- Email: [email protected]
Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions
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: [email protected] (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 < 6k < ; 1 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
matrix and is defined as follows [3,6]: 3 ... 0 7 ... 07 7 .. 7 .. 7 . .7 5 ... 0
also Ei on the ith 2 0 6 60 6 16. Ei ¼ 6 .. N6 6 6 40
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 e2 C .. .
... ... .. .
0
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 cðn; mÞ ¼ m ð2:2Þ 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
804
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
BðsÞB ðsÞ ds Y T
0
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
2 4 5 s 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 [1] K.B. Datta, B.M. Mohan, Orthogonal Function in Systems and Control (1995). [2] L.M. Delves, J.L. Mohammed, Computational Methods for Integral Equations, Cambridge University Press, 1983. [3] Z.H. Jung, W. Schanfelberger, Block-Pulse functions and their applications in control systems, Springer-Verlag, Berlin, 1992. [4] K. Maleknejad, M. Hadizadeh, A new computational method for Volterra–Hammerstein integral equations, Computers Mathematics and Applications 37 (1999) 1–8. [5] K. Maleknejad, M.K. Tavassoli, Y. Mahmoudi, Numerical solution of linear Fredholm and Voltera integral equation of the second kind by using Legandre wavelets, Journal of Sciences, Islamic Republic of Iran (to appear). [6] M. Razzaghi, A. Arabshahi, Optimal control of linear distributed-parameter system via polynomial series, International Journal of System Science 20 (1989) 1141–1148.
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: [email protected] (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 < 6k < ; 1 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
matrix and is defined as follows [3,6]: 3 ... 0 7 ... 07 7 .. 7 .. 7 . .7 5 ... 0
also Ei on the ith 2 0 6 60 6 16. Ei ¼ 6 .. N6 6 6 40
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 e2 C .. .
... ... .. .
0
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 cðn; mÞ ¼ m ð2:2Þ 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
804
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
BðsÞB ðsÞ ds Y T
0
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
2 4 5 s 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 [1] K.B. Datta, B.M. Mohan, Orthogonal Function in Systems and Control (1995). [2] L.M. Delves, J.L. Mohammed, Computational Methods for Integral Equations, Cambridge University Press, 1983. [3] Z.H. Jung, W. Schanfelberger, Block-Pulse functions and their applications in control systems, Springer-Verlag, Berlin, 1992. [4] K. Maleknejad, M. Hadizadeh, A new computational method for Volterra–Hammerstein integral equations, Computers Mathematics and Applications 37 (1999) 1–8. [5] K. Maleknejad, M.K. Tavassoli, Y. Mahmoudi, Numerical solution of linear Fredholm and Voltera integral equation of the second kind by using Legandre wavelets, Journal of Sciences, Islamic Republic of Iran (to appear). [6] M. Razzaghi, A. Arabshahi, Optimal control of linear distributed-parameter system via polynomial series, International Journal of System Science 20 (1989) 1141–1148.