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Computers and Mathematics with Applications
journal homepage:www.elsevier.com/locate/camwa
Finite difference method for multipoint nonlocal elliptic–parabolic
problems
Allaberen Ashyralyev
a,b, Okan Gercek
c,d,∗ aDepartment of Mathematics, Fatih University, Istanbul, Turkey bDepartment of Mathematics, ITTU, Ashgabat, Turkmenistan cVocational School, Fatih University, Istanbul, TurkeydDepartment of Mathematics, Yildiz Technical University, Istanbul, Turkey
a r t i c l e i n f o
Article history: Received 30 May 2010
Received in revised form 23 July 2010 Accepted 23 July 2010
Keywords: Difference schemes Elliptic–parabolic equation Nonlocal boundary value problems Stability
a b s t r a c t
A finite difference method for solving the multipoint elliptic–parabolic partial differential equation with a nonlocal boundary condition is considered. Stable difference schemes accurate to first and second orders for this problem are presented. Stability, almost coercive stability and coercive stability for the solution of these difference schemes are obtained. The theoretical statements for the solution of these difference schemes are supported by numerical examples.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Nonlocal boundary value problems for partial differential equations have been applied by various researchers in order to model numerous processes in different fields of applied sciences when they are unable to determine the boundary values of the unknown function. Many authors have investigated methods of this kind (see, for example, [1–16] and the references therein). Our interest is in the study of well-posedness of difference schemes of elliptic–parabolic problem with nonlocal boundary value problems. Numerical methods and theory of solutions of the nonlocal boundary value problems for partial differential equations of variable type were carried out in [17–29].
In the present paper, we consider the multipoint nonlocal boundary value problem for the multi-dimensional elliptic– parabolic equation
−
utt−
nX
r=1(
ar(
x)
uxr)
xr=
g(
t,
x),
0<
t<
1,
x∈
Ω,
ut+
nX
r=1(
ar(
x)
uxr)
xr=
f(
t,
x), −
1<
t<
0,
x∈
Ω,
u(
t,
x) =
0, −
1≤
t≤
1,
x∈
S,
u(
1,
x) =
nX
k=1α
ku(λ
k,
x) + ϕ(
x),
nX
k=1|
α
k| ≤
1,
−
1≤
λ
1< λ
2< · · · < λ
k< · · · < λ
n≤
0,
x∈
Ω,
u(
0+,
x) =
u(
0−,
x),
ut(
0+,
x) =
ut(
0−,
x),
x∈
Ω,
(1.1)∗Corresponding author at: Vocational School, Fatih University, Istanbul, Turkey. Tel.: +90 2128663300; fax: +90 2128663431. E-mail addresses:[email protected](A. Ashyralyev),[email protected](O. Gercek).
0898-1221/$ – see front matter©2010 Elsevier Ltd. All rights reserved.
where g
(
t,
x)(
t∈
(
0,
1),
x∈
Ω),
f(
t,
x) (
t∈
(−
1,
0),
x∈
Ω),
ar(
x)(
x∈
Ω)
, andϕ(
x)(
x∈
Ω)
are given smooth functions in[0
,
1] ×Ω. HereΩis the unit open cube in the n-dimensional Euclidean space Rn(
0<
xk
<
1,
1≤
k≤
n)
with boundaryS
,
Ω=
Ω∪
S, and ar(
x) ≥
a>
0.In this paper, we are interested in the study of stable difference schemes for the numerical solution of the multipoint nonlocal boundary value problem for the elliptic–parabolic differential equation (1.1). Difference schemes which are accurate to first and second orders are presented. The stability, almost coercive stability and coercive stability of these difference schemes are obtained. For the numerical study, procedure of modified Gauss elimination method is used to solve these difference schemes for the one-dimensional elliptic–parabolic differential equation.
2. Difference schemes and stability estimates
The discretization of problem(1.1)is carried out in two steps. In the first step, let us define the grid sets
Ωh
= {
x=
xm=
(
h1m1, . . . ,
hnmn),
m=
(
m1, . . . ,
mn),
0≤
mr≤
Nr,
hrNr=
1,
r=
1, . . . ,
n}
,
Ωh
=
Ωh∩
Ω,
Sh=
Ωh∩
S.
To the differential operator A generated by problem(1.1)we assign the difference operator Axhby the formula
Axhuhx
= −
nX
r=1 ar(
x)
uh− xr xr,mr (2.1) 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
−
d 2uh(
t,
x)
dt2+
A x hu h(
t,
x) =
gh(
t,
x),
0<
t<
1,
x∈
Ω h,
duh(
t,
x)
dt−
A x hu h(
t,
x) =
fh(
t,
x), −
1<
t<
0,
x∈
Ω h,
uh(
1,
x) =
nX
k=1α
kuh(λ
k,
x) + ϕ
h(
x),
nX
k=1|
α
k| ≤
1,
x∈
Ωh,
uh(
0+,
x) =
uh(
0−,
x),
du h(
0+
,
x)
dt=
duh(
0−
,
x)
dt,
x∈
Ωh (2.2)for an infinite system of ordinary differential equations.
In the second step, we replace problem(2.2)by the difference scheme accurate to first order (see [30])
−
u h k+1(
x) −
2uhk(
x) +
uhk−1(
x)
τ
2+
A x hu h k(
x) =
g h k(
x),
gkh(
x) =
gh(
tk,
x),
tk=
kτ,
1≤
k≤
N−
1,
Nτ =
1,
x∈
Ωh,
uhk(
x) −
uhk−1(
x)
τ
−
Axhu h k−1(
x) =
f h k(
x),
fkh(
x) =
fh(
tk−1,
x),
tk−1=
(
k−
1) τ, −
N+
1≤
k≤
0,
x∈
Ωh,
uhN(
x) =
nX
k=1α
kuhhλ k τ i(
x) + ϕ
h(
x),
x∈
Ω h,
uh1(
x) −
uh0(
x) =
u0h(
x) −
uh−1(
x),
x∈
Ωh (2.3)and the difference scheme accurate to second order (see [30,31])
−
u h k+1(
x) −
2uhk(
x) +
uhk−1(
x)
τ
2+
A x hu h k(
x) =
g h k(
x),
gkh(
x) =
gh(
tk,
x),
tk=
kτ,
1≤
k≤
N−
1,
Nτ =
1,
x∈
Ωh,
uhk(
x) −
uhk−1(
x)
τ
−
Ax h 2 u h k(
x) + ϕ
h k−1(
x) =
f h k(
x),
fkh(
x) =
fh tk−1 2,
x,
tk− 1 2=
k−
1 2τ, −
N+
1≤
k≤
0,
x∈
Ωh,
uhN(
x) =
NX
k=1α
k uhhλ k τ i(
x) +
λ
k−
λ
kτ
hfhhλ k τ i
+
Ax hu h hλ k τ i(
x)
+
ϕ
h(
x),
x∈
Ωh,
−
uh2(
x) +
4uh1(
x) −
3uh0(
x) =
3uh0(
x) −
4uh−1(
x) +
uh−2(
x),
x∈
Ωh.
(2.4)To formulate the results, we introduce the spaces L2h
=
L2(
Ωh),
W2h1=
W 12
(
Ωh)
, and W2h2=
W 22
(
Ωh)
of the grid functionsϕ
h(
x) = {ϕ(
h1m1, . . . ,
hnmn)}
defined onΩh, equipped with the normsϕ
hL2h
=
X
x∈Ωhϕ
h(
x)
2 h1
· · ·
hn
1/2,
ϕ
hW2h1
=
ϕ
hL2h
+
X
x∈Ωh nX
r=1(ϕ
h)
xr2 h1
· · ·
hn
1/2,
andϕ
hW2h2
=
ϕ
hL2h
+
X
x∈Ωh nX
r=1(ϕ
h)
xr2 h1
· · ·
hn
1/2+
X
x∈Ωh nX
r=1(ϕ
h)
xrxr,mr2 h1
· · ·
hn
1/2.
Moreover, let Fτ(
H) =
F([
a,
b]
τ,
H)
be the linear space of mesh functionsϕ
τ= {
ϕ
k}
eeNe
Ndefined on
[
a,
b]
τ= {
tk=
kh,
e
N≤
k
≤ e
e
N,
e
Nτ =
a, e
e
Nτ =
b}
with values in the Hilbert space H. Next on Fτ(
H)
we denote Banach spaces C([
a,
b]
τ,
H)
andC0α,1
([−
1,
1]
τ,
H),
C0α,1([−
1,
0]
τ,
H),
C0α([
0,
1]
τ,
H),
e
C0α,1([−
1,
1]
τ,
H),
e
C0α([−
1,
0]
τ,
H),
0< α <
1 with the normsk
ϕ
τk
C([a,b]τ,H)=
max Na≤k≤Nbk
ϕ
kk
H,
k
ϕ
τk
C α 0,1([−1,1]τ,H)= k
ϕ
τk
C([−1,1]τ,H)+
sup −N≤k<k+r≤0k
ϕ
k+r−
ϕ
kk
E(−
k)
α rα+
sup 1≤k<k+r≤N−1k
ϕ
k+r−
ϕ
kk
E((
k+
r)τ)
α(
N−
k)
α rα,
k
ϕ
τk
C α 0([−1,0]τ,H)= k
ϕ
τk
C([−1,0]τ,H)+
sup −N≤k<k+r≤0k
ϕ
k+r−
ϕ
kk
E(−
k)
α rα,
k
ϕ
τk
C α 0,1([0,1]τ,H)= k
ϕ
τk
C([0,1]τ,H)+
sup 1≤k<k+r≤N−1k
ϕ
k+r−
ϕ
kk
E((
k+
r)τ)
α(
N−
k)
α rαk
ϕ
τk
eC α0,1([−1,1]τ,H)= k
ϕ
τk
C([−1,1]τ,H)+
sup −N≤k<k+2r≤0k
ϕ
k+2r−
ϕ
kk
E(−
k)
α(
2r)
α+
sup 1≤k<k+r≤N−1k
ϕ
k+r−
ϕ
kk
E((
k+
r)τ)
α(
N−
k)
α rα,
k
ϕ
τk
eC α0([−1,0]τ,H)= k
ϕ
τk
C([−1,0]τ,H)+
sup −N≤k<k+2r≤0k
ϕ
k+2r−
ϕ
kk
E(−
k)
α(
2r)
α.
Theorem 2.1. Let
τ
and|
h| =
q
h2
1
+ · · · +
h2nbe sufficiently small numbers. Then, solutions of difference schemes(2.3)and(2.4)satisfy the following stability estimate:
uhk
N−−N1
C([−1,1]τ,L2h)
≤
M2h
{
fkh}
−−1N+1C([−1,0]τ,L2h)
+
{
gkh}
N1−1C([0,1]τ,L2h)
+
ϕ
hL2h
i .
Here M1is independent of not only
τ,
h, ϕ
h(
x)
but also fkh, −
N+
1≤
k≤
0 and gkh(
x),
1≤
k≤
N−
1.Theorem 2.2. Let
τ
and|
h| =
q
h2
1
+ · · · +
h2nbe sufficiently small numbers. Then, solutions of difference scheme(2.3)satisfythe following almost coercivity estimate:
{
τ
−2 uhk+1−
2ukh+
uhk−1}
N1−1C([0,1]τ,L2h)
+
{
τ
−1(
uhk−
uhk−1)}
0−N+1C([−1,0]τ,L2h)
+
uhk
N−1 −N
C([−1,1]τ,W2h2)
≤
M2ϕ
hW2h2
+
τ
f h 0
W2h1
+
ln 1τ + |
h|
{
fkh}
−1 −N+1C([−1,0]τ,L2h)
+
gkh
N1−1
C([0,1]τ,L2h)
,
where M2is independent of
τ,
h, ϕ
h(
x),
gkh(
x),
1≤
k≤
N−
1, and f hTheorem 2.3. Let
τ
and|
h| =
q
h21
+ · · · +
h2nbe sufficiently small numbers. Then, solutions of difference scheme(2.4)satisfy
the following almost coercivity estimate:
k{
τ
−2(
uhk+1−
2uhk+
uhk−1)}
N1−1k
C([0,1]τ,L2h)+
{
uhk}
N1−1C([0,1]τ,W2h2)
+ k{
τ
−1(
uhk−
uhk−1)}
−0N+1k
C([−1,0]τ,L2h)+
(
uh k+
uhk−1 2)
0 −N+1C([−1,0]τ,W2h2)
≤
M3f0h
L2h
+
f−h1
L2h
+
g1h
L2h
+
µ
hW2h2
+
τ
f0h
W2h1
+
τ
f−h1
W2h1
+
τ
g1h
W2h1
+
ln 1τ + |
h|
k{
f h k}
−1 −N+1k
C([−1,0]τ,L2h)+ k{
g h k}
N−1 1k
C([0,1]τ,L2h).
Here, M3is independent of
τ,
h, ϕ
h(
x),
gkh(
x),
1≤
k≤
N−
1, and fkh, −
N+
1≤
k≤
0.Proofs ofTheorems 2.1–2.3are based on the symmetry properties of the difference operator Axhdefined by formula(2.1)
in L2hand on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.
Theorem 2.4. For the solution of elliptic difference problem
Axhuh
(
x) = ω
h(
x),
x∈
Ωh,
(2.5)uh
(
x) =
0,
x∈
Sh,
the following coercivity inequality holds [32]: n
X
r=1u h − xrxr,mr
L2h
≤
M4ω
hL2h
.
Here M4is independent of h and
w
h(
x)
.Theorem 2.5. Let
τ
and|
h|
be sufficiently small positive numbers. Then, solutions of difference scheme(2.3)satisfy the following coercivity stability estimate:k{
τ
−2(
uhk+1−
2ukh+
uhk−1)}
N1−1k
C α 0,1([0,1]τ,L2h)+ k{
τ
−1(
uh k−
u h k−1)}
0 −N+1k
C α0([−1,0]τ,L2h)+ k{
u h k}
N−1 −Nk
C α0,1([−1,1]τ,W2h2)≤
M5ϕ
hW2 2h
+
τ
f0h
W1 2h
+
1α(
1−
α)
h
k{
fkh}
−−1N+1k
C α 0([−1,0]τ,L2h)+ k{
g h k}
N−1 1k
C α0,1([0,1]τ,L2h)i
,
where M5is independent of not only
τ,
h, ϕ
h(
x)
but also fkh, −
N+
1≤
k≤
0 and g hk
(
x),
1≤
k≤
N−
1.Theorem 2.6. Let
τ
and|
h|
be sufficiently small positive numbers. Then, solutions of difference scheme(2.4)satisfy the following coercivity stability estimates:k{
τ
−2(
uhk+1−
2ukh+
uhk−1)}
N1−1k
C α 0,1([0,1]τ,L2h)+ k{
τ
−1(
uh k−
u h k−1)}
0 −N+1k
eC α0([−1,0]τ,L2h)+
{
uhk}
N1−1C α0,1([0,1]τ,W2 2h)
+
(
uh k+
uhk−1 2)
0 −N+1eC α0([−1,0]τ,W2h2)
≤
M6"
ϕ
hW2h2
+
τ
f h 0
W2h1
+
τ
f h −1
W2h1
+
τ
g h 1
W2h1
+
1α(
1−
α)
h
k{
fkh}
−−1N+1k
C α 0([−1,0]τ,L2h)+ k{
g h k}
N−1 1k
C α0,1([0,1]τ,L2h)i
#
,
k{
τ
−2(
uhk+1−
2ukh+
uhk−1)}
N1−1k
C α 0,1([0,1]τ,L2h)+
(
uh k+
uhk−1 2)
0 −N+1eC α0([−1,0]τ,W2h2)
+ k{
τ
−1(
uhk−
uhk−1)}
0−N+1k
eC α0([−1,0]τ,L2h)+
{
uhk}
N −1 1C α0,1([0,1]τ,W2h2)
≤
M6"
ϕ
hW2h2
+
τ
f h 0
W2h1
+
τ
f h −1
W2h1
+
τ
g h 1
W2h1
+
1α(
1−
α)
h
k{
fkh}
−−1N+1k
eC α0([−1,0]τ,L2h)+ k{
g h k}
N−1 1k
C α0,1([0,1]τ,L2h)i
#
.
Here M6is independent of
τ,
h, ϕ
h(
x),
gkh(
x),
1≤
k≤
N−
1, and fkh, −
N+
1≤
k≤
0.Proofs ofTheorems 2.5and2.6are based on the symmetry properties of the difference operator Axhdefined by formula
(2.1)and onTheorem 2.4.
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 nonlocal boundary value problem
∂
2u∂
t2+
∂
u∂
x(
1+
x)
∂
u∂
x=
g(
t,
x),
g
(
t,
x) = −
t sin x+
(
e−t+
t)(
cos x−
x sin x)
0
<
t<
1,
0<
x< π,
∂
u∂
t+
∂
u∂
x(
1+
x)
∂
u∂
x=
f(
t,
x),
f
(
t,
x) = (−
2e−t+
1−
t)
sin x+
(
e−t+
t)(
cos x−
x sin x),
−1
<
t≤
0,
0<
x< π,
u(
1,
x) =
1 2u(−
1,
x) +
1 2u−
1 2,
x+
ϕ(
x),
ϕ(
x) =
e−1−
e 2−
1 2e 1 2+
7 4 sin x,
0≤
x≤
π,
u(
t,
0) =
u(
t, π) =
0, −
1≤
t≤
1 (3.1)for the elliptic–parabolic equation. The exact solution of this problem is
u
(
t,
x) = (
e−t+
t)
sin x.
First, applying formulae
u(
xn+1) −
u(
xn−1)
2h−
u 0(
xn) =
O(
h2),
u(
xn+1) −
2u(
xn) +
u(
xn−1)
h2−
u 00(
xn) =
O(
h2),
(3.2)and using difference scheme accurate to first order(2.3)for the approximate solutions of the nonlocal boundary value problem(3.1), we get the following system of equations
ukn+1−
2ukn+
ukn−1τ
2+
(
1+
xn)(
unk+1−
2ukn+
u k n−1)
h2+
ukn+1−
ukn−1 2h=
g(
tk,
xn),
1≤
k≤
N−
1,
1≤
n≤
M−
1,
uk n−
uk −1 nτ
+
(
1+
xn)(
ukn−+11−
2uk −1 n+
u k−1 n−1)
h2+
ukn−+11−
ukn−−11 2h=
f(
tk−1,
xn),
−
N+
1≤
k≤
0,
1≤
n≤
M−
1,
u1n−
u0nτ
+
(
1+
xn)
u−n+11−
2u −1 n+
u −1 n−1 h2+
u−n+11−
u−n−11 2h= −
(
1+
xn)
sin xn+
cos xn,
xn=
nh,
1≤
n≤
M−
1,
uNn=
1 2u −N n+
1 2u −N2 n+
ϕ (
xn) ,
xn=
nh,
0≤
n≤
M,
uk0=
ukM=
0, −
N≤
k≤
N.
(3.3)We have
(
2N+
1) × (
M+
1)
system of linear equations in(3.3)which can be written in matrix form. We can rewrite this system as the following form
1+
xn h2+
1 2h ukn+1+
1τ
2 ukn+1+
−
2τ
2−
2(
1+
xn)
h2 ukn+
1τ
2 ukn−1+
1+
xn h2−
1 2h ukn−1=
g(
tk,
xn),
1≤
k≤
N−
1,
1≤
n≤
M−
1,
1+
xn h2+
1 2h ukn−+11+
1τ
ukn+
−
1τ
−
2(
1+
xn)
h2 ukn−1+
1+
xn h2−
1 2h ukn−−11=
f(
tk−1,
xn), −
N+
1≤
k≤
0,
1≤
n≤
M−
1,
1+
xn h2+
1 2h u−n+11+
−
2(
1+
xn)
h2 u−n1+
−
1τ
u0n+
1τ
u1n+
1+
xn h2−
1 2h u−n−11= −
(
1+
xn)
sin xn+
cos xn,
xn=
nh,
1≤
n≤
M−
1,
uNn−
1 2u −N n−
1 2u −N2 n=
ϕ (
xn) ,
xn=
nh,
0≤
n≤
M,
uk0=
ukM=
0, −
N≤
k≤
N.
(3.4)In this first step, applying difference scheme accurate to first order(2.3), we obtain a system of equations in matrix form
AnUn+1
+
BnUn+
CnUn−1=
Dϕ
n,
1≤
n≤
M−
1,
U0
=
e
0,
UM=
e
0,
(3.5)
where An
,
Bn,
Cnare(
2N+
1) × (
2N+
1)
matrices defined byAn
=
an 0.
0 0 0 0 0.
0 0 0 an.
0 0 0 0 0.
0 0.
.
. .
.
.
.
.
.
.
.
0 0.
0 an 0 0 0.
0 0 0 0.
0 an 0 0 0.
0 0 0 0.
0 0 0 an 0.
0 0 0 0.
0 0 0 0 an.
0 0.
.
. .
.
.
.
.
.
.
.
0 0.
0 0 0 0 0.
an 0 0 0.
0 0 0 0 0.
0 0
,
Bn=
bn c 0.
0.
0 0 0 0 0 0.
0 0 0 0 bn c.
0.
0 0 0 0 0 0.
0 0 0.
.
. .
.
. .
.
.
.
.
. . .
.
.
0 0 0.
0.
0 bn c 0 0 0.
0 0 0 0 0 0.
0.
0 f g c 0 0.
0 0 0 0 0 0.
0.
0 0 d en d 0.
0 0 0 0 0 0.
0.
0 0 0 d en d.
0 0 0.
.
. .
.
. .
.
.
.
.
. . .
.
.
0 0 0.
0.
0 0 0 0 0 0.
d en d−1
/
2 0 0. −
1/
2.
0 0 0 0 0 0.
0 0 1
,
and Cn=
rn 0.
0 0 0 0 0.
0 0 0 rn.
0 0 0 0 0.
0 0.
. . .
.
.
.
. . .
.
0 0.
0 rn 0 0 0.
0 0 0 0.
0 rn 0 0 0.
0 0 0 0.
0 0 0 rn 0.
0 0 0 0.
0 0 0 0 rn.
0 0.
. . .
.
.
.
. . .
.
0 0.
0 0 0 0 0.
rn 0 0 0.
0 0 0 0 0.
0 0
,
and D is
(
2N+
1) × (
2N+
1)
identity matrix,ϕ
n,
Usare(
2N+
1) ×
1 column vectors asϕ
n=
ϕ
−N n...
ϕ
0 n...
ϕ
N n
,
Us=
Us−N...
Us0...
UsN
for s=
n±
1,
n withϕ
k n=
f(
tk−1,
xn), −
N+
1≤
k≤
0,
−
(
1+
xn)
sin xn+
cos xn,
k=
0,
g(
tk,
xn),
1≤
k≤
N−
1,
ϕ (
xn) ,
k=
N.
Here, we denote an=
1+
xn h2+
1 2h,
rn=
1+
xn h2−
1 2h,
bn= −
1τ
−
2(
1+
xn)
h2,
c=
1τ
,
d=
1τ
2,
en= −
2τ
2−
2(
1+
xn)
h2,
f= −
2(
1+
xn)
h2,
g= −
1τ
.
So, we have second order difference Eq.(3.5)with respect to n with matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method. This type of system was used by Samarskii and Nikolaev [33] for difference equations. Hence, we obtain a solution of the matrix equation in the following form
Uj
=
α
j+1Uj+1+
β
j+1,
j=
M−
1, . . . ,
2,
1,
UM
=
0,
where
α
j(
j=
1, . . . ,
M)
are(
2N+
1) × (
2N+
1)
square matrices andβ
j(
j=
1, . . . ,
M)
are(
2N+
1) ×
1 column matrices defined byα
j+1= −
(
B+
Cα
j)
−1A,
β
j+1=
(
B+
Cα
j)
−1(
Dϕ
j−
Cβ
j),
where j
=
1, . . . ,
M−
1, α
1is the(
2N+
1) × (
2N+
1)
zero matrix andβ
1is the(
2N+
1) ×
1 zero matrix.Second, applying formulae(3.2)and using the second order of accuracy difference scheme(2.4)for the approximate solutions of problem(3.1), we obtain the following system of equations
uk+1 n−
2ukn+
uk −1 nτ
2+
(
1+
xn)(
ukn+1−
2ukn+
ukn−1)
h2+
uk n+1−
ukn−1 2h=
g(
tk,
xn),
tk=
kτ,
xn=
nh,
1≤
k≤
N−
1,
1≤
n≤
M−
1,
uk n−
uk −1 nτ
+
(
1+
xn)(
ukn+1−
2u k n+
ukn−1)
2h2+
ukn+1−
ukn−1 4h+
(
1+
xn)(
u k−1 n+1−
2uk −1 n+
u k−1 n−1)
2h2+
ukn−+11−
ukn−−11 4h=
f tk−
τ
2,
xn,
tk=
kτ,
xn=
nh, −
N+
1≤
k≤
0,
1≤
n≤
M−
1,
−
u2n+
4u1n−
3u0n=
3u0n−
4un−1+
u−n2,
xn=
nh,
1≤
n≤
M−
1,
uNn=
1 2u −N n+
1 2u −N2 n+
ϕ (
xn) ,
xn=
nh,
0≤
n≤
M,
uk0=
ukM=
0, −
N≤
k≤
N.
(3.6)Again, we have the
(
2N+
1) × (
M+
1)
system of linear equations and we will write them in the matrix form. We can rewrite this system in the following form
1+
xn h2+
1 2h ukn+1+
1τ
2 ukn+1+
−
2τ
2−
2(
1+
xn)
h2 ukn+
1τ
2 ukn−1+
1+
xn h2−
1 2h ukn−1=
g(
tk,
xn),
1≤
k≤
N−
1,
1≤
n≤
M−
1,
1+
xn 2h2+
1 4h ukn−+11+
1+
xn 2h2+
1 4h ukn+1+
−
1τ
−
1+
xn h2 ukn−1+
1τ
−
1+
xn h2 ukn+
1+
xn 2h2−
1 4h ukn−−11+
1+
xn 2h2−
1 4h ukn−1=
f tk−
τ
2,
xn, −
N+
1≤
k≤
0,
1≤
n≤
M−
1,
u−n2−
4u−n1+
6u0n−
4un1+
u2n=
0,
1≤
n≤
M−
1,
uNn−
1 2u −N n−
1 2u −N2 n=
ϕ (
xn) ,
xn=
nh,
0≤
n≤
M,
uk0=
ukM=
0, −
N≤
k≤
N (3.7)for the approximate solutions of problem(3.1).
In the second step, we apply second order difference scheme(2.4)to get the system of linear equations in matrix form
AnUn+1
+
BnUn+
CnUn−1=
Dϕ
n,
1≤
n≤
M−
1,
U0
=
e
0,
UM=
e
0,
(3.8)
where An
,
Bn,
Cnare(
2N+
1) × (
2N+
1)
matrices defined byAn
=
v
nv
n 0.
0 0 0 0.
0 0 0v
nv
n.
0 0 0 0.
0 0.
.
.
.
.
.
.
.
.
.
.
0 0 0. v
nv
n 0 0.
0 0 0 0 0.
0 0 0 0.
0 0 0 0 0.
0 0 an 0.
0 0 0 0 0.
0 0 0 an.
0 0.
.
.
.
.
.
.
.
.
.
.
0 0 0.
0 0 0 0.
an 0 0 0 0.
0 0 0 0.
0 0
,
Bn=
yn zn 0.
0.
0 0 0 0 0 0 0.
0 0 0 0 yn zn.
0.
0 0 0 0 0 0 0.
0 0 0.
.
.
.
.
. . .
.
.
.
.
. . .
.
.
0 0 0.
0.
0 0 yn zn 0 0 0.
0 0 0 0 0 0.
0.
0 1−4
6−4
1 0.
0 0 0 0 0 0.
0.
0 0 0 d en d 0.
0 0 0 0 0 0.
0.
0 0 0 0 d en d.
0 0 0.
.
.
.
.
. . .
.
.
.
.
. . .
.
.
0 0 0.
0.
0 0 0 0 0 0 0.
d en d−1
/
2 0 0. −
1/
2.
0 0 0 0 0 0 0.
0 0 1
,
and Cn=
w
nw
n 0.
0 0 0 0 0.
0 0 0w
nw
n.
0 0 0 0 0.
0 0.
.
.
. .
.
.
.
. . .
.
0 0 0.
0w
nw
n 0 0.
0 0 0 0 0.
0 0 0 0 0.
0 0 0 0 0.
0 0 0 rn 0.
0 0 0 0 0.
0 0 0 0 rn.
0 0.
.
.
. .
.
.
.
. . .
.
0 0 0.
0 0 0 0 0.
rn 0 0 0 0.
0 0 0 0 0.
0 0
,
Table 1
Error analysis for the solution u(t,x).
Method N=M=30 N=M=60 N=M=90 Difference scheme(2.3) 0.015167 0.007318 0.004822 Difference scheme(2.4) 0.000908 0.000227 0.000101
ϕ
n=
ϕ
−N n...
ϕ
0 n...
ϕ
N n
,
Us=
Us−N...
Us0...
UsN
for s=
n±
1,
n.
Here, we denotev
n=
1+
xn 2h2+
1 4h,
w
n=
1+
xn 2h2−
1 4h,
an=
1+
xn h2+
1 2h,
rn=
1+
xn h2−
1 2h,
yn= −
1τ
−
1+
xn h2,
zn=
1τ
−
1+
xn h2,
d=
1τ
2,
en= −
2τ
2−
2(
1+
xn)
h2 withϕ
k n=
f tk−
τ
2,
xn, −
N+
1≤
k≤
0,
0,
k=
0,
g(
tk,
xn),
1≤
k≤
N−
1,
ϕ (
xn) ,
k=
N.
Hence, we have second order difference equation(3.8)with respect to n with matrix coefficients. To solve this difference equation, we have applied the same procedure of modified Gauss elimination method.
Finally, we give the results of the numerical analysis. The numerical solutions are recorded for different values of N and
M and uk
nrepresents the numerical solutions of these difference schemes at
(
tk,
xn)
.Table 1is constructed for N=
M=
30, 60 and 90, respectively and the error is computed by the following formula.EMN
=
max −N≤k≤N 1≤n≤M−1u
(
tk,
xn) −
ukn.
Thus, by using the second order of accuracy difference scheme, the accuracy of solution increases faster than the first order of accuracy difference scheme.
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