- Email: [email protected]
Numerical solution of Fredholm integral equations by using CAS wavelets
Applied Mathematics and Computation 183 (2006) 458–463 www.elsevier.com/locate/amc
Numerical solution of Fredholm integral equations by using CAS wavelets S. Yousefi
a,*
, A. Banifatemi
b
a
b
Department of Mathematics, Shahid Beheshti University, Tehran, Iran Department of Applied Mathematics, Amirkabir University of Technology, Tehran, Iran
Abstract A numerical method for solving the Fredholm integral equations is presented. The method is based upon CAS wavelet approximations. The properties of CAS wavelet are first presented. CAS wavelet approximations method are then utilized to reduce the Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. 2006 Elsevier Inc. All rights reserved. Keywords: CAS wavelets; Fredholm integral equations; Numerical method
1. Introduction Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representation and segmentations, time–frequency analysis and fast algorithms for easy implementation [1]. Wavelets permit the accurate representation of a variety of functions and operators. Moreover wavelets establish a connection with fast numerical algorithms [2]. Several numerical methods for approximating the solution of integral equations are known. For Fredholm– Hammerstein integral equations, the classical method of successive approximations was introduced in [3]. A variation of the Nystrom method was presented in [4]. A collocation type method was developed in [5]. In [6], Yalcinbas applied a Taylor polynomial solution to Volterra–Fredholm integral equations. In the present article, we are concerned with the application of CAS wavelets to the numerical solution of a Fredholm integral equation of the form Z 1 yðxÞ ¼ f ðxÞ þ k Kðx; tÞyðtÞdt 0 6 x; t 6 1; ð1Þ 0
*
Corresponding author. E-mail address: s-yousefi@sbu.ac.ir (S. Yousefi).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.081
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
459
where f(x) and the kernels K(x, t) are assumed to be in L2(R) on the interval 0 6 x, t 6 1. We assume that Eq. (1) has a unique solution y to be determined. In this paper, we introduce a new numerical method to solve Fredholm integral equations. The method consists of reducing the integral equations to a set of algebraic equations by expanding the solution as CAS wavelets with unknown coefficients. The CAS wavelets are first given. The product operational matrix and orthonormality property of CAS wavelets basis are then utilized to evaluate the coefficients of CAS wavelets expansion. The article is organized as follows: In Section 2, we describe the basic formulation of wavelets and CAS wavelets required for our subsequent development. Section 3 is devoted to the solution of Eq. (1) by using CAS wavelets. In Section 4, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples. 2. Properties of CAS wavelets 2.1. Wavelets and CAS wavelets Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets as [7] tb 12 wa;b ðtÞ ¼ jaj w ; a; b 2 R; a 6¼ 0: a k If we restrict the parameters a and b to discrete values as a ¼ ak 0 , b ¼ nb0 a0 , a0 > 1, b0 > 0 and n, and k positive integers, we have the following family of discrete wavelets: k=2 wk;n ðtÞ ¼ ja0 j w ak0 t nb0 ;
where wk,n(t) form a wavelet basis for L2(R). In particular, when a0 = 2 and b0 = 1 then wk,n(t) forms an orthonormal basis [7]. CAS wavelets wnm(t) = w(k, n, m, t) have four arguments n = 0, 1, 2, . . . , 2k 1, k can assume any nonnegative integer, m is any integer and t is the normalized time. They are defined on the interval [0, 1) as ( k ; 22 CASm ð2k t nÞ; for 2nk 6 t < nþ1 2k wnm ðtÞ ¼ ð2Þ 0; otherwise; where CASm ðtÞ ¼ cosð2mptÞ þ sinð2mptÞ:
ð3Þ
The dilation parameter is a = 2k and translation parameter is b = n 2k. The set of CAS wavelets are an orthonormal basis. 2.2. Function approximation A function f(t) defined over [0, 1) may be expanded as 1 X X f ðtÞ ¼ cnm wnm ðtÞ;
ð4Þ
n¼1 m2Z
where cnm = (f(t), wnm(t)), in which (. , .) denotes the inner product. If the infinite series in Eq. (4) is truncated, then Eq. (4) can be written as f ðtÞ ’
2k M X X n¼1 m¼M
cnm wnm ðtÞ ¼ C T WðtÞ;
ð5Þ
460
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
where C and W(t) are 2k(2M + 1) · 1 matrices given by h iT C ¼ c1ðMÞ ; c1ðMþ1Þ ; . . . ; c1ðMÞ ; c2ðMÞ ; . . . ; c2ðMÞ ; . . . ; c2k ðMÞ ; . . . ; c2k ðMÞ ; h iT WðtÞ ¼ w1ðMÞ ðtÞ; w1ðMþ1Þ ðtÞ; . . . ; w1M ðtÞ; w2ðMÞ ðtÞ; . . . ; w2M ðtÞ; . . . ; w2k ðMÞ ðtÞ; . . . ; w2k M ðtÞ :
ð6Þ ð7Þ
2.3. The product operational matrix of the CAS wavelet Let e WðtÞWT ðtÞC ’ CWðtÞ;
ð8Þ
e is a 2k(2M + 1) · 2k(2M + 1) product operational matrix. To illustrate the calculation procedure we where C choose M = 1 and k = 1. Thus we have T C ¼ c1ð1Þ ; c10 ; c11 ; c2ð1Þ ; c20 ; c21 ; ð9Þ h iT WðtÞ ¼ w1ð1Þ ðtÞ; w10 ðtÞ; w11 ðtÞ; w2ð1Þ ðtÞ; w20 ðtÞ; w21 ðtÞ : ð10Þ In Eq. (10) we have 9 w1ð1Þ ðtÞ ¼ 21=2 ðcosð4ptÞ sinð4ptÞÞ > =
w10 ðtÞ ¼ 21=2 w11 ðtÞ ¼ 21=2 ðcosð4ptÞ þ sinð4ptÞÞ
> ;
9 w2ð1Þ ðtÞ ¼ 21=2 ðcosð4ptÞ sinð4ptÞÞ > = w20 ðtÞ ¼ 21=2
w21 ðtÞ ¼ 2
1=2
ðcosð4ptÞ þ sinð4ptÞÞ
> ;
1 06t< ; 2
ð11Þ
1 6 t < 1; 2
ð12Þ
we also get 0
w1ð1Þ w1ð1Þ
B w w B 10 1ð1Þ B B w11 w1ð1Þ T WðtÞW ðtÞ ¼ B Bw B 2ð1Þ w1ð1Þ B @ w20 w1ð1Þ w21 w1ð1Þ
w1ð1Þ w10
w1ð1Þ w11
w1ð1Þ w2ð1Þ
w1ð1Þ w20
w10 w10 w11 w10
w10 w11 w11 w11
w10 w2ð1Þ w11 w2ð1Þ
w10 w20 w11 w20
w2ð1Þ w10 w20 w10
w2ð1Þ w11 w20 w11
w2ð1Þ w2ð1Þ w20 w2ð1Þ
w2ð1Þ w20 w20 w20
w21 w10
w21 w11
w21 w2ð1Þ
w21 w20
Expanding each product by CAS wavelet basis we get pffiffiffi 0 pffiffiffi 0 0 2w1ð1Þ 2w10 pffiffiffi pffiffiffi B pffiffiffi B 2w1ð1Þ 0 2w10 2w11 B pffiffiffi pffiffiffi B B 0 2w10 0 2w11 WðtÞWT ðtÞ ’ B pffiffiffi B 0 0 0 2w20 B B pffiffiffi B 0 0 0 2w2ð1Þ @ 0
0
0
0
0 0 0 pffiffiffi 2w2ð1Þ pffiffiffi 2w20 pffiffiffi 2w21
e in Eq. (8) is By using the vector C in Eq. (9) the 6 · 6 matrix C ! pffiffiffi e e ¼ 2 C1 0 ; C e2 0 C
0
w1ð1Þ w21
1
C C C C C: w2ð1Þ w21 C C C w20 w21 A w10 w21 w11 w21
ð13Þ
w21 w21 1
C C C C 0 C C: 0 C C C pffiffiffi 2w21 C A pffiffiffi 2w20 0
ð14Þ
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
461
e i , i = 1, 2 are 3 · 3 matrices given by where C 0 1 cið1Þ 0 ci0 C ei ¼ B C ci0 ci1 A: @ cið1Þ 0 ci1 ci0 3. Solution of the Fredholm integral equations Consider the Fredholm integral equations given in Eq. (1). In order to use CAS wavelets, we first approximate y(x) as yðxÞ ¼ C T WðxÞ;
ð15Þ
T
f ðxÞ ¼ d WðxÞ;
ð16Þ
T
ð17Þ
and Kðx; tÞ ¼ WðxÞ KWðtÞ;
where C, and W(x) are defined similarly to Eqs. (6) and (7). Also K is 2k(2M + 1) · 2k(2M + 1) matrices where the elements of K calculated as follows: Z 1Z 1 Wni ðxÞWlj ðtÞKðx; tÞdtdx; 0
0
where n ¼ 1; . . . ; 2k ;
i ¼ M; . . . ; M;
l ¼ 1; . . . ; 2k ;
j ¼ M; . . . ; M:
Then we have C T wðxÞ ¼ d T WðxÞ þ k
Z
1 T
T
WðxÞ KWðtÞwðtÞ Cdt:
ð18Þ
0
Thus with the orthonormality of CAS wavelets we have wðxÞT C ¼ WðxÞT d þ kWðxÞT KC:
ð19Þ
Eq. (19) is a linear systems interms of C and the answer is 1
C ¼ ðI KÞ d: where I is identity matrix. 4. Illustrative examples We applied the method presented in this paper and solved three examples. This method differs from the Taylor series method used in [6], Adomian approach considered in [8] and collocation-type method approximation given in [5] and thus could be used as a basis for comparison. 4.1. Example 1 Consider the Fredholm integral equation Z 1 yðxÞ ¼ cosð4pxÞ þ tyðtÞdt: 0
We applied the method presented in this paper and solved Eq. (20) with k = 1 and M = 1. We have
ð20Þ
462
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
d¼
1 1 1 1 pffiffiffi ; 0; pffiffiffi ; pffiffiffi ; 0; pffiffiffi 2 2 2 2 2 2 2 2
T
and 2
K 66
1 8p
0
0
0:125 0 1 0 8p
0 0
0
60 6 6 60 ¼6 60 6 6 40
1 8p
0 0:375 0 1 8p
0
0
0 0
1 8p
0
1 8p
0
3
0:125 0 7 7 7 1 7 0 8p 7 1 07 7 8p 7 0:375 0 5
0
and then 1
C ¼ ðI KÞ d ¼
1 1 1 1 pffiffiffi ; 0; pffiffiffi ; pffiffiffi ; 0; pffiffiffi 2 2 2 2 2 2 2 2
T :
Therefore yðxÞ ¼ C T WðxÞ ¼ cosð4pxÞ; which is the exact solution. 4.2. Example 2 Consider the Fredholm integral equation by Z 1 1 yðxÞ ¼ x þ ð3t 6x2 ÞyðtÞdt 4 0
ð21Þ
which has the exact solution y(x) = x2 x. We applied the CAS wavelets approach and solved Eq. (21). We have
T 1 1 1 1 1 1 1 ;...; ; ; ; ; ;...; d¼ 2Mp 4p 2p 4 2p 4p 2Mp and the answer is
T 1 1 1 1 1 1 1 ; ; . . . ; ; ; ; ; . . . ; : C¼ 8p2 2p2 6 2p2 8p2 2M 2 p2 2M 2 p2 Thus yðxÞ ¼ lim C T WðxÞ ¼ x2 x M>1
which is the exact solution. 4.3. Example 3 Consider the nonlinear Fredholm integral equation by Z 1 1 yðxÞ ¼ x þ cosð2pxÞ ðxtÞy 2 ðtÞdt: 4 0 Let y(x) = CTw(x), 14 x þ cosð2pxÞ ¼ d T WðxÞ, x = hTW(x) and t = hTW(t). Thus Eq. (22) becomes Z 1 C T WðxÞ ¼ d T WðxÞ WT ðxÞhhT WðtÞC T WðtÞWT ðtÞCdt: 0
ð22Þ
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
463
Using product operational matrix (8) we have Z 1 e WT ðxÞC ¼ WT ðxÞd WT ðxÞh hT WðtÞWT ðtÞ CCdt 0
and by orthonormality we have e WT ðxÞC ¼ WT ðxÞd WT ðxÞhhT CC and then e ¼ 0: C d þ hhT CC The unknown vector C obtain with solution of the above algebraic system. For M = 1 and k = 0 we have c1ð1Þ ¼ 0:5;
c10 ¼ 0;
c11 ¼ 0:5
and then y(x) = cos(2px) which is exact solution. 5. Conclusion The aim of present work is to develope an efficient and accurate method for solving the Fredholm integral equations. The present method reduces an integral equation into a set of algebraic equations. The integration of the product of two CAS wavelet function vectors is an identity matrix, hence making CAS wavelet computationally attractive. It is also shown that the CAS wavelets provide an exact solution. Illustrative examples are included to demonstrate the validity and applicability of the technique. References [1] C.K. Chui, Wavelets: A mathematical Tool for Signal Analysis, SIAM, Philadelphia PA, 1997. [2] G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Commun. Pure Appl. Math. 44 (1991) 141– 183. [3] F.G. Tricomi, Integral Equations, Dover publications, 1982. [4] L.J. Lardy, A variation of Nystrom’s method for Hammerstein equations, J. Integral Equat. 3 (1981) 43–60. [5] S. Kumar, I.H. Sloan, A new collocation-type method for Hammerstein integral equations, J. Math. Comp. 48 (1987) 123–129. [6] S. Yalcinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comp. 127 (2002) 195–206. [7] J.S. Gu, W.S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Sys. Sci. 27 (1996) 623–628. [8] A.M. Wazwaz, A First Course in Integral Equations, World scientific Publishing Company, New Jersey, 1997.
Numerical solution of Fredholm integral equations by using CAS wavelets S. Yousefi
a,*
, A. Banifatemi
b
a
b
Department of Mathematics, Shahid Beheshti University, Tehran, Iran Department of Applied Mathematics, Amirkabir University of Technology, Tehran, Iran
Abstract A numerical method for solving the Fredholm integral equations is presented. The method is based upon CAS wavelet approximations. The properties of CAS wavelet are first presented. CAS wavelet approximations method are then utilized to reduce the Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. 2006 Elsevier Inc. All rights reserved. Keywords: CAS wavelets; Fredholm integral equations; Numerical method
1. Introduction Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representation and segmentations, time–frequency analysis and fast algorithms for easy implementation [1]. Wavelets permit the accurate representation of a variety of functions and operators. Moreover wavelets establish a connection with fast numerical algorithms [2]. Several numerical methods for approximating the solution of integral equations are known. For Fredholm– Hammerstein integral equations, the classical method of successive approximations was introduced in [3]. A variation of the Nystrom method was presented in [4]. A collocation type method was developed in [5]. In [6], Yalcinbas applied a Taylor polynomial solution to Volterra–Fredholm integral equations. In the present article, we are concerned with the application of CAS wavelets to the numerical solution of a Fredholm integral equation of the form Z 1 yðxÞ ¼ f ðxÞ þ k Kðx; tÞyðtÞdt 0 6 x; t 6 1; ð1Þ 0
*
Corresponding author. E-mail address: s-yousefi@sbu.ac.ir (S. Yousefi).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.081
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
459
where f(x) and the kernels K(x, t) are assumed to be in L2(R) on the interval 0 6 x, t 6 1. We assume that Eq. (1) has a unique solution y to be determined. In this paper, we introduce a new numerical method to solve Fredholm integral equations. The method consists of reducing the integral equations to a set of algebraic equations by expanding the solution as CAS wavelets with unknown coefficients. The CAS wavelets are first given. The product operational matrix and orthonormality property of CAS wavelets basis are then utilized to evaluate the coefficients of CAS wavelets expansion. The article is organized as follows: In Section 2, we describe the basic formulation of wavelets and CAS wavelets required for our subsequent development. Section 3 is devoted to the solution of Eq. (1) by using CAS wavelets. In Section 4, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples. 2. Properties of CAS wavelets 2.1. Wavelets and CAS wavelets Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets as [7] tb 12 wa;b ðtÞ ¼ jaj w ; a; b 2 R; a 6¼ 0: a k If we restrict the parameters a and b to discrete values as a ¼ ak 0 , b ¼ nb0 a0 , a0 > 1, b0 > 0 and n, and k positive integers, we have the following family of discrete wavelets: k=2 wk;n ðtÞ ¼ ja0 j w ak0 t nb0 ;
where wk,n(t) form a wavelet basis for L2(R). In particular, when a0 = 2 and b0 = 1 then wk,n(t) forms an orthonormal basis [7]. CAS wavelets wnm(t) = w(k, n, m, t) have four arguments n = 0, 1, 2, . . . , 2k 1, k can assume any nonnegative integer, m is any integer and t is the normalized time. They are defined on the interval [0, 1) as ( k ; 22 CASm ð2k t nÞ; for 2nk 6 t < nþ1 2k wnm ðtÞ ¼ ð2Þ 0; otherwise; where CASm ðtÞ ¼ cosð2mptÞ þ sinð2mptÞ:
ð3Þ
The dilation parameter is a = 2k and translation parameter is b = n 2k. The set of CAS wavelets are an orthonormal basis. 2.2. Function approximation A function f(t) defined over [0, 1) may be expanded as 1 X X f ðtÞ ¼ cnm wnm ðtÞ;
ð4Þ
n¼1 m2Z
where cnm = (f(t), wnm(t)), in which (. , .) denotes the inner product. If the infinite series in Eq. (4) is truncated, then Eq. (4) can be written as f ðtÞ ’
2k M X X n¼1 m¼M
cnm wnm ðtÞ ¼ C T WðtÞ;
ð5Þ
460
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
where C and W(t) are 2k(2M + 1) · 1 matrices given by h iT C ¼ c1ðMÞ ; c1ðMþ1Þ ; . . . ; c1ðMÞ ; c2ðMÞ ; . . . ; c2ðMÞ ; . . . ; c2k ðMÞ ; . . . ; c2k ðMÞ ; h iT WðtÞ ¼ w1ðMÞ ðtÞ; w1ðMþ1Þ ðtÞ; . . . ; w1M ðtÞ; w2ðMÞ ðtÞ; . . . ; w2M ðtÞ; . . . ; w2k ðMÞ ðtÞ; . . . ; w2k M ðtÞ :
ð6Þ ð7Þ
2.3. The product operational matrix of the CAS wavelet Let e WðtÞWT ðtÞC ’ CWðtÞ;
ð8Þ
e is a 2k(2M + 1) · 2k(2M + 1) product operational matrix. To illustrate the calculation procedure we where C choose M = 1 and k = 1. Thus we have T C ¼ c1ð1Þ ; c10 ; c11 ; c2ð1Þ ; c20 ; c21 ; ð9Þ h iT WðtÞ ¼ w1ð1Þ ðtÞ; w10 ðtÞ; w11 ðtÞ; w2ð1Þ ðtÞ; w20 ðtÞ; w21 ðtÞ : ð10Þ In Eq. (10) we have 9 w1ð1Þ ðtÞ ¼ 21=2 ðcosð4ptÞ sinð4ptÞÞ > =
w10 ðtÞ ¼ 21=2 w11 ðtÞ ¼ 21=2 ðcosð4ptÞ þ sinð4ptÞÞ
> ;
9 w2ð1Þ ðtÞ ¼ 21=2 ðcosð4ptÞ sinð4ptÞÞ > = w20 ðtÞ ¼ 21=2
w21 ðtÞ ¼ 2
1=2
ðcosð4ptÞ þ sinð4ptÞÞ
> ;
1 06t< ; 2
ð11Þ
1 6 t < 1; 2
ð12Þ
we also get 0
w1ð1Þ w1ð1Þ
B w w B 10 1ð1Þ B B w11 w1ð1Þ T WðtÞW ðtÞ ¼ B Bw B 2ð1Þ w1ð1Þ B @ w20 w1ð1Þ w21 w1ð1Þ
w1ð1Þ w10
w1ð1Þ w11
w1ð1Þ w2ð1Þ
w1ð1Þ w20
w10 w10 w11 w10
w10 w11 w11 w11
w10 w2ð1Þ w11 w2ð1Þ
w10 w20 w11 w20
w2ð1Þ w10 w20 w10
w2ð1Þ w11 w20 w11
w2ð1Þ w2ð1Þ w20 w2ð1Þ
w2ð1Þ w20 w20 w20
w21 w10
w21 w11
w21 w2ð1Þ
w21 w20
Expanding each product by CAS wavelet basis we get pffiffiffi 0 pffiffiffi 0 0 2w1ð1Þ 2w10 pffiffiffi pffiffiffi B pffiffiffi B 2w1ð1Þ 0 2w10 2w11 B pffiffiffi pffiffiffi B B 0 2w10 0 2w11 WðtÞWT ðtÞ ’ B pffiffiffi B 0 0 0 2w20 B B pffiffiffi B 0 0 0 2w2ð1Þ @ 0
0
0
0
0 0 0 pffiffiffi 2w2ð1Þ pffiffiffi 2w20 pffiffiffi 2w21
e in Eq. (8) is By using the vector C in Eq. (9) the 6 · 6 matrix C ! pffiffiffi e e ¼ 2 C1 0 ; C e2 0 C
0
w1ð1Þ w21
1
C C C C C: w2ð1Þ w21 C C C w20 w21 A w10 w21 w11 w21
ð13Þ
w21 w21 1
C C C C 0 C C: 0 C C C pffiffiffi 2w21 C A pffiffiffi 2w20 0
ð14Þ
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
461
e i , i = 1, 2 are 3 · 3 matrices given by where C 0 1 cið1Þ 0 ci0 C ei ¼ B C ci0 ci1 A: @ cið1Þ 0 ci1 ci0 3. Solution of the Fredholm integral equations Consider the Fredholm integral equations given in Eq. (1). In order to use CAS wavelets, we first approximate y(x) as yðxÞ ¼ C T WðxÞ;
ð15Þ
T
f ðxÞ ¼ d WðxÞ;
ð16Þ
T
ð17Þ
and Kðx; tÞ ¼ WðxÞ KWðtÞ;
where C, and W(x) are defined similarly to Eqs. (6) and (7). Also K is 2k(2M + 1) · 2k(2M + 1) matrices where the elements of K calculated as follows: Z 1Z 1 Wni ðxÞWlj ðtÞKðx; tÞdtdx; 0
0
where n ¼ 1; . . . ; 2k ;
i ¼ M; . . . ; M;
l ¼ 1; . . . ; 2k ;
j ¼ M; . . . ; M:
Then we have C T wðxÞ ¼ d T WðxÞ þ k
Z
1 T
T
WðxÞ KWðtÞwðtÞ Cdt:
ð18Þ
0
Thus with the orthonormality of CAS wavelets we have wðxÞT C ¼ WðxÞT d þ kWðxÞT KC:
ð19Þ
Eq. (19) is a linear systems interms of C and the answer is 1
C ¼ ðI KÞ d: where I is identity matrix. 4. Illustrative examples We applied the method presented in this paper and solved three examples. This method differs from the Taylor series method used in [6], Adomian approach considered in [8] and collocation-type method approximation given in [5] and thus could be used as a basis for comparison. 4.1. Example 1 Consider the Fredholm integral equation Z 1 yðxÞ ¼ cosð4pxÞ þ tyðtÞdt: 0
We applied the method presented in this paper and solved Eq. (20) with k = 1 and M = 1. We have
ð20Þ
462
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
d¼
1 1 1 1 pffiffiffi ; 0; pffiffiffi ; pffiffiffi ; 0; pffiffiffi 2 2 2 2 2 2 2 2
T
and 2
K 66
1 8p
0
0
0:125 0 1 0 8p
0 0
0
60 6 6 60 ¼6 60 6 6 40
1 8p
0 0:375 0 1 8p
0
0
0 0
1 8p
0
1 8p
0
3
0:125 0 7 7 7 1 7 0 8p 7 1 07 7 8p 7 0:375 0 5
0
and then 1
C ¼ ðI KÞ d ¼
1 1 1 1 pffiffiffi ; 0; pffiffiffi ; pffiffiffi ; 0; pffiffiffi 2 2 2 2 2 2 2 2
T :
Therefore yðxÞ ¼ C T WðxÞ ¼ cosð4pxÞ; which is the exact solution. 4.2. Example 2 Consider the Fredholm integral equation by Z 1 1 yðxÞ ¼ x þ ð3t 6x2 ÞyðtÞdt 4 0
ð21Þ
which has the exact solution y(x) = x2 x. We applied the CAS wavelets approach and solved Eq. (21). We have
T 1 1 1 1 1 1 1 ;...; ; ; ; ; ;...; d¼ 2Mp 4p 2p 4 2p 4p 2Mp and the answer is
T 1 1 1 1 1 1 1 ; ; . . . ; ; ; ; ; . . . ; : C¼ 8p2 2p2 6 2p2 8p2 2M 2 p2 2M 2 p2 Thus yðxÞ ¼ lim C T WðxÞ ¼ x2 x M>1
which is the exact solution. 4.3. Example 3 Consider the nonlinear Fredholm integral equation by Z 1 1 yðxÞ ¼ x þ cosð2pxÞ ðxtÞy 2 ðtÞdt: 4 0 Let y(x) = CTw(x), 14 x þ cosð2pxÞ ¼ d T WðxÞ, x = hTW(x) and t = hTW(t). Thus Eq. (22) becomes Z 1 C T WðxÞ ¼ d T WðxÞ WT ðxÞhhT WðtÞC T WðtÞWT ðtÞCdt: 0
ð22Þ
S. Yousefi, A. Banifatemi / Applied Mathematics and Computation 183 (2006) 458–463
463
Using product operational matrix (8) we have Z 1 e WT ðxÞC ¼ WT ðxÞd WT ðxÞh hT WðtÞWT ðtÞ CCdt 0
and by orthonormality we have e WT ðxÞC ¼ WT ðxÞd WT ðxÞhhT CC and then e ¼ 0: C d þ hhT CC The unknown vector C obtain with solution of the above algebraic system. For M = 1 and k = 0 we have c1ð1Þ ¼ 0:5;
c10 ¼ 0;
c11 ¼ 0:5
and then y(x) = cos(2px) which is exact solution. 5. Conclusion The aim of present work is to develope an efficient and accurate method for solving the Fredholm integral equations. The present method reduces an integral equation into a set of algebraic equations. The integration of the product of two CAS wavelet function vectors is an identity matrix, hence making CAS wavelet computationally attractive. It is also shown that the CAS wavelets provide an exact solution. Illustrative examples are included to demonstrate the validity and applicability of the technique. References [1] C.K. Chui, Wavelets: A mathematical Tool for Signal Analysis, SIAM, Philadelphia PA, 1997. [2] G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Commun. Pure Appl. Math. 44 (1991) 141– 183. [3] F.G. Tricomi, Integral Equations, Dover publications, 1982. [4] L.J. Lardy, A variation of Nystrom’s method for Hammerstein equations, J. Integral Equat. 3 (1981) 43–60. [5] S. Kumar, I.H. Sloan, A new collocation-type method for Hammerstein integral equations, J. Math. Comp. 48 (1987) 123–129. [6] S. Yalcinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comp. 127 (2002) 195–206. [7] J.S. Gu, W.S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Sys. Sci. 27 (1996) 623–628. [8] A.M. Wazwaz, A First Course in Integral Equations, World scientific Publishing Company, New Jersey, 1997.