
method. Therefore, we seek a solution of the matrix equation by using the following iteration
formula:
un αn1 un1 βn1 ,
n M − 1, . . . , 2, 1, 0,
4.6
22
Abstract and Applied Analysis
where αj , βj j 1, . . . , M are N 1 × N 1 square matrices and γj are N 1 × 1 column
matrices. Now, we obtain formula of αn1 , βn1 ,
βn1
αn1 −B Cαn −1 A,
B Cαn −1 Dϕn − Cβn , n 1, 2, 3, . . . M − 1.
4.7
Note that
⎡
⎤
0 · 0
α1 ⎣ · · · ⎦
,
0 · 0 N1×N1
⎡
⎤
0 · 0
β1 ⎣ · · · ⎦
,
0 · 0 N1×N1
4.8
and UM 0.
Thus, using the formulas and matrices above we obtain the difference scheme first
order of accuracy in t and second order of accuracy in x for approximate solution of nonlocal
boundary value problem 4.1.
Second, let us consider the second order of accuracy in time implicit difference scheme
3.1 for the approximate solution of problem 4.1. Using difference scheme 3.1, we obtain
k−1
k−1
k
k
k1
k1
Un1
Un1
− 2Unk Un−1
− 2Unk1 Un−1
− 2Unk−1 Un−1
Unk1 − 2Unk Unk−1 Un1
−
−
−
τ2
2h2
4h2
4h2
6tk 2tk 3 sinxn , tk kτ, 1 ≤ k ≤ N − 1, Nτ 1,
xn nh,
1 ≤ n ≤ M − 1,
Mh π,
1 N 1 N/21 7
u − u
sin x, 1 ≤ n ≤ M − 1,
−
4
4
32
τ
τ 2 Axh
τ 2 Axh h
h
h
x h
I
I
u1 xn − u0 xn f 0, xn − Ah u0 xn 4
4
2
u0n 1
2τ−1 −uhN−2 xn 4uhN−1 xn − 3uhN xn 4
9
1
− 2τ−1 −3uhN/21 xn 4uhN/21 xn −uhN/23 xn − sinxn ,
4
16
uk0 ukM 0,
1 ≤ n ≤ M − 1,
0 ≤ k ≤ N,
4.9
Abstract and Applied Analysis
23
for the approximate solution of problem 4.1. We have again the same linear system 4.3;
therefore, we use exactly the same method that we have used for the solution of difference
equation 4.2, but matrices are different as follows:
⎡
0
⎢z
⎢
⎢0
⎢
A⎢
⎢·
⎢
⎣0
0
0
w
z
·
0
0
⎡
0
0
⎢ 1
⎢
⎢ b
c
d
⎢
⎢
⎢ 0
b
c
⎢
⎢ ·
·
·
B⎢
⎢ 0
0
0
⎢
⎢ 0
0
0
⎢
⎢ 0
0
0
⎢
⎢
⎣ 3 4
1
− τ τ − τ
2 2
2
0 ·
0
d
·
0
0
0
·
·
·
·
·
·
0
z
w
·
0
0
1
4
0
0
·
0
0
0
0
0
z
·
0
0
·
·
·
·
·
·
0
0
0
·
z
0
0
0
0
·
w
0
⎤
0
0⎥
⎥
0⎥
⎥
,
⎥
·⎥
⎥
z⎦
0 N1×N1
0
0
·
0
0
·
0
0
0
0
0
·
0
0
0
·
·
·
·
·
·
3 4
0 · − τ τ
8 8
⎡
1
⎢0
C A,
D⎢
⎣·
0
0
0
1
−
4
0
0
·
0
0
d
⎤
⎥
⎥
⎥
0
0
⎥
⎥
⎥
0
0
⎥
⎥
·
·
⎥
,
⎥
0
0
⎥
⎥
c
d
⎥
⎥
b
c
⎥
⎥
1 4
3 ⎦
− τ τ − τ
8 8
8 N1×N1
4.10
1
− τ ·
8
⎤
0 · 0
1 · 0⎥
⎥
.
· · ·⎦
0 · 1 N1×N1
Here,
x
1
1
1
,
τ 2 2h2 4
y−
2
1
1
,
τ 2 h2 2
z
−1
,
4h2
w−
1
.
2h2
4.11
Thus, using the above matrices we obtain the difference scheme second order of accuracy in
t and x for approximate solution of nonlocal boundary value problem 4.1.
Next, let us consider the second order of accuracy in time implicit difference scheme
3.1 for the approximate solution of problem 4.1. Using difference scheme 3.1, we obtain
k
k−1
ukn1 − 2ukn ukn−1
uk1
n − 2un un
−
ukn
τ2
h2
k1
k1
k1
k1
k1
k1
uk1
τ 2 un2 − 4un1 6uk1
n − 4un−1 un−2
n1 − 2un un−1
−2
4
h2
h4
6tk 2tk 3 sinxn ,
24
Abstract and Applied Analysis
tk kτ,
1 ≤ k ≤ N − 1,
Nτ 1,
xn nh,
1 ≤ n ≤ M − 1,
Mh π,
1 N 1 N/21 7
u − u
sin x, 1 ≤ n ≤ M − 1,
−
4
4
32
τ
τ 2 Axh
τ 2 Axh h
h
h
x h
I
I
u1 xn − u0 xn f 0, xn − Ah u0 xn 4
4
2
u0n 1
2τ−1 −uhN−2 xn 4uhN−1 xn − 3uhN xn 4
1
− 2τ−1 −3uhN/21 xn 4uhN/21 xn − uhN/23 xn 4
−
9
sinxn ,
16
1 ≤ n ≤ M − 1,
uk0 ukM 0,
uk1 4 k 1 k
u − u ,
5 2 5 3
ukM−1 0 ≤ k ≤ N.
4 k
1
uM−2 − ukM−3 ,
5
5
0 ≤ k ≤ N,
4.12
for the approximate solution of problem 4.1. We have N 1 × N 1 system of linear
equation in 4.12 and we can write in the matrix form as
AUn2 BUn1 CUn DUn−1 EUn−2 Rϕn ,
U0 UM 0,
U3k 4U2k − 5U1k ,
2 ≤ n ≤ M − 2,
k
k
k
UM−3
4UM−2
− 5UM−1
,
where
⎡
0 0
⎢0 0
⎢
⎢0 0
⎢
A⎢
⎢· ·
⎢
⎣0 0
0 0
⎡
0 0
⎢0 w
⎢
⎢0 0
⎢
B⎢
⎢· ·
⎢
⎣0 0
0 0
0
x
0
·
0
0
0
0
x
·
0
0
·
·
·
·
·
·
0
0
0
·
0
0
0
y
w
·
0
0
0
0
y
·
0
0
·
·
·
·
·
·
0
0
0
·
w
0
⎤
0
0⎥
⎥
0⎥
⎥
,
⎥
·⎥
⎥
0⎦
0 N1×N1
⎤
0 0
0 0⎥
⎥
0 0⎥
⎥
,
⎥
· ·⎥
⎥
y 0⎦
0 0 N1×N1
0
0
0
·
x
0
0 ≤ k ≤ N,
4.13
Abstract and Applied Analysis
⎡
0
0
⎢ 1
⎢
⎢ v
t
z
⎢
⎢ 0
v
t
⎢
⎢ ·
·
·
⎢
C⎢
⎢ 0
0
0
⎢
⎢ 0
0
0
⎢
⎢ 0
0
0
⎢
⎣ 3 4
1
− τ τ − τ
2 2
2
25
⎤
1
1
0
0 · 0
0 − ⎥
4
4⎥
0 · 0
0
0 · 0
0
0 ⎥
⎥
z · 0
0
0 · 0
0
0 ⎥
⎥
· ·
·
·
·
·
·
·
· ⎥
⎥
,
⎥
0 · 0
0
0 · 0
0
0 ⎥
⎥
0 · 0
0
0 ·
t
z
0 ⎥
⎥
0 · 0
0
0 · v
t
z ⎥
⎥
3 4
1
1 4
3 ⎦
0 · − τ τ − τ · − τ τ − τ
8 8
8
8 8
8 N1×N1
⎡
⎤
1 0 · 0
⎢0 1 · 0⎥
⎥
,
D B,
E A,
R⎢
⎣· · · · ⎦
0 0 · 1 N1×N1
⎡ 0⎤
us
⎢ 1⎥
⎢ us ⎥
⎥
Usk ⎢
, 0 ≤ k ≤ N, s n ± 2, n ± 1, n.
⎢ .. ⎥
⎣ . ⎦
uN
s
N1×1
0 ·
4.14
Here,
x
τ2
,
4h4
d
y−
τ2
,
h4
1
2
1,
τ 2 h2
1
3τ 2
−2
,
t 2,
τ 2 2h4
τ
1
1
v 2,
w − 2.
τ
h
z
4.15
For the solution of the linear system 4.13, we use the modified variant Gauss elimination
method and seek a solution of the matrix equation by the following form:
un αn1 un1 βn1 un2 γn1 , n M − 2, . . . , 2, 1, 0,
4.16
where αj , βj j 1 : M−1 are N 1×N 1 square matrices and γj -s are N 1×1 column
matrices. We obtain the following formulas of αn1 , βn1 , γn1 from linear system 4.13 by
using formula 4.16:
.−1
αn1 C Dαn Eαn−1 αn Eβn−1
.
× −B − Dβn − Eαn−1 βn ,
.−1
βn1 − C Dαn Eαn−1 αn Eβn−1 A,
.−1
γn1 C Dαn Eαn−1 αn Eβn−1
.
× Rϕn − Dγn Eαn−1 γn Eγn−1 .
4.17
26
Abstract and Applied Analysis
We have α1 , β1 , γ1 , α2 , β2 , γ2 :
⎡
⎤
0 · 0
α1 ⎣ · · · ⎦
,
0 · 0 N1×N1
⎡ ⎤
0
⎢0⎥
⎢ ⎥
γ1 ⎢ . ⎥
,
⎣ .. ⎦
0 N1×1
⎡
4
⎢5
⎢
⎢
⎢0
α2 ⎢
⎢·
⎢
⎣
0
UM 0,
⎡
⎤
0 · 0
β1 ⎣ · · · ⎦
,
0 · 0 N1×N1
⎡ ⎤
0
⎢0⎥
⎢ ⎥
γ2 ⎢ . ⎥
,
⎣ .. ⎦
0 N1×1
⎡
⎤
1
−
· 0⎥
0
·
0
⎢ 5
⎥
⎢
⎥
⎥
1
⎢
⎥
⎥
· 0⎥
⎢ 0 − · 0 ⎥
,
β2 ⎢
,
⎥
⎥
5
⎢ ·
· ·⎥
· · · ⎥
⎢
⎥
⎥
⎣
4⎦
1⎦
·
0 0 · −
5 N1×N1
5 N1×N1
-
.−1 .
βM−2 5I − 4I − αM−2 αM−1
4I − αM−2 γM−1 − γM−2 .
⎤
0
4
5
·
0
UM−1
4.18
Thus, using the above matrices we obtain the difference scheme second order of accuracy in
t and x for approximate solution of nonlocal boundary value problem 4.1.
Matlab is a programming language for numeric scientific computation. One of its
characteristic features is the use of matrices as the only data type. Therefore, the implementations of numerical examples are carried out by Matlab.
Now, we will give some numerical results for the solutions of 4.2, 4.9, and 4.12 for
different M N values where N and M are the step numbers for the time and space
variables, respectively. Note that the grid step numbers N and M in the given examples are
chosen equal for clarity and this is not necessary for the stability and solutions of the difference schemes.
The errors are computed by the following formula:
N
EM
max
1≤k≤M−1
1/2
2
k
.
utk , xn − un h
N−1
4.19
k1
Here, utk , xn represents the exact solution and ukn represents the approximate solution at
tk , xn . The errors and the related CPU times are presented in Tables 1 and 2, respectively, for
N M 20, 30, 40, 80, 300, and 500. The implementations are carried out by MATLAB 7.1
software package and obtained by a PC System 64 bit, Pentium R CoreTM i5 CPU, 3.20
6 Hz, 3.19 Hz, 4000 Mb of RAM.
Let us denote the first order of accuracy difference scheme 4.2 as F, the second order
of accuracy difference scheme generated by three points 4.9 as S1, and the second order of
accuracy difference scheme generated by five points 4.12 as S2.
In order to get accurate results, CPU times are recorded by running each program 100
times for small values N M 20, 30, 40, 80 and taking the average of the elapsed time.
Abstract and Applied Analysis
27
Table 1: Comparison of errors for approximate solutions.
Difference Scheme
F
S1
S2
20
0.0494
0.0020
0.0031
30
0.0324
0.0008
0.0015
40
0.0241
0.0004918
0.0007875
80
0.0119
0.0001230
0.0001971
300
0.0032
0.0000087
0.0000140
500
0.0019
0.0000031
80
0.4518
0.4631
0.9842
300
69.384
69.933
169.880
500
545.411
532.852
Table 2: CPU times.
Difference Scheme
F
S1
S2
20
0.0080
0.0086
0.0112
30
0.0261
0.0267
0.0441
40
0.0542
0.0572
0.1042
The following conclusions can be noted for the comparison of the numerical results
presented in the tables above.
i In the tables, it is noted that almost the same accuracy is achieved by S1 with data
error 0.0020, N 20 and by F with data error 0.0019, N 500 in different CPU
times; 545 s and 0.0086 s, respectively. This means the use of the difference scheme
S1 accelerates the computation with a ratio of more than 545/0.0086 ∼
63372 times,
that is, S1 is considerably faster than F.
ii It is also noted that almost the same accuracy is achieved by the difference scheme
S2 with data error 0.0031, N 20, and by the difference scheme F with data
error 0.0032, N 300 in different CPU times; 0.0112 s and 69 s, respectively. Thus,
the use of the difference scheme S2 accelerates the computation with a ratio of more
than 69/0.0112 ∼
6160 times, that is, S2 is considerably faster than F.
iii When we consider almost the same CPU times for F and S1 as 0.0080 s ∼
0.0086 s, F computes the solution with an error 0.0494, for N 20, where the
difference scheme S1 computes the solution with an error 0.0020, which is almost
25 times smaller error than the computation error of F. Namely, S1 yields 25 times
more accurate results than F. For S2, this ratio reduces to 0.0112/0.0080 ∼
1.4 times.
iv While both types of the second order difference schemes reach approximately the
same accuracy, the CPU time for the difference scheme S2 is always greater than
S1.
v The CPU times of difference schemes F, S1, and S2 recorder for N ≥ 100 exceed
ones. Matlab gives “out of memory” error for S2 with values N 500 which means
the memory of the computer is not enough. It is not necessary to have results for S2
when N 500, since it is obvious that CPU time of S2 is more than that of F and
S1.
vi It is observed from the tables that for larger N values the numerical results become
approximately the same for each difference scheme in the reliable range of the CPU
times. This indicates that the approximation made for the solution of problem 4.1
is valid.
In conclusion, the second order difference schemes are much more accurate than the
first order difference scheme, and the second order difference scheme generated by three
points is more convenient than the second order difference scheme generated by five points
28
Abstract and Applied Analysis
when considering the CPU times and the error levels. Comparing with many other numerical
methods, our method is not based on the relationship between the grid step sizes of time and
space variables see 25–30 and the references therein.
Acknowledgment
The authors would like to thank Professor P. E. Sobolevskii for his helpful suggestions to the
improvement of this paper.
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