Applied Mathematics and Computation 149 (2004) 799–806
www.elsevier.com/locate/amc
Numerical solution of linear
Fredholm integral equation by using
hybrid Taylor and Block-Pulse functions
K. Maleknejad
a
a,*
, Y. Mahmoudi
b
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
b
Science and Research Campus, Islamic Azad University of Tehran, Tehran, Iran
Abstract
In this paper, we use a combination of Taylor and Block-Pulse functions on the
interval [0,1], that is called Hybrid functions, to estimate the solution of a linear
Fredholm integral equation of the second kind. We convert the integral equation to a
system of linear equations, and by using numerical examples we show our estimation
have a good degree of accuracy.
Ó 2003 Published by Elsevier Science Inc.
Keywords: Block-Pulse functions; Fredholm integral equation; Operational matrix; Product
operation; Taylor polynomials
1. Introduction
In recent years, many different basic functions have used to estimate the
solution of integral equations, such as orthonormal bases and wavelets [3–6].
In this paper we use a simple bases, a combination of Block-Pulse functions on
½0; 1, and Taylor polynomials, that is called the Hybrid Taylor Block-Pulse
functions, to solve the linear Fredholm integral equation of the second kind.
One of the advantages of this method is that, the coefficients of expansion of
*
Corresponding author.
E-mail address: maleknejad@iust.ac.ir (K. Maleknejad).
URL: http://www.iust.ac.ir/members/maleknejad.
0096-3003/$ - see front matter Ó 2003 Published by Elsevier Science Inc.
doi:10.1016/S0096-3003(03)00180-2
800
K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
each function in this bases, could be compute directly without estimation. The
method is used to solve some examples and the numerical results are represented in the last section.
1.1. Hybrid Taylor Block-Pulse functions
Definition. A set of Block-Pulse functions bi ðkÞ, i ¼ 1; 2; . . . ; m on the interval
½0; 1 is defined as follows:
8
i1
i
<
1
6k < ;
ð1:1Þ
bi ðkÞ ¼
m
m
:
0 otherwise:
The Block-Pulse functions on ½0; 1 are disjoint, that is, for i ¼ 1; 2; . . . ; m,
j ¼ 1; 2; . . . ; m we have bi ðtÞbj ðtÞ ¼ dij bi ðtÞ, also these functions have the
property of orthogonality on ½0; 1, see [3].
Consider Taylor polynomials Tm ðtÞ ¼ tm on the interval ½0; 1. The Hybrid
Taylor Block-Pulse functions are defined as follows.
Definition. For m ¼ 0; 1; 2; . . . ; M 1 and n ¼ 0; 1; 2; . . . ; N the Hybrid Taylor
Block-Pulse functions is defined as
(
n1
n
6t < ;
bðn; m; tÞ ¼ Tm ðNt ðn 1ÞÞ
ð1:2Þ
N
N
0
otherwise:
Suppose we estimate function f ðtÞ 2 L2 ½0; 1, using Taylor polynomials of
the order M 1, on the interval ½a; b, then using TaylorÕs Residual Theorem,
the truncation error is
en ðtÞ ¼ f ðtÞ
M 1
X
ðt aÞi ðiÞ
ðt aÞM ðMÞ
f ðaÞ ¼
f ðnÞ;
i!
M!
i¼0
where n lies between a and t. Then
ken ðtÞk1 6
ðb aÞM ðMÞ
kf ðtÞk1 :
M!
ð1:3Þ
If we use the Hybrid
Block-Pulse functions on the interval ½0; 1, then
Taylor
i
for ith sub interval i1
;
,
we
will have [3]
N N
ken ðtÞk1 6
1
kf ðMÞ ðtÞk1 ;
N M M!
ð1:4Þ
where the infinity norm is computed on the ith sub interval. It shows that the
error improves while N and M are increased.
K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
801
1.2. The operational matrix
T
If BðtÞ ¼ ½bð1; 0; tÞ; bð1; 1; tÞ; . . . ; bð1; M 1; tÞ; bð2; 0; tÞ; . . . ; bðN ; M 1; tÞ ,
be the vector function of Hybrid Taylor and Block-Pulse functions on ½0; 1, the
integration of this vector BðtÞ follows:
Z t
Bðt0 Þ dt0 ’ PBðtÞ;
ð1:5Þ
0
where P is an MN MN matrix, that is called the operation matrix for Hybrid
Taylor and Block-Pulse functions [3].
Suppose that Ei , i ¼ 1; 2;. . . ; M be the operation matrix of Taylor polynomials on ith sub interval i1
; i , then the operation matrix P has the folN N
lowing form [6]:
2
3
E1 H12 . . . H1N
6 0 E2 . . . H2N 7
6
7
P ¼ 6 ..
ð1:6Þ
.. 7;
..
..
4 .
. 5
.
.
0
0 . . . EN
where Hlj is an M M
2
1 0
61
0
6
2
16
6
Hlj ¼ 6 .
..
N 6 ..
.
41
0
M
also Ei on the ith
2
0
6
60
6
16.
Ei ¼ 6
..
N6
6
6
40
matrix and is defined as follows [3,6]:
3
... 0
7
... 07
7
.. 7
..
7
. .7
5
... 0
interval is defined as follows:
3
1 0 ...
0
1
7
...
0 7
0
7
2
.. 7
.. .. . .
7
. 7:
.
. .
1 7
7
0 0 ...
5
M 1
0 0 0 ...
0
ð1:7Þ
ð1:8Þ
1.3. The product operation matrix
The following property of the product of two Hybrid Taylor and BlockPulse vector functions will also be used:
e T BðtÞ;
BðtÞBT ðtÞC ’ C
ð1:9Þ
802
K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
e is an MN MN matrix, which is
where C is a given MN column vector and C
called the product operation matrix of Hybrid Taylor and Block-Pulse functions.
Consider T ðtÞ ¼ ½T0 ðtÞ; T1 ðtÞ; . . . ; TM1 ðtÞT and A ¼ ½a0 ; a1 ; . . . ; aM1 T , where
Ti ðtÞ; i ¼ 0; 1; . . . ; M 1 is ith Taylor polynomial, then
e ðtÞ;
T ðtÞT T ðtÞA ’ AT
e is an M M matrix,
where A
2
a0 a1 a2 . . .
6 0 a0 a1 . . .
6
6 0 0 a0 . . .
6
e
A ¼ 6 ..
.. . .
..
6 .
.
.
.
6
4 0 0 0 ...
0 0 0 ...
and defined as follows:
3
aM2 aM1
aM3 aM2 7
7
aM4 aM3 7
7
.. 7:
..
. 7
.
7
a0
a1 5
0
a0
ð1:10Þ
The product operation matrix for Hybrid Taylor and Block-Pulse functions
is defined as follows:
e T BðtÞ;
BðtÞBT ðtÞC ’ C
such that
2
e1
C
6 0
e ¼6
C
6 .
4 ..
0
0
e2
C
..
.
0
...
...
..
.
...
3
0
0 7
7
.. 7
. 5
e
CN
ð1:11Þ
e i is the operation matrix of transferred Taylor polynomials on the ith sub
and C
e1 ¼ C
e2 ¼ ¼ C
e N [1,3,6].
interval and C
2. Function approximation
A function f 2 L2 ½0; 1Þ can be approximated as
f ðtÞ ’
N X
M 1
X
cðn; mÞbðn; m; tÞ ¼ C T BðtÞ;
ð2:1Þ
n¼1 m¼0
where BðtÞ is the vector function defined before and cðn; mÞ is defined as follows
[3]:
m
1
d f ðtÞ
ð2:2Þ
cðn; mÞ ¼ m
N m!
dtm
t¼n1
N
for n ¼ 1; 2; . . . ; N and m ¼ 0; 1; . . . ; M 1.
K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
803
We can also approximate the function kðt; sÞ 2 L2 ð½0; 1 ½0; 1Þ as follows:
kðt; sÞ ¼ BT ðtÞKBðsÞ;
ð2:3Þ
where K is an MN MN matrix that
iþj
1
o kðt; sÞ
Kij ¼ uþv
N u!v!
oti osj
ðt;sÞ¼ð i ; j Þ
ð2:4Þ
N N
for i; j ¼ 0; 1; . . . ; MN 1, u ¼ i Ni N , v ¼ j Nj N .
We also define the matrix D as follows
Z 1
D¼
BðtÞBT ðtÞ dt:
ð2:5Þ
0
For the Hybrid Taylor and Block-Pulse functions, D has the following form:
2
3
0
D1 0 . . .
6 0 D2 . . .
0 7
6
7
D ¼ 6 ..
.. 7;
.. . .
4 .
. 5
.
.
0
0
...
DN
where Di is defined as follows:
Z 1
1
T ðtÞT T ðtÞ dt:
Di ¼
N 0
3. Fredholm integral equation of the second kind
Consider the following integral equation:
Z 1
kðt; sÞyðsÞ ds;
qðtÞyðtÞ ¼ xðtÞ þ k
ð3:1Þ
0
where x 2 L2 ½0; 1, q 2 L2 ½0; 1, k 2 L2 ð½0; 1 ½0; 1Þ and y is an unknown
function [2].
Let approximate x; q; y and k by (2.1)–(2.4) as follows:
xðtÞ ’ X T BðtÞ;
qðtÞ ’ QT BðtÞ;
yðtÞ ’ Y T BðtÞ;
kðt; sÞ ’ BT ðtÞKBðsÞ:
With substituting in (3.1)
QT BðtÞBT ðtÞY ¼ BT ðtÞX þ k
Z
0
1
BT ðtÞKBðsÞBT ðsÞY ds
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K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
with (1.7) we have
e T Y ¼ BT ðtÞX þ kBT ðtÞK
B ðtÞ Q
T
Z
1
0
BðsÞB ðsÞ ds Y
T
¼ BT ðtÞX þ kBT ðtÞKDY ¼ BT ðtÞð X þ kKDY Þ;
then
e T Y ¼ ðX þ kKDY Þ ) ð Q
e T kKDÞY ¼ X :
Q
ð3:2Þ
This is a linear system of equations that gives the numerical solution of the
integral equation, yn .
4. Error estimation
Consider the following Fredholm integral equation of the second kind:
Z 1
yðtÞ ¼ k
kðt; sÞyðsÞ ds þ xðtÞ:
0
Let en ðtÞ ¼ yðtÞ yn ðtÞ be the error function, where yn ðtÞ is the estimation of the
true solution yðtÞ. Then
Z 1
yn ðtÞ ¼ k
kðt; sÞyn ðsÞ ds þ xðtÞ þ Hn ðtÞ;
ð4:1Þ
0
where Hn ðtÞ is the perturbation function that depends only on yn ðtÞ, and is given
with
Z 1
Hn ðtÞ ¼ yn ðtÞ k
kðt; sÞyn ðsÞ ds xðtÞ:
ð4:2Þ
0
With (4.1) and (4.2) we have
Z 1
en ðtÞ k
kðt; sÞen ðsÞ ds ¼ Hn ðtÞ:
ð4:3Þ
0
This is a Fredholm integral equation of the second kind. We can solve this
integral equation, using the method mentioned before as an estimation of the
error function of the method.
5. Numerical examples
Consider the following three examples. We solve them with different M and
N Õs, using the method represented before. The result improves when we use
larger M and N Õs as shown in tables.
K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
805
Example 1
yðtÞ ¼
Z
1
ðt þ sÞyðsÞ ds þ et þ ð1 eÞt 1:
0
With exact solution yðtÞ ¼ et . Table 1 shows the numerical results of Example 1.
Example 2
yðtÞ ¼
Z
1
0
4
5
s2 t ð3=2Þst2 yðsÞ ds þ ð3=4Þt2 lnð2Þt þ t:
3
9
With exact solution yðtÞ ¼ 2 lnðt þ 1Þ. Table 2 shows the numerical results of
Example 2.
Example 3
1
yðtÞ ¼
3
Z
1
e2tð5=3Þs yðsÞ ds þ e2tþð1=3Þ :
0
With exact solution yðtÞ ¼ e2t . Table 3 shows the numerical results of Example 3.
Table 1
M
N
ky yn k1
CondðI KDÞ
3
3
3
3
10
20
40
80
1.383039 103
1.777834 104
2.255183 105
2.840509 106
102.5481
114.2248
122.7133
128.4338
4
4
4
4
10
20
40
80
1.383039 103
1.777834 104
2.255183 105
2.840509 106
130.2973
144.1830
154.0536
160.6114
N
ky yn k1
CondðI KDÞ
Table 2
M
3
3
3
3
3
10
20
40
80
6.145928 10
1.527655 103
3.808887 104
9.509965 105
2.2860
2.5913
2.7694
2.8656
4
4
4
4
10
20
40
80
4.557614 103
1.139760 103
2.849499 104
7.123710 105
2.5172
2.8776
3.0860
3.1981
806
K. Maleknejad, Y. Mahmoudi / Appl. Math. Comput. 149 (2004) 799–806
Table 3
M
N
ky yn k1
CondðI KDÞ
2
3
3
3
3
10
20
40
80
3.005651 10
8.668870 103
2.316608 103
5.981580 104
6.0920
7.0734
7.6292
7.9248
4
4
4
4
10
20
40
80
2.892984 102
7.351252 103
1.847061 103
4.625381 104
7.1086
8.3168
9.0048
9.3718
6. Conclusion
The Block-Pulse functions have the property of orthogonality on the
interval ½0; 1, then we can combine various bases with Block-Pulse functions to
produce Hybrid functions, with the property of semi-orthogonality (orthogonality on disjoint sub intervals). The Hybrid Taylor and Block-Pulse functions are used to solve the Fredholm integral equation. The same approach can
be used to solve other problems. The numerical examples shows that the accuracy improves with increasing the M and N , then for better results, using the
larger M, specially larger N is recommended.
References
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integral equations, Computers Mathematics and Applications 37 (1999) 1–8.
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K. Maleknejad