Applied Mathematics E-Notes, 13(2013), 136-140 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Maximum Norm Analysis For Nonlinear Two-Point
Boundary Value Problems.
Messaoud Boulbracheney, Fatma Al Kharousiz
Received 24 January 2013
Abstract
In this paper, we propose a new approach for the …nite di¤erence approximation on non-uniform mesh of the nonlinear two-point boundary value problem
(p(x)u0 )0 = f (x; u); a < x < b; u(a) = u(b) = 0. Under a realistic assumption on the nonlinearity and a C 3;1 [a; b] regularity of the solution, we show that
the approximation is O(h2 ) accurate in the maximum norm, making use of the
Banach’s …xed point principle.
1
Introduction
We are interested in the error estimate of the …nite di¤erence approximation of the
nonlinear boundary value problem
(p(x)u0 )0 = f (x; u); a < x < b;
u(a) = u(b) = 0;
(1)
and its discretization by …nite di¤erence methods (FDM) on (nonuniform) nodes
4 : a = x0 < x1 < ::: < xn < xn+1 < b,
hi = xi
xi+1 ; h = max hi :
i
Throughout the paper, we assume that p(x) 2 C 1 [a; b]; p(x)
p , f is a nonlinearity satisfying the following condition:
0
@f
@u
p > 0 with a constant
K
(2)
where K is a positive constant, and put
D = u 2 C 2 [a; b] such that u(a) = u(b) = 0 :
(3)
Mathematics Subject Classi…cations: 65L10, 65L12, 65L60.
of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Muscat 123,
Oman. E-mail: boulbrac@squ.edu.om.
z College of Applied Sciences, Rustaq, Oman. E-mail: fatma.ahmed.rus@cas.edu.om
y Department
136
M. Boulbrachene and F. A. Kharousi
137
The usual discretization of (1) by the …nite di¤erence method is
8
Ui+1 Ui
Ui Ui 1
1
<
pi+ 12
pi 12
= f (xi ; Ui );
Lh Ui =
!i
hi+1
hi
:
U0 = Un+1 = 0; 1 i n;
(4)
where
! i = 12 (hi + hi+1 ); xi+ 12 = 12 (xi + xi+1 ); pi+ 12 = p(xi+ 21 )
and Ui denote the approximation of the exact value u(xi ). In a matrix form, problem
(4) reads as follows
HAU = F (U )
(5)
where
H = diag
1
1
; :::::::;
!1
!n
;
t
U = (U1 ; :::; Un )t ; F (U ) = [f (x1 ; U1 ); :::; f (xn ; Un )] ;
2
3
a1 + a2
a2
7
6
a2
a2 + a3
a3
1
7
6
A=6
7 and ai = pi
.
.
..
..
h
5
4
i
a
n
an
1
2
:
an + an+1
Thanks to Yamamoto [5], the inverse matrix of A enjoys the following interesting
properties:
A 1 = G(xi ; xj )
(6)
where G(x; ) denotes the Green’s function for the operator
L=
(pu0 )0 : D ! C[a; b]
and
jG(xi ; xj )j
M
(7)
where M is a is a constant independent of h.
Error estimates of the approximation by …nite di¤erence methods for problem (1)
have been extensively discussed by several authors since the early sixties (cf.,e.g., references [1–6]).
In this paper, we propose a new and simple approach to derive error estimate of
the above …nite di¤erence approximation. Indeed, combining assumption (2) with the
properties (6) and (7), we characterize the solution of the discrete problem (5) as the
…xed point of a contraction. As a result of this, we also show that, if the exact solution
is in C 3;1 [a; b], then the approximation is O(h2 ) accurate in the maximum norm, that
is
max jUi ui j Ch2
1
i
n
where ui = u(xi ) and C is a constant independent of h.
For existence and uniqueness of solutions to the continuous and discrete problems
(1) and (5), we refer to [5]. Next, we shall characterize the solution of the discrete
problem (5) as the unique …xed point of a contraction.
138
Maximum Norm Analysis for Two-Point Boundary Value Problems
2
Characterization of the Discrete Solution as a Fixed
Point of a Contraction
Consider the mapping
T : Rn ! R n
V ! TV =
where
is the unique solution of the linear system
HA = F (V ):
(8)
LEMMA 1. Let (6) and (7) hold. Then for h su¢ ciently small, we have
K A 1 1 H 1 1 < 1:
=
PROOF. We know that
H
1
= diagf! 1 ; :::; ! n g:
Then
1
H
Since A
1
1
h:
= G(xi ; yj ) is tridiagonal and jG(xi ; yj )j
A
1
1
M , we observe that
3M:
So, choosing h su¢ ciently small, we get the desired result.
THEOREM 1. Let conditions of Lemma 1 hold. Then T is a contraction with rate
of contraction equal to :
PROOF. Let U; V in Rn and = T U; = T V be the respective solutions of equation
(8) and W = U V . In order to simplify notations, we denote f (xi ; Ui ) by f (Ui ) and
@f
by f 0 . Then, we have
@u
F (U )
F (V ) = [f (U1 )
and
1
[f (Ui )
Wi
f (Vi )] =
So
f (V1 ); :::; f (Un )
Z
t
f (Vn )]
1
f 0 (Vi + (Ui
Vi ))d :
0
jf (Ui )
f (Vi )j
K jUi
Vi j
max jf (Ui )
f (Vi )j
Kmax jUi
and
i
i
Vi j
M. Boulbrachene and F. A. Kharousi
139
or
kF (U )
F (V )k1
On the other hand, we have
kT U
T V k1 = k
= A
1
A
1
A
1
k1
1
H
K kU
(F (U )
H
1
1
H
1
kU
1
V k1 :
F (V ))
kF (U )
1
1
V k1 :
K kU
1
F (V )k1
V k1
Thus, making use of Lemma 1, T is a contraction.
3
Error Estimate
Let U = T [u] be the unique solution of the linear system
HAU = F ([u])
(9)
t
where u is the solution of problem (1), [u] = [u1 ; :::; un ] ; and
t
F ([u]) = [f (x1 ; u1 ); :::; f (xn ; un )] :
Then we have the following error estimate.
LEMMA 2. Assume u 2 C 3;1 [a; b]. Then, there exists a constant C independent of
h such that
(10)
U [u] 1 Ch2
where
U
[u]
1
= max Ui
1 i n
ui :
For more details regarding the regularity assumption C 3;1 [a; b] of u, we refer to [5].
PROOF. Since F ([u]) is the restriction of f (u) to nodes fxi g, then problem (9) is
the discrete counterpart of (1). So, making use of [6], we get the desired result.
THEOREM 2. Let u and U be the solutions of (1) and (5), respectively. Then
there exists a constant C independent of h such that
k[u]
Ch2 :
U k1
PROOF. Since U = T U and U = T [u], making use of (10) and the fact that T is a
contraction, we get
k[u]
U k1
k[u]
U
2
T [u]k1 + kT [u]
[u]
1
+ kT [u]
Ch + k[u]
U k1 :
U k1
T U k1
140
Maximum Norm Analysis for Two-Point Boundary Value Problems
So,
k[u]
U k1
Ch2
:
1
References
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degree of accuracy nets, U.S.S.R. Comp. Math. Phys., (1961), 1465–1486.
[2] F. de Hoog and D. Jackett, On the rate of convergence of …nite di¤erence schemes
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[3] H. O. Kreiss, T. A. Manteu¤el, B. Swarts, B. Wendro¤ and A. B. White, Jr.,
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[7] F. A. M. Al Kharousi, Mathematical and Numerical Analysis of Nonlinear TwoPoint Boundary Value Problems, Master Thesis, Sultan Qaboos University, (2008).