Using the Petrov–Galerkin elements for solving Hammerstein integral equations

Using the Petrov–Galerkin elements for solving Hammerstein integral equations

Applied Mathematics and Computation 172 (2006) 831–845 www.elsevier.com/locate/amc Using the Petrov–Galerkin elements for solving Hammerstein integra...

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Applied Mathematics and Computation 172 (2006) 831–845 www.elsevier.com/locate/amc

Using the Petrov–Galerkin elements for solving Hammerstein integral equations K. Maleknejad a

a,*

, M. Karami b, M. Rabbani

a

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran b Department of Mathematics, Technical Faculty, Islamic Azad University (South Unit), Ahang Blvd., Tehran, Iran

Abstract In this paper, we use several specific constructions of the Petrov–Galerkin elements that constructed from piecewise polynomials on interval [0, 1] for solving a class of nonlinear Hammerstien equations and for showing efficiency of method, we use numerical examples.  2005 Elsevier Inc. All rights reserved. Keywords: Nonlinear integral equations; The Petrov–Galerkin method; Regular pairs; Trial space; Test space

1. Introduction In this paper, we solve Hammerstein equations given in the form uðxÞ  ðKWuÞðxÞ ¼ f ðxÞ;

x 2 ½0; 1;

*

ð1Þ

Corresponding author. E-mail addresses: [email protected] (K. Maleknejad), [email protected] (M. Karami), [email protected] (M. Rabbani). 0096-3003/$ - see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.041

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where ðKWuÞðxÞ ¼

Z

1

kðx; sÞwðs; uðsÞÞds 0

the functions f ; w, the kernel k are given and u is the unknown function to be determined. The Petrov–Galerkin method for linear Fredholm integral equations of the second kind uðxÞ 

Z

1

kðx; sÞuðsÞds ¼ f ðxÞ, x 2 ½0; 1

ð2Þ

0

has been studied in [1]. In [2], we used continuous and discontinuous Lagrange type k–0 elements with 1 6 k 6 5 for Eq. (2). In [3], we used Hermite-type 3–1 elements for Eq. (2). In [4], Alpert constructed a class of wavelet basis in L2 ½0; 1 and applied it to approximate the solution of Eq. (2). The numerical method employed in [3] was the Galerkin method. In [5], the Wavelet Petrov–Galerkin (WPG) Schemes based on discontinuous orthogonal multiwavelets were described. The results of this method yields integrals that arc not solved easily and for this problem in [6] Discrete Wavelet Petrov–Galerkin (DWPG) method was described. In [7], we used AlpertÕs multiwavelets for Eq. (2). In this paper we use continuous and discontinuous Lagrange-type k–0 elements and Hemitetype 3–1 elements for solving Eq. (1). We organize this paper as follows. In Section 2, we review the Petrov–Galerkin method for Eq. (1). In Section 3, we describe continuous Lagrange-type k–0 elements. In Section 4, we describe discontinuous Lagrange-type k–0 elements. In Section 5, we describe Hermite-type 3–1 elements. In Section 6, we use these elements for solving Eq. (1) and finally in Section 7, we will show efficiency of this method with solving two examples.

2. The Petrov–Galerkin method In this section we follow the paper [1] a brief review of the Petrov–Galerkin method. Let X be a Banach space and X  be its dual space of continuous linear functional. For each positive integer n, we assume that X n  X , Y n  X  and X n ; Y n are finite dimensional vector spaces with dimX n ¼ dimY n ;

n ¼ 1; 2; . . .

ð3Þ

also X n ; Y n satisfying condition ðH Þ: For each x 2 X and y 2 X  , there exist xn 2 X n and y n 2 Y n such that kxn  xk ! 0 and ky n  yk ! 0 as n ! 1.

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

833

Definition. for x 2 X , an element P n x 2 X n called the generalized best approximation from X n to x with respect to Y n , by the equation ðx  P n x; y n Þ ¼ 0

for all y n 2 Y n .

ð4Þ

It is proved in [1], that for each x 2 X , the generalized best approximation from X n to x with respect to Y n exists uniquely if and only if Yn \ X? n ¼ f0g

ð5Þ

under this condition, P n is a projection, i.e. P 2n ¼ P n . Assume that, for each n, there is a linear operator Pn : X n ! Y n with Pn X n ¼ Y n satisfying the follow two conditions ðH-1Þ

for all xn 2 X n ; kxn k 6 C 1 ðxn ; Pn xn Þ1=2 ;

ðH-2Þ

for all xn 2 X n ; kPn xn k 6 C 2 kxn k;

where C 1 ; C 2 are positive constants independent of n. If a pair of space sequences fX n g and fY n g satisfy ðH-1Þ and ðH-2Þ, we call fX n ; Y n g a regular Pair. It is Proved in [1], that, if a regular Pair fX n ; Y n g satisfies dimX n ¼ dimY n and condition ðHÞ, then the corresponding generalized projection P n satisfies: (1) for all x 2 X ; kP n x  xk ! 0 as n ! 1; (2) there is a constant C > 0 such that, kP n k < C; n ¼ 1; 2; . . .; (3) for some constant C > 0 independent of n; kP n x  xk 6 CkQn x  xk; where Qn x is the best approximation from X n to x. The Petrov–Galerkin method for Eq. (1) is a numerical method for finding un 2 X n such that ðun  KWun ; y n Þ ¼ ðf ; y n Þ

for all y n 2 Y n .

ð6Þ

Eq. (6) is equivalent to un  P n KWun ¼ P n f .

ð7Þ

Eq. (7) can also be derived from the fact that P n x ¼ 0 for an x 2 X if and if ðx; y n Þ ¼ 0 for all y n 2 Y n . The Petrov–Galerkin methods using regular pairs fX n ; Y n g of piecewise polynomial spaces are called Petrov–Galerkin elements. If we use piecewise polynomials of degree k and k 0 for spaces X n and Y n respectively, we call the corresponding Petrov–Galerkin elements k  k 0 elements.

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3. Continuous Lagrange-type k–0 elements We subdivide the interval [0, 1] into n subinterval by a sequence of points 0 ¼ t0 < t1 <    < tn ¼ 1. Denote I i ¼ ½ti1 ; ti  and hi ¼ ti  ti1 for i ¼ 1; . . . ; n and let X n be the space of continuous piecewise polynomials of degree 6 k with knots at ti ; i ¼ 1; 2; . . . ; n  1. We construct a basis for X n . For this purpose, let sj ¼ kj ; j ¼ 0; 1; . . . ; k and define ðiÞ

tj ¼ ti1 þ sj hi ;

j ¼ 0; 1; . . . ; k; i ¼ 1; . . . ; n

ðiÞ

ðiÞ

ðiÞ

then, clearly ti1 ¼ t0 <    < tk ¼ ti . We now define nk þ 1 function Uj ðtÞ by letting 2 i ¼ 1; 2; . . . ; n; 8 ðiÞ Q tt < k‘¼0 ‘ 6 t 2 I i 6 j ¼ 1; 2; . . . ; k  1; ðiÞ ðiÞ ðiÞ t t‘ Uj ðtÞ ¼ 6 ‘6¼j j : 4 i ¼ 1; j ¼ 0; 0 t 62 I i i ¼ n; j ¼ k; 8 Qk1 ttðiÞ > ‘ > > ‘¼0 tðiÞ tðiÞ > < k ‘ ðiÞ ðiþ1Þ tt‘ Uk ðtÞ ¼ Qk > ‘¼1 ðtðiþ1Þ tðiþ1Þ > > 0 ‘ > : 0

t 2 I i; t 2 I iþ1 ;

i ¼ 1; 2; . . . ; n  1;

t 62 I i [ I iþ1 .

Note that for any xn 2 X n , we have xn ðtÞ ¼

k X

ðiÞ

ðiÞ

xn ðtj ÞUj ðtÞ;

t 2 I i ; i ¼ 1; . . . ; n.

j¼0

To construct the test space Y n , we define ( h1 1 0 6 t 6 2k ; ð1Þ W0 ðtÞ ¼ 0 otherwise; ( ðiÞ Wj ðtÞ

ðiÞ Wk ðtÞ

ðnÞ

¼  ¼

Wk ðtÞ ¼



1 0

ti1 þ 2j1 hi 6 t 6 ti1 þ 2jþ1 hi 2k 2k otherwise

1

hi 6 t 6 ti þ 2k1 hiþ1 ti1 þ 2k1 2k

0

otherwise

1 0

hn 6 t 6 1; tn1 þ 2k1 2k otherwise;

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; k  1; i ¼ 1; 2; . . . ; n  1;

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

835

and define a linear operator Pn : X n ! Y n as follows: Pn xn ðtÞ ¼

k X

ðiÞ

ðiÞ

xn ðtj ÞWj ðtÞ;

t 2 I i ; i ¼ 1; . . . ; n:

j¼0

Then dimX n ¼ dimY n ¼ nk þ 1 and Pn X n ¼ Y n and in [1] is proved that for 1 6 k 6 5 these two space sequences form a regular pair. nkþ1 We arrange nk þ 1 basis functions for X n by fbj gj¼1 and nk þ 1 basis func nkþ1 tions for Y n by fbj gj¼1 and let B be a ðnk þ 1Þ  ðnk þ 1Þ matrix with entries bij ¼ ðbi ðtÞ; bj ðtÞÞ;

i; j ¼ 1; 2; . . . ; nk þ 1.

A direct computation gives 0 L1 N 1 0 0 B B M 1 L2 N 2 0 B B 0 M L3 N 3 2 B B B .. .. .. .. B¼B . . . . B B B 0 0 0 0 B B 0 0 0 0 @ 0 0 0 0

0

...

0

0

0

0

...

0

0

0

0 .. .

... .. .

0 .. .

0 .. .

0 .. .

0

. . . M n2

Ln1

N n1

0

...

0

M n1

Ln

0

...

0

0

Wn

0

1

C 0 C C 0 C C C .. C ; . C C C 0 C C VnC A Zn

where Li , M i and N i are k  k matrices and V n and W n are respectively k  1 and 1  k vectors and Z n is a scaler. For k ¼ 1; . . . ; 5 and h0 ¼ 0 we have k¼1 Li ¼ 38ðhi1 þ hi Þ; i ¼ 1; 2; . . . ; n; hi M i ¼ N i ¼ ; i ¼ 1; . . . ; n  1; 8 hn 3hn V n ¼ W n ¼ ; Zn ¼ ; 8 8 k¼2

 8ðhi1 þ hi Þ hi ; i ¼ 1; 2; . . . ; n; 5hi 22hi   hi 1 1 Mi ¼ ; i ¼ 1; . . . ; n  1; 48 0 0   hi 1 0 Ni ¼ ; i ¼ 1; . . . ; n  1; 48 5 0   hn 1 hn 8hn ; ; W n ¼ ð 1 1 Þ; Z n ¼ Vn ¼ 48 5 48 48

Li ¼

1 48



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K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

k¼3

1 119ðhi1 þ hi Þ 16hi 0 1 B C Li ¼ 352hi 16hi A; i ¼ 1; 2; . . . ; n; 107hi @ 1152 43hi 16hi 352hi 0 1 9 0 16 hi B C Mi ¼ @ 0 0 0 A; i ¼ 1; . . . ; n  1; 1152 0 0 0 0 1 9 0 0 hi B C Ni ¼ @ 43 0 0 A; i ¼ 1; . . . ; n  1; 1152 107 0 0 0 1 9 hn B hn C ð 9 0 16 Þ; Vn ¼ @ 43 A; W n ¼ 1152 1152 107 119hn Zn ¼ ; 1152

k¼4

0

0

1 1694ðhi1 þ hi Þ 223hi 17hi 17hi 1969hi 5348hi 308hi 68hi C 1 B B C Li ¼ B C; 23040 @ 1191hi 138hi 5178hi 138hi A 499hi 68hi 308hi 5348hi 0 1 91 17 17 223 0 0 0 C hi B B 0 C Mi ¼ B C; i ¼ 1; . . . ; n  1; 23040 @ 0 0 0 0 A 0 0 91 0 0 hi B B 499 Mi ¼ B @ 23040 1191 0 1969 0 0 1 91 C hn B B 499 C Vn ¼ B C; 23040 @ 1191 A 0

0 0 0 0 0

1 0 0C C C; 0A 0

0

i ¼ 1; . . . ; n  1;

1969 hn ð 91 23040 1694hn ; Zn ¼ 23040 Wn ¼

17 17

223 Þ;

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

837

k¼5 0

1

13013ðhi1 þ hi Þ

1648hi

136hi

0

18447hi

43464hi

2464hi

136hi

14918hi

256hi

41424hi

2464hi

9382hi

1904hi

2464hi

41424hi

816hi C C C 1904hi C C; C 256hi A

3423hi

816hi

136hi 1

2464hi

43464hi

B B 1 B B Li ¼ 230400 B B @ 0

539 136 0 136 1648 B 0 0 0 0 0 C C B C hi B B Mi ¼ 0 0 0 0 0 C C; B 230400 B C @ 0 0 0 0 0 A 0

0

0

539 B 3423 B hi B B 9382 Ni ¼ 230400 B B @ 14918 0

18447 539

0

0

0 0 0 0

1

0 0 0 0C C C 0 0 0 0C C; C 0 0 0 0A

136hi

i ¼ 1; . . . ; n  1;

0

i ¼ 1; . . . ; n  1;

0 0 0 0 1

B 3423 C C B C hn B B 9382 C; Vn ¼ C B 230400 B C @ 14918 A 18447 hn ð 539 136 0 136 1648 Þ; 230400 13013hn Zn ¼ . 230400

Wn ¼

4. Discontinuous Lagrange-type k–0 elements As before we subdivide the interval [0, 1] into n subinterval by a sequence of points 0 ¼ t0 < t1 <    < tn ¼ 1. Denote I i ¼ ½ti1 ; ti  and hi ¼ ti  ti1 for i ¼ 1; . . . ; n and let X n be the space of piecewise polynomials of degree 6 k with 2jþ1 ; j ¼ 0; 1; . . . ; k and define knots at ti ; i ¼ 1; . . . ; n  1. Let sj ¼ 2kþ2 ðiÞ

tj ¼ ti1 þ sj hi ;

j ¼ 0; 1; . . . ; k; i ¼ 1; . . . ; n

838

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845 ðiÞ

also we define nðk þ 1Þ functions Uj ðtÞ by letting 8 ðiÞ < Qk‘¼0 tt‘ t 2 I i i ¼ 1; . . . ; n; ðiÞ ðiÞ ðiÞ t t‘ Uj ðtÞ ¼ ‘6¼j j : 0 t 62 I i j ¼ 0; 1; . . . ; k. Then, for each xn 2 X n , we have xn ðtÞ ¼

k X

ðiÞ

ðiÞ

xn ðtj ÞUj ðtÞ;

t 2 I i ; i ¼ 1; . . . ; n.

j¼0

We then construct the test space Y n by ( jhi i 6 t 6 ti1 þ ðjþ1Þh 1 ti1 þ kþ1 ðiÞ kþ1 Wj ðtÞ ¼ 0 otherwise

j ¼ 0; 1; . . . ; k; i ¼ 1; . . . ; n.

Now, we define a linear operator Pn : X n ! Y n as follows Pn xn ðtÞ ¼

k X

ðiÞ

ðiÞ

xn ðtj ÞWj ðtÞ;

t 2 I i ; i ¼ 1; . . . ; n.

j¼0

Then dimX n ¼ dimY n ¼ nðk þ 1Þ and Pn X n ¼ Y n and in [1] is proved that for 1 6 k 6 5 these two space sequences form a regular pair. nðkþ1Þ We arrange nðk þ 1Þ basis functions for X n by fbj gj¼1 and nðk þ 1Þ basis  nðkþ1Þ functions for Y n by fbj gj¼1 ans let B be a nðk þ 1Þ  nðk þ 1Þ matrix with entries bij ¼ ðbi ðtÞ; bj ðtÞÞ; A direct computation 0 L1 0 0 B B 0 L2 0 B B B 0 0 L3 B B B . .. .. B ¼ B .. . . B B B0 0 0 B B B0 0 0 @ 0

0

0

i; j ¼ 1; 2; . . . ; nðk þ 1Þ. gives 0

...

0

0

0

...

0

0

0

...

0

0

.. .

.. .

.. .

.. .

0

...

Ln2

0

0

...

0

Ln1

0

...

0

0

0

1

C 0C C C 0C C C .. C ; . C C C 0C C C 0C A Ln

where Li are ðk þ 1Þ  ðk þ 1Þ matrices. For k ¼ 1; . . . ; 5 we have k¼1   hi 1 0 Li ¼ ; i ¼ 1; . . . ; n; 2 0 1

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

k¼2

0 Li ¼

25

1

839

1

1

hi B C @ 2 22 2 A; 72 1 1 25

i ¼ 1; . . . ; n;

k¼3 0

1 1 4 C C C; 5 A

26 1 0 B hi B 5 22 1 Li ¼ B 96 @ 4 1 22 1 0 1 k¼4

0

26

223

17

5348 138

308 5178

68 17

308 17

206 5433

17 308

0 17

17 102

32

5178

308

238

238 102

308 17

5178 308

32 5433

17

0

17

206

6463

B B 2092 hi B B 2298 Li ¼ 28800 B B @ 1132 223

i ¼ 1; . . . ; n;

17

223

1

C 1132 C C 2298 C C; C 5348 2092 A 223 6463 68 138

i ¼ 1; . . . ; n;

k¼5 0

6669 B 3122 B B 4358 hi B B Li ¼ B 34560 B 3192 B @ 1253 206

1 206 1253 C C C 3192 C C. 4358 C C C 3122 A 6669

5. Hermite-type 3–1 elements We subdivide the interval [0, 1] into n subintervals by a sequence of points 0 ¼ t0 < t1 <    < tn ¼ 1. Denote I i ¼ ½ti1 ; ti  and hi ¼ ti  ti1 and let X n is the space of piecewise Hermite-type cubic polynomials, that is:  X n ¼ xn 2 C 1 ½0; 1 : xn jI i is a cubic polynomial determined by xðlÞ n ðt i1 Þ;  ðlÞ xn ðti Þ; l ¼ 0; 1; i ¼ 1; . . . ; n ¼ spanfb1 ðtÞ; b2 ðtÞ; . . . ; b2nþ2 ðtÞg. Using Hermite interpolation, for each xn 2 X n it holds that xn ðtÞ ¼

nþ1 X j¼1

fxn ðtj1 Þb2j1 ðtÞ þ x0n ðtj1 Þb2j ðtÞg;

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K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

where ( bj ðtÞ ¼

/j ðsÞðh1 Þ

j1

0 s ¼ tt ; t 2 I1 h1

j ¼ 1; 2;

t 62 I 1 ;

0

8 j1 > s ¼ tthi1 ; t 2 I i; /jþ2 ðsÞðhi Þ > i > > > ( > < i ¼ 1; . . . ; n  1; j1 tt b2iþj ðtÞ ¼ /j ðsÞðhiþ1 Þ s ¼ hiþ1i ; t 2 I iþ1 > > j ¼ 1; 2; > > > > : 0 t 62 I i [ I iþ1 ; ( j1 s ¼ tthn1 ; t 2 I n j ¼ 1; 2; /jþ2 ðsÞðhn Þ n b2nþj ðtÞ ¼ 0 t 62 I n ; and 2

/1 ðsÞ ¼ ð1  sÞ ð2s þ 1Þ; 2

/2 ðsÞ ¼ sð1  sÞ ; /3 ðsÞ ¼ s2 ð3  2sÞ; /4 ðsÞ ¼ ðs  1Þs2 . Now, let Y n be the space of piecewise linear polynomials, that is: Y n ¼ spanfb1 ðtÞ; b2 ðtÞ; . . . ; b2nþ2 ðtÞg; where ( b2iþ1 ðtÞ

¼ (

b2iþ2 ðtÞ

¼

1

  t 2 ti  h2i ; ti þ hiþ1 2

0

otherwise

i ¼ 0; 1; . . . ; n;

t  ti

  t 2 ti  h2i ; ti þ hiþ1 2

0

otherwise

i ¼ 0; 1; . . . ; n;

h0 ¼ hnþ1 ¼ 0. Then dimX n ¼ dimY n ¼ 2n þ 2 and in [1] is proved that fX n ; Y n g form a regular pair. Let B be a ð2n þ 2Þ  ð2n þ 2Þ matrix with entries bij ¼ ðbi ðtÞ; bj ðtÞÞ

i; j ¼ 1; . . . ; 2n þ 2.

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

A direct computation gives 0 L0 N 1 0 0 B B M 1 L1 N 2 0 B B 0 M 2 L2 N 3 B B . .. .. .. B¼B . . . B .. B B 0 0 0 0 B B 0 0 0 @ 0 0 0 0 0 where 29 2 h 320 1 1 3 h 60 1

L0 ¼

Li ¼

13 h þ 13 h 32 i 32 iþ1 11 2 11 2 h þ 192 hiþ1 192 i

Mi ¼

...

0

0

0

0

...

0

0

0

0 .. . 0

... 0 .. .. . . . . . M n2

0 .. .

0 .. .

Ln2

N n1

0 0

... ...

M n1 0

Ln1 Mn

!

13 h 32 1 11 2 h 192 1

11 2 h 320 i 3 3 h 320 i

3 h 32 i 5 2 h 192 i

0

Ln ¼

;

0 0

13 h 32 n 11 2 h 192 n

29 2 29 2 h þ 320 hiþ1 320 i 1 3 1 3 h þ 60 hiþ1 60 i

! ;

Ni ¼

3 h 32 i 5 h2 192 i

29 2 h 320 n 1 3 h 60 n

0

841

1

C 0 C C 0 C C .. C C . C; C 0 C C C Nn A Ln

! ;

! ;

i ¼ 1; 2; . . . ; n  1;

11 2 h 320 i 3 3 h 320 i

! i ¼ 1; . . . ; n.

6. Numerical method N

N

Assume un 2 X n and fbi gi¼1 is a basis for X n and fbj gj¼1 is a basis for Y n . Note that N ¼ nk þ 1 for continuous Langrange-type k–0 elements, N ¼ nðk þ 1Þ for discontinuous Langrange-type k–0 elements and N ¼ 2n þ 2 for Hermite-type k–0 elements. For solving Eq. (1), we let zðsÞ ¼ wðs; uðsÞÞ then uðxÞ ¼ f ðxÞ þ

Z

ð8Þ 1

kðx; sÞzðsÞds

ð9Þ

0

substituting (9) in (8),  Z zðxÞ ¼ w x; f ðxÞ þ

1

 kðx; sÞzðsÞds .

0

We approximate Eq. (10) by   Z 1 zn ðxÞ ¼ w x; f ðxÞ þ kðx; sÞzn ðsÞds ; 0

ð10Þ

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K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

where zn ðxÞ ¼

PN

j¼1 cj bj ðxÞ

2 X n is computed via     Z 1 ðzn ðxÞ; bi ðxÞÞ ¼ w x; f ðxÞ þ kðx; sÞzn ðsÞds ; bi ðxÞ ; 0

i ¼ 1; . . . ; N .

ð11Þ

Now define Z 1 bji ¼ bj ðxÞbi ðxÞdx 0

d j ðxÞ ¼

Z

1

kðx; sÞbj ðsÞds

0

then Eq. (11) is equivalent to ! Z 1 N N X X cj bji ¼ w x; f ðxÞ þ cj d j ðxÞ bi ðxÞdx i ¼ 1; . . . ; N . 0

j¼1

ð12Þ

j¼1

We solve non-linear system (12) by Newton method.For this purpose, let ! Z 1 N N X X gi ðc1 ; . . . ; cN Þ ¼ cj bji  w x; f ðxÞ þ cj d j ðxÞ bi ðxÞdx; 0

j¼1

j¼1

i ¼ 1; . . . ; N so that ogi ¼ bli  ocl

Z

1 0

! N X ow x; f ðxÞ þ cj d j ðxÞ d l ðxÞbi ðxÞdx; ou j¼1 T

i ¼ 1; . . . ; N . T

Let ca ¼ ðc1;a ; c2;a ; . . . ; cN ;a Þ ; F ðca Þ ¼ ðf1 ðca Þ; f2 ðca Þ; . . . ; fN ðca ÞÞ and J ðca Þ ¼ N ogi ð c Þ . With c0 2 RN given and using the Newton method, a ocl



i;l¼1

caþ1 ¼ ca  J ðca Þ1 F ðca Þ

8a ¼ 0; 1; 2; . . .

7. Numerical results In the follow examples, we use the Newton method by accuracy  ¼ 104 and then from Eq. (9), we comput approximate solution un ðxÞ. Example 1 uðxÞ þ

Z 0

1 3

ex2s  uðsÞ ds ¼ exþ1 ;

06x61

K. Maleknejad et al. / Appl. Math. Comput. 172 (2006) 831–845

843

with unique exact solution uðxÞ ¼ ex . In Tables 1 and 2 we computed ðiÞ ðiÞ kun ðtj Þ  uðtj Þk2 respectively by using continuous Langrange-type k–0 elements and discontinuous Langrange-type k–0 elements and in Table 3, we computed kun ðti Þ  uðti Þk2 by using Hermite-type 3–1 elements for n ¼ 1; 2; 4; 10 with equally spaced points. Example 2 Z

p x  2xInð3Þ; 0 6 x 6 1 2 uðsÞ þ s2 þ 1 0

with unique exact solution uðxÞ ¼ sin p2 x . In Tables 4 and 5 we computed ðiÞ ðiÞ kun ðtj Þ  uðtj Þk2 respectively by using continuous Lagrange-type k–0 elements and discontinuous Lagrange-type k–0 elements and in Table 3, we uðxÞ 

1

4xs þ px sinðpsÞ 2

ds ¼ sin

Table 1 Error for Example 1 by using continuous Langrange-type k–0 elements k

1 2 3 4 5

n 1

2

4

10

0.0711171 0.0318487 0.00718256 0.000379042 0.0000644195

0.188107 0.00217291 0.000482825 7.01533 · 106 1.11348 · 106

0.054749 0.000169031 0.0000381322 1.43578 · 107 2.25054 · 108

0.0120867 6.33462 · 106 1.45472 · 106 8.91439 · 1010 1.39802 · 1010

Table 2 Error for Example 1 by using discontinuous Lagrange-type k–0 elements k

1 2 3 4 5

n 1

2

4

10

0.0711045 0.00960519 0.00287936 0.000238635 0.0000458096

0.0224796 0.000812911 0.000216985 4.94556 · 106 8.48186 · 107

0.0071885 0.0000710505 0.0000183828 1.07522 · 107 1.78763 · 108

0.00193661 2.86684 · 106 7.35154 · 107 6.93185 · 1010 1.14221 · 1010

Table 3 Error for Examples 1, 2 by using Hermite-type 3–1 elements n 1 2 4 10

Example 1

Example 2 15

4.66082 · 10 2.28836 · 1014 1.17916 · 1013 4.89515 · 1013

0.00902821 0.00061158 0.0000443491 1.552805 · 106

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Table 4 Error for Example 2 by using continuous Lagrange-type k–0 elements k

1 2 3 4 5

n 1

2

4

10

0.281718 0.000615015 0.000389197 8.64184 · 107 8.14611 · 107

0.0551084 0.0000388461 0.0000289194 1.56978 · 108 1.61389 · 108

0.0143986 2.63723 · 106 2.30127 · 106 3.08519 · 1010 3.34152 · 1010

0.00298249 8.69589 · 108 8.68816 · 108 1.85981 · 1012 2.07412 · 1012

Table 5 Error for Example 2 by using discontinuous Lagrange-type k–0 elements k

1 2 3 4 5

n 1

2

4

10

0.0116123 0.0000668994 0.000128848 4.01056 · 107 5.10969 · 107

0.00483185 8.79569 · 106 0.0000119047 9.61518 · 109 1.16155 · 108

0.0017779 8.47232 · 107 1.09386 · 106 2.17294 · 1010 2.58529 · 1010

0.0004548 3.51021 · 108 4.31958 · 108 1.41735 · 1012 1.67806 · 1012

computed kun ðti Þ  uðti Þk2 by using Hermite-type 3–1 elements for n = 1, 2, 4, 10 with equally spaced points.

8. Conclusion One of the advantages of the Petrov–Galerkin method is that it allows us to achieve the same order of convergence as the Galerkin method with much less computational cost by choosing the test spaces to be spaces of piecewise polynomials of lower degree than the trial space. Therefore, in this paper we supposed the trial space constructed from piecewise continuous and discontinuous Lagrange polynomials of degree k ¼ 1; . . . ; 5 and the test space constructed from piecewise polynomials of degree zero and then in following we supposed the trial space constructed from piecewise Hermite cubic polynomials and the test space constructed from piecewise polynomials of degree one and we used this method for solving Eq. (1).

References [1] Z. Chen, Y. Xu, Thc Petrov–Galerkin and iterated Petrov–Galerkin methods for second-kind integral equations, SIAM J. Num. Anal. 35 (1) (1998) 406–434.

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[2] K. Maleknejad, M. Karami, The Petrov–Galerkin method for solving second kind Fredholm integral equations, in: 34th Iranian Mathematics Conference, Aug 30–Sep 2 (2003), Shahrood university, Shahrood, Iran, 2003. [3] M. Karami, Solving integral equations by Petrov–Galerkin method and using Hermite-type 3–1 elements, in: Dynamic Systems and Applications Conference, July 05–10 (2004), Antalya, Turkey, 2004. [4] B.K. Alpert, A class of bases in L2 for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993) 246–262. [5] Z. Chen, C.A. Micehelli, Y. Xu, The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes, Adv. Comput. Math. 7 (1997) 199–233. [6] Z. Chen, C.A. Micehelli, Y. Xu, Discrete wavelet Petrov–Galerkin mcthods, Adv. Comput. Math. 16 (2002) 1–28. [7] K. Malcknejad, M. Karami, Using the WPG method for solving integral equations of the second kind, Appl. Math. Comput., in press, doi:10.1016/j.amc.2004.04.109.