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

Ma**the**matical Sciences Vol. 2, No. 3 (2008) 273-280

**S inc**

**the** **Second** K**in**d

M. Rabbani a,1 , R. Mollapourasl b

a Islamic Azad University, Sari Branch, Sari, Iran

b School **of** Ma**the**matics, Iran University **of** Science & Technology, Narmak, Tehran 1684613114, Iran

Abstract

In this paper, we use **S inc** approximation

**of** **the** second k**in**d **in** L α space **for** every positive α, **the**n by us**in**g collocation

method we estimate a solution **for** second k**in**d **Fredholm** **in**tegral equation. F**in**ally

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

by some examples.

Keywords: **Integral** equation **of** **the** second k**in**d, Collocation method, CG method,

**S inc** approximation, Romberg method.

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

1 Introduction

Consider **Fredholm** **in**tegral equation **of** **the** second k**in**d [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 **for**m **of** this equation we have

1 Correspond**in**g Author. E-mail Address: mrabbani@iust.ac.ir

(I − K)f = g (2)

274 Ma**the**matical Sciences Vol. 2, No. 3 (2008)

where I is **the** identity operator and K is an **in**tegral operator def**in**ed 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

**in**jective. Then **the** **in**verse operator (I − K) −1 : X → X exist and is bounded.

pro**of** [4]

By regard to **the** above **the**orem we conclude **the** uniqueness **of** solution **of** **Fredholm**

**in**tegral equation **of** **the** second k**in**d.

In projection methods such as collocation method **for** solv**in**g 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 **in**tegral equation **of** **the** second k**in**d **in** a space with f**in**ite elements. In

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

constructed by **S inc** basis functions. By us

every function **in** this space.

2 Card**in**al Whittaker Expansion

**S inc** function is def

⎧

⎪⎨ s**in**(πx)

πx

, x ≠ 0.

s**in**c(x) =

⎪⎩ 1, x = 0;

Now, we def**in**e **the** sequence **of** **S inc** basis function

where k is an **in**teger [3].

S(k, h)(x) =

s**in**(π(x − kh)/h)

π(x − kh)/h

Def**in**ition 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** card**in**al series representation

pro**of** [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 card**in**al expansion **of** f

whenever this series converges.

3 Collocation Method

Consider **Fredholm** **in**tegral equation **of** **the** second k**in**d

f(s) =

∫ b

a

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

∞ < a ≤ s ≤ b < ∞

we can write this equation as a **for**m **of**

(I − K)f = g

where I : X → X is **the** identity operator and K : X → X is **in**tegral 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 sampl**in**g **the**orem we have

f(t) =

N∑

f(kh)S(k, h)(t) ∀h > 0

k=−N

**the**n by substitut**in**g this **in** **in**tegral 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 Ma**the**matical Sciences Vol. 2, No. 3 (2008)

Now we def**in**e 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** determ**in****in**g **the** unknown coefficients f(kh)’s we select some collocation po**in**ts such

that

R N (s i ) = 0

i = 0, ..., 2N

**in** this paper collocation po**in**ts are

s i = a +

i(b − a)

2N

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

so that we have a system **of** l**in**ear 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** l**in**ear equation we need to determ**in**e ∫ b

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

some alternatives **for** this problem. First, we can determ**in**e this quantity by us**in**g **S inc**

**in**tegration **for**mula [9]. In [6] authors have used **S inc** approximation on Γ = [0, 1] and

**S inc**

classify and know that **the** exact solution is satisfied **in** some conditions or not. O**the**r

ways to determ**in**e above quantity are traditional numerical **in**tegration methods such

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

**the** exact solution **of** **the** **in**tegral equation.

Now,we should discuss about solvability **of** this system **of** l**in**ear equations. Accord**in**g

to **the** [2], if basis functions that construct **the** solution space be **in**dependent **the**n

**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 **in**tegral equation (1). In all examples

we use relations (3),(4),(5) to convert **in**tegral equation to system **of** l**in**ear 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**

l**in**ear equations is solved by conjugate gradient (CG) method [8]. In **the**se examples

E N is def**in**ed by

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

i

where t i ’s are **the** collocation po**in**ts.

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 )arcs**in**h(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 Ma**the**matical 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 s**in**(t + s) − t cos(t − s)

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

where **the** exact solution is f(t) = s**in**(t).

Conclusion

In [6, 7] **Fredholm** **in**tegral equation **of** **the** second k**in**d by us**in**g **S inc** approximation

and **S inc**

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

not, so that, we should classify **the** **in**tegral 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** **in**tegral equation is satisfied **in** some conditions

or not.

An o**the**r po**in**t that we can mention is **the** order **of** **S inc** 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

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