Sinc Function in Strip for Fredholm Integral Equation of the Second ...

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Sinc Function in Strip for Fredholm Integral Equation of the Second ...

Mathematical Sciences Vol. 2, No. 3 (2008) 273-280

Sinc Function in Strip for Fredholm Integral Equation of

the Second Kind

M. Rabbani a,1 , R. Mollapourasl b

a Islamic Azad University, Sari Branch, Sari, Iran

b School of Mathematics, Iran University of Science & Technology, Narmak, Tehran 1684613114, Iran

Abstract

In this paper, we use Sinc approximation in strip for Fredholm integral equation

of the second kind in L α space for every positive α, then by using collocation

method we estimate a solution for second kind Fredholm integral equation. Finally

convergence of this method will be discussed and efficiency of this method is shown

by some examples.

Keywords: Integral equation of the second kind, Collocation method, CG method,

Sinc approximation, Romberg method.

c○ 2008 Published by Islamic Azad University-Karaj Branch.

1 Introduction

Consider Fredholm integral equation of the second kind [5]

f(s) =

∫ b

a

k(s, t)f(t)dt + g(s) ∞ < a ≤ s ≤ b < ∞ (1)

where k(s, t) and g(s) are known functions, but f(t) is an unknown function. In the

operator form of this equation we have

1 Corresponding Author. E-mail Address: mrabbani@iust.ac.ir

(I − K)f = g (2)


274 Mathematical Sciences Vol. 2, No. 3 (2008)

where I is the identity operator and K is an integral operator defined by

K : X → X

Kf =

∫ b

a

k(s, t)f(t)dt.

Theorem 1.1 Let X be a normed space, K : X → X a compact and I − K be

injective. Then the inverse operator (I − K) −1 : X → X exist and is bounded.

proof [4]

By regard to the above theorem we conclude the uniqueness of solution of Fredholm

integral equation of the second kind.

In projection methods such as collocation method for solving equation (2), we assume

that X n ⊂ X and {φ i } i = 1, ..., n are basis functions for X n . By this assumption

we try to solve integral equation of the second kind in a space with finite elements. In

this paper X is L α for every positive α which is defined in next section and X n is a space

constructed by Sinc basis functions. By using sampling theorem we can approximate

every function in this space.

2 Cardinal Whittaker Expansion

Sinc function is defined by


⎪⎨ sin(πx)

πx

, x ≠ 0.

sinc(x) =

⎪⎩ 1, x = 0;

Now, we define the sequence of Sinc basis function for positive h by

where k is an integer [3].

S(k, h)(x) =

sin(π(x − kh)/h)

π(x − kh)/h

Definition 2.1 Let h > 0, and let W ( π h

) denote the family of functions f that are

analytic in C, such that


|f(t)| 2 dt < ∞


M. Rabbani and R. Mollapourasl 275

and such that for all z in C

|f(z)| ≤ ce π|z|

h

with c a positive constant.

Theorem 2.2 Let h > 0, the sequence {h − 1 2 S(k, h)} ∞ −∞ is complete orthonormal

sequence in W ( π h ). Every f in W ( π h

) has the cardinal series representation

proof [9]

f(x) = C(f, h)(x) =

∞∑

k=−∞

f(kh)S(k, h)(x)

For h > 0 the series C(f, h)(x) is called the whittaker cardinal expansion of f

whenever this series converges.

3 Collocation Method

Consider Fredholm integral equation of the second kind

f(s) =

∫ b

a

k(s, t)f(t)dt + g(s)

∞ < a ≤ s ≤ b < ∞

we can write this equation as a form of

(I − K)f = g

where I : X → X is the identity operator and K : X → X is integral operator. Now,

we assume X in this paper is L α (D d ) and X n is a subspace constructed by orthonormal

sequence {h 1/2 S(k, h)} N k=−N

[9]. Now we approximate an unknown function f(s) by

this orthonormal sequence, so by sampling theorem we have

f(t) =

N∑

f(kh)S(k, h)(t) ∀h > 0

k=−N

then by substituting this in integral equation we get

N∑

∫ b

N∑

f(kh)S(k, h)(s) = k(s, t) f(kh)S(k, h)(t)dt + g(s)

k=−N

a

k=−N


276 Mathematical Sciences Vol. 2, No. 3 (2008)

Now we define residual equation by

R N (s) =

N∑

k=−N

∫ b

f(kh)S(k, h)(s) −

a

k(s, t)

N∑

k=−N

f(kh)S(k, h)(t)dt − g(s)

for determining the unknown coefficients f(kh)’s we select some collocation points such

that

R N (s i ) = 0

i = 0, ..., 2N

in this paper collocation points are

s i = a +

i(b − a)

2N

i = 0, 1, ..., 2N

so that we have a system of linear equations A N X = b N where

∫ b

A N = [S(k, h)(s i ) − k(s i , t)S(k, h)(t)dt] N k=−N i = 0, 1, ..., 2N (3)

a

b N = [g(s i )] i = 0, 1, ..., 2N (4)

X T = [f(kh)] N k=−N (5)

In this system of linear equation we need to determine ∫ b

a k(s i, t)S(k, h)(t)dt. We have

some alternatives for this problem. First, we can determine this quantity by using Sinc

integration formula [9]. In [6] authors have used Sinc approximation on Γ = [0, 1] and

Sinc integration formula to estimate the above quantity but in this strategy we need to

classify and know that the exact solution is satisfied in some conditions or not. Other

ways to determine above quantity are traditional numerical integration methods such

as Romberg method [1], that we use in this paper so that, we can skip from classifying

the exact solution of the integral equation.

Now,we should discuss about solvability of this system of linear equations. According

to the [2], if basis functions that construct the solution space be independent then

the stated system has unique solution.


M. Rabbani and R. Mollapourasl 277

4 Numerical Examples

In this section we want to use this method to solve integral equation (1). In all examples

we use relations (3),(4),(5) to convert integral equation to system of linear equations,

for this result we choose α = 1 and d = π so that h =

π √

N

, in this procedure we use

Romberg method to estimate ∫ b

a k(s i, t)S(k, h)(t)dt numerically. At the end, system of

linear equations is solved by conjugate gradient (CG) method [8]. In these examples

E N is defined by

E N = max |f(t i ) − PN num (f)(t i )| i = 0, 1, ..., 2N

i

where t i ’s are the collocation points.

Example 1. In this example we solve equation (1) with a = 1 2 , b = 1 and

k(s, t) = t 3 + s 2

g(s) = − 48 − π(−24 + π(24 + π + 8πs2 ))

8π 4

where the exact solution is f(t) = cos(πt).

+ cos(πs)

Example 2. In this example we solve equation (1) with a = 0 , b = 1 and

k(s, t) =

(s − t)2

1 + t 2

g(s) = √ 1 + s 2 + 2( √ 2 − 1)s + ( 1 2 − x2 )arcsinh(1) − 1 √

2

where the exact solution is f(t) = √ 1 + t 2 .

Example 3. In this example we solve equation (1) with a = 0 , b = 1 and

k(s, t) =

exp(s − t)

1 + s 2

g(s) = exp(−s 2 ) −

where the exact solution is f(t) = exp(t 2 ).

0.85 exp(x)

1 + s 2


278 Mathematical Sciences Vol. 2, No. 3 (2008)

Table 1: Numerical results for Examples

N E N for Ex.1 E N for Ex.2 E N for Ex.3 E N for Ex.4

3 5.4467E-3 3.8315E-3 2.2966E-2 5.6884E-2

5 3.4743E-8 8.8190E-7 2.4965E-6 5.1365E-6

7 6.0758E-12 7.8724E-11 5.5715E-7 1.7283E-12

9 3.9635E-14 3.2124E-11 5.5360E-7 1.2213E-15

Example 4. In this example we solve equation (1) with a = 0 , b = π 2 and

k(s, t) = s sin(t + s) − t cos(t − s)

g(s) = sin(s) + 1 16 (−2π(2s − 1) cos(s) + (4 + π2 − 8s) sin(s))

where the exact solution is f(t) = sin(t).

Conclusion

In [6, 7] Fredholm integral equation of the second kind by using Sinc approximation

and Sinc integration formula on Γ = [0, 1] has been solved. In this method we need to

know that the exact solution of the integral equation is satisfied in some conditions or

not, so that, we should classify the integral equation. In this paper, we use Romberg

method to skip this problem and estimate ∫ b

a k(s i, t)S(k, h)(t)dt . In this case we do

not need to know the exact solution of integral equation is satisfied in some conditions

or not.

An other point that we can mention is the order of Sinc approximation that is

O(exp(−cN 1/2 )) and accuracy of this method is clear in numerical examples.


M. Rabbani and R. Mollapourasl 279

References

[1] Atkinson K., An Introduction to Numerical Analysis, John Wiley and Sons, 1978.

[2] Atkinson K., The Numerical Solution of Integral Equations of the Second Kind,

Cambridge University Press, 1997.

[3] Bialecki B., Stenger F. (1988) ”Sinc-Nystrm method for numerical solution of

one-dimensional Cauchy singular integral equations given on a smooth arc in the

complex plane,” Math. Comp., 51, 133165.

[4] Kress R., Linear Integral Equation, Springer-Verlag, New York, 1998.

[5] Rashed M.T. (2003) ”An expansion method to treat integral equations,” Appl.

Math. Comput., 135, 6572.

[6] Rashidinia J., Zarebnia M. (2007) ”Solution of a Volterra integral equation by the

Sinc-collocation method,” Journal of Computational and Applied Mathematics,

206 (2), 801-813.

[7] Rashidinia J., Zarebnia M. (2007) ”Convergence of approximate solution of system

of Fredholm integral equations,” Journal of Mathematical Analysis and Applications,

333 (2), 1216-1227.

[8] Saad Y., Iterative Methods for Sparse Linear Systems, PWS Publishing Co.,

Boston, MA, 1996.

[9] Stenger F., Numerical Methods Based on Sinc and Analytic Functions, Springer,

New York, 1993.


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