research article the approximate solution of fredholm ...due to the highly oscillatory kernels of...

Research ArticleThe Approximate Solution of Fredholm Integral Equations withOscillatory Trigonometric Kernels

Qinghua Wu

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Qinghua Wu; [email protected]

Received 9 March 2014; Accepted 4 April 2014; Published 17 April 2014

Academic Editor: Turgut Öziş

Copyright © 2014 Qinghua Wu.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A method for approximating the solution of weakly singular Fredholm integral equation of the second kind with highly oscillatorytrigonometric kernel is presented. The unknown function is approximated by expansion of Chebychev polynomial and thecoefficients are determinated by classical collocation method. Due to the highly oscillatory kernels of integral equation, thediscretised collocation equation will give rise to the computation of oscillatory integrals. These integrals are calculated by usingrecursion formula derived from the fundamental recurrence relation of Chebyshev polynomial. The effectiveness and accuracy ofthe proposed method are tested by numerical examples.

1. Introduction

The Fredholm integral equations of second kind,

𝑦 (𝑥) + 𝜆∫

𝑏

𝑎

𝐾 (𝑥, 𝑡)

|𝑥 − 𝑡|𝛼𝑦 (𝑡) 𝑑𝑡 = 𝑓 (𝑥) ,

𝑥 ∈ [𝑎, 𝑏] , 0 < 𝛼 < 1,

(1)

where 𝐾(𝑥, 𝑡) is a continuous function and 𝑓(𝑥) is a givenfunction, have many applications in mathematical physicsand engineering, such as heat conduction problem, potentialproblems, quantum mechanics, and seismology image pro-cessing [1–4]. Particularly, when 𝐾(𝑥, 𝑥) ̸= 0 and 0 < 𝛼 < 1,(1) is called weakly singular.

In most of the cases, the integral equation cannot bedone analytically and one has to resort to numericalmethods.Many numerical methods, such as collocation method andGalerkinmethod, have been developed to solve (1); for detailssee [2, 5, 6]. These methods are well-established numericalalgorithms; however, standard version of these classicalmeth-ods may suffer from difficulty for computation of (1), con-taining highly oscillatory kernels since the computation of thehighly oscillatory integrals by standard quadrature methodsis exceedingly difficult and the cost steeply increases withthe frequency. Furthermore, Galerkin method requires manydouble integrals when approximating the solution of integral

equation. Specially, when the kernel is highly oscillatory, itrequires the evaluation of many highly oscillatory doubleintegrals, which can become computationally expensive.

Recently, for weakly singular Volterra integral equationsof the second kind with highly oscillatory Bessel kernels, itwas found that the collocationmethods aremuchmore easilyimplemented and can get higher accuracy than discontinuousGalerkin methods under the same piecewise polynomialsspace; for details see [7–10]. In addition, collocation methodonly involves single integrals which are a little easier to evalu-ate. Motivated by this fact, here we investigate the applicationof collocation method for the solution of Fredholm integralequation

𝑦 (𝑥) + 𝜆∫

𝑏

𝑎

𝑒𝑖𝜔(𝑥−𝑡)

|𝑥 − 𝑡|𝛼𝑦 (𝑡) 𝑑𝑡 = 𝑓 (𝑥) ,

𝑥 ∈ [𝑎, 𝑏] , 𝛼 < 1, 𝜔 ≫ 1.

(2)

Here 𝑦(𝑥) is the unknown function and 𝑓(𝑥) is a givenfunction and assume that 𝜆 is not the eigenvalue of integralequation, since for any finite interval [𝑎, 𝑏] it can be trans-formed to interval [−1, 1] by linear transformation. In thispaper, we only consider the case of [𝑎, 𝑏] = [−1, 1].

On the other hand, when the unknown function isapproximated byChebychev polynomial, the discretization of

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 172327, 7 pageshttp://dx.doi.org/10.1155/2014/172327

2 Journal of Applied Mathematics

the integral equation (2) by collocation method will give riseto highly oscillatory integral which can be formulated as

∫

𝑏

𝑎

𝑒(𝑖𝜔(𝑥−𝑡))

|𝑥 − 𝑡|𝛼𝑇𝑗 (𝑡) 𝑑𝑥, 𝑥 ∈ [𝑎, 𝑏] , 0 < 𝛼 < 1, (3)

where 𝑇𝑗(𝑥) denotes the Chebychev polynomials of the firstkind.

In last few years many efficient methods have beendevised for the evaluation of oscillatory integral, such asasymptotic method [11], Filon-type method [12, 13], Levin’scollocation method [14], modified Clenshaw-Curtis method,Clenshaw-Curtis-Filon-type method [15], and generalizedquadrature rule [16], although some of these methods do notinvolve Chebyshev polynomial. Piessens and Poleunis [17]consider a simpler case of 𝛼 = 0 and use Chebyshev polyno-mials to evaluate the integral by somewhat indirect methodinvolving a truncated infinite series of Bessel functions. Theevaluation of this infinite summation suggests a defect of thismethod. An alternative procedure which avoids this infiniteseries is the original Bakhvalo and Vasil’eva-Legendre workinvolved in the evaluation of the integral

𝑀𝑖 (𝜔) = ∫

1

−1

𝑃𝑖 (𝑥) 𝑒𝑖𝜔𝑥𝑑𝑥, (4)

by recurrence relation, where𝑃𝑖(𝑥) denotes Legendre polyno-mial. Sadly, this relation proved to be unstable in the forwardsdirection for small 𝜔. In [18], Alaylioglu et al proposeda simple alternative approach analogous to Newton-Cotesbased formulae. This method avoids the instability to acertain degree compared with these two methods mentionedabove. We used the same idea and generalized it to the caseof weakly singular and calculated oscillatory integral (3).

This paper is organized as follows: in Section 2 we derivesome basic formulae and introduce some mathematicalpreliminaries of the proposed method. In Section 3 we dis-cuss the evaluation of the integrals occurring in collocationequation. In Section 4 numerical experiments are conductedto illustrate the performance of the proposed method.

2. Fundamental Relations

Firstly, by separation of real and imaginary part of 𝑦(𝑥) and𝑒𝑖𝑤(𝑥−𝑡), we transform the integral equation (2) into equivalentsystems of two linear integral equations of Fredholm in theforms

𝑦1 (𝑥) + 𝜆∫

1

−1

cos (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼𝑦1 (𝑡) 𝑑𝑡

− 𝜆∫

1

−1

sin (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼𝑦2 (𝑡) 𝑑𝑡 = 𝑓1 (𝑥) ,

𝑦2 (𝑥) + 𝜆∫

1

−1

sin (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼𝑦1 (𝑡) 𝑑𝑡

+ 𝜆∫

1

−1

cos (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼𝑦2 (𝑡) 𝑑𝑡 = 𝑓2 (𝑥) ,

(5)

where 𝑦(𝑥) = 𝑦1(𝑥) + 𝑖𝑦2(𝑥), 𝑓(𝑥) = 𝑓1(𝑥) + 𝑖𝑓2(𝑥), and 𝑖 =√−1.

Assume that y(𝑥) = (𝑦1(𝑥), 𝑦2(𝑥))𝑇 and f(𝑥) = (𝑓1(𝑥),

𝑓2(𝑥))𝑇; then, systems of linear integral equations (5) can be

written in the matrix form

Iy (𝑥) + 𝜆∫1

−1

K (𝑥, 𝑡) y (𝑡) 𝑑𝑡 = f (𝑥) , (6)

where

I = (1 00 1) ,

K (𝑥, 𝑡) = (

cos (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼

− sin (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼

sin (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼

cos (𝜔 (𝑥 − 𝑡))|𝑥 − 𝑡|

𝛼

).

(7)

Set 𝑦𝑖(𝑥) = T(𝑥)b𝑖, 𝑖 = 1, 2, where T(𝑥) = [𝑇0(𝑥),𝑇1(𝑥), . . . , 𝑇𝑁(𝑥)], b𝑖 = [𝑏𝑖0, 𝑏𝑖1, . . . , 𝑏𝑖𝑁]

𝑇.Hence, unknown functions can be expressed by

y (𝑥) = T𝐵, (8)where

y (𝑥) = (𝑦1 (𝑥)𝑦2 (𝑥)) , T (𝑥) = (

T (𝑥) 00 T (𝑥)) ,

𝐵 = (b1b2) .

(9)

Then, the aim is to find Chebyshev coefficients, that is,the matrix 𝐵. We first substitute the Chebyshev collocationpoints, which are defined by 𝑥𝑖 = cos(𝑖𝜋/𝑁), 𝑖 = 0, . . . , 𝑁,into (6) and then rearrange a new matrix form to determine𝐵:

IY + 𝜆K = F (10)

in whichK is the integral part of (6) and

I = (

I 0 ⋅ ⋅ ⋅ 00 I ⋅ ⋅ ⋅ 0...

... d...

0 0 ⋅ ⋅ ⋅ I

), Y =(

y (𝑥0)

y (𝑥1)...

y (𝑥𝑁)

),

F =(

f (𝑥0)

f (𝑥1)...

f (𝑥𝑁)

), K =(

IK (𝑥0)

IK (𝑥1)...

IK (𝑥𝑁)

).

(11)

By substituting (8) into (10), the unknown coefficients canbe easily computed from this linear algebraic equations andtherefore we find the solution of integral equation (2).

3. Evaluation of the Integral

𝐼[𝜔, 𝛼, 𝑗, 𝑥]= ∫1

−1(𝑒(𝑖𝜔(𝑥−𝑡))

/|𝑥−𝑡|𝛼)𝑇𝑗(𝑡)𝑑𝑡

The discretization of integral equation will lead to thecalculation of integral 𝐼[𝜔, 𝛼, 𝑗, 𝑥]. In this section, we willderive the recursion formula to compute it efficiently from thefundamental recurrence relation of Chebyshev polynomial.

Journal of Applied Mathematics 3

100

10−2

10−4

10−6

10−8

10−10

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x

Abso

lute

erro

rApproximate the solution of example 4.1 with

𝛼 = 0.5, = 1000, N = 4𝜔

(a)

100

10−2

10−4

10−6

10−8

10−10

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x

Abso

lute

erro

r

Approximate the solution of example 4.1 with𝛼 = 0.5, = 1000, N = 8𝜔

(b)

100

10−2

10−4

10−6

10−8

10−10

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x

Abso

lute

erro

r

Approximate the solution of example 4.1 with𝛼 = 0.5, = 1000, N = 16𝜔

(c)

Figure 1: Example 1: the absolute error versus the 𝑥-coordinate for𝑁 = 4 (a),𝑁 = 8 (b), and𝑁 = 16 (c) with negative data ignored.

The Chebyshev polynomials are of the form

𝑇0 (𝑥) = 1

𝑇1 (𝑥) = 𝑥

...𝑇𝑛 (𝑥) = 2𝑥𝑇𝑛−1 (𝑥) − 𝑇𝑛−2 (𝑥) , 𝑛 = 2, 3, . . . ,

(12)

and the coefficients 𝐶𝑖,j of 𝑥𝑗 in 𝑇𝑖(𝑥) can be easily calculated

by the means of the recurrence relation

𝐶𝑖,𝑗 = 2𝐶𝑖−1,𝑗−1 − 𝐶𝑖−2,𝑗, 𝑖 ≥ 2, 𝑗 ≤ 𝑖. (13)

By expanding the Chebyshev polynomial 𝑇𝑖(𝑥) in terms ofpowers of 𝑥, the integral 𝐼[𝜔, 𝛼, 𝑗, 𝑥] would be transformedinto the form of

𝐼 [𝜔, 𝛼, 𝑗, 𝑥] =

𝑗

∑

𝑘=0

𝐶𝑗,𝑘𝐼1 [𝜔, 𝛼, 𝑘, 𝑥] , (14)

where 𝐼1[𝜔, 𝛼, 𝑘, 𝑥] = ∫1−1(𝑒(𝑖𝜔(𝑥−𝑡))

/|𝑥 − 𝑡|𝛼)𝑡𝑘𝑑𝑡.

By separation of real and imaginary part of 𝑒𝑖𝑤(𝑥−𝑡), (14) istransformed into

𝐼1𝑐 [𝜔, 𝛼, 𝑗, 𝑥] = ∫

1

−1

𝑡𝑗

|𝑥 − 𝑡|𝛼cos (𝜔 (𝑥 − 𝑡)) 𝑑𝑡,

𝐼1𝑠 [𝜔, 𝛼, 𝑗, 𝑥] = ∫

1

−1

𝑡𝑗

|𝑥 − 𝑡|𝛼sin (𝜔 (𝑥 − 𝑡)) 𝑑𝑡,

(15)

whose main difficulty now is turning to the evaluation of thefollowing basic integrals:

𝑀1𝑐 [𝜔, 𝛼, 𝑗, 𝑥] = ∫

1

−1

𝑡𝑗

|𝑥 − 𝑡|𝛼cos (𝜔𝑡) 𝑑𝑡,

𝑀1𝑠 [𝜔, 𝛼, 𝑗, 𝑥] = ∫

1

−1

𝑡𝑗

|𝑥 − 𝑡|𝛼sin (𝜔𝑡) 𝑑𝑡.

(16)


100

10−2

10−4

10−5

10−6

10−7

10−3

10−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x

Abso

lute

erro

rApproximate the solution of example 4.2 with

𝛼 = 1/3, = 1000, N = 4𝜔

(a)

100

10−2

10−4

10−6

10−8

10−10

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x

Abso

lute

erro

r

Approximate the solution of example 4.2 with𝛼 = 1/3, = 1000, N = 8𝜔

(b)

100

10−2

10−4

10−6

10−8

10−10

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x

Abso

lute

erro

r

Approximate the solution of example 4.2 with𝛼 = 1/3, = 1000, N = 16𝜔

(c)

Figure 2: Example 2: the absolute error versus the 𝑥-coordinate for𝑁 = 4 (a),𝑁 = 8 (b), and𝑁 = 16 (c) with negative data ignored.

With substitution for definite integral and binomial the-orem, (16) is equivalent to

𝑀1𝑐 [𝜔, 𝛼, 𝑗, 𝑥]

=

𝑗

∑

𝑘=0

𝐶𝑘

𝑗𝑥𝑗−𝑘[(−1)

𝑘∫

𝑥+1

0

𝑡𝑘−𝛼 cos (𝜔 (𝑥 − 𝑡)) 𝑑𝑡

+∫

1−𝑥

0

𝑡𝑘−𝛼 cos (𝜔 (𝑥 − 𝑡)) 𝑑𝑡] ,

(17)

where 𝐶𝑘𝑗is number of combination and𝑀1𝑠[𝜔, 𝛼, 𝑗, 𝑥] can

be formulated analogy.Efficient evaluation of (18) is based on accurate calcu-

lation of integrals in the formula which can be computed

explicitly by the incomplete Gamma function Γ(𝑧, 𝛼) [19];that is,

∫ 𝑡𝜇−1 cos (𝜔𝑡) 𝑑𝑡

=1

2[(𝑖𝜔)−𝜇Γ (𝜇, 𝑖𝜔𝑡) + (−𝑖𝜔)

−𝜇Γ (𝜇, −𝑖𝜔𝑡)] ,

(18)

∫ 𝑡𝜇−1 sin (𝜔𝑡) 𝑑𝑡

=𝑖

2[(𝑖𝜔)−𝜇Γ (𝜇, 𝑖𝜔𝑡) − (−𝑖𝜔)

−𝜇Γ (𝜇, −𝑖𝜔𝑡)] ,

(19)

in which 𝜇 > 0, 𝑥 > 0; for details see ([20], pp 215).Once the integral 𝐼[𝜔, 𝛼, 𝑗, 𝑥] is obtained, by substituting

into (10), the coefficient matrix is derived; then, we cancompute the solution of integral by solving these linearalgebraic equations.


Table 1: Approximations for 𝑦(𝑥) + ∫1−1(𝑒𝑖𝜔(𝑥−𝑡)/|𝑥 − 𝑡|

0.5)𝑦(𝑡)𝑑𝑡 = 𝑒

𝑥 with 𝜔 = 103.

𝑥 cos(𝜋) cos(3𝜋/4) cos(0)𝑦4

10.351696035495271 0.459037927481918 0.924223909091303

𝑦8

10.351696947581043 0.459033811940511 0.924224547405338

𝑦16

10.351741610883315 0.459036527929146 0.924226969620880

𝑦4

20.0141286409694892 0.000146310591227839 −0.00107727880701281

𝑦8

20.0141193392298137 0.000145019893750479 −0.00107593754108643

𝑦16

20.0140583682955762 0.000144178555797097 −0.00106881635211410


0.5)𝑦(𝑡)𝑑𝑡 = 𝑒


𝑥 cos(𝜋/4) 1𝑦4

11.88249404678421 2.61059423521207

𝑦8

11.88248275737555 2.61059120470589

𝑦16

11.88248361661370 2.61073031676337

𝑦4

20.00323202027752064 −0.0995508315886191

𝑦8

20.00321634589900894 −0.0994917961267987

𝑦16

20.00319500282444326 −0.0993143176430846

4. Numerical Examples

In this section, we give some numerical examples to illustratethe performance of proposed method. In all the followingexamples,𝑁+1 is the number of mesh points, 𝑦𝑁

𝑖(𝑥) denotes

the approximate solution, where 𝑁 is the number of termsof the Chebyshev series, and 𝑦(𝑥) denotes the exact solution,respectively. All the computations have been performed byusing Matlab R2012a on a 2.5GHz PC with 2GB of RAM.

Example 1. We consider 𝑦(𝑥) + ∫1−1(𝑒𝑖𝜔(𝑥−𝑡)/|𝑥 − 𝑡|

𝛼)𝑦(𝑡)𝑑𝑡 =

𝑓(𝑥) and set 𝑓(𝑥) = 1 + ∫1−1(𝑒𝑖𝜔(𝑥−𝑡)/|𝑥 − 𝑡|

𝛼)𝑑𝑡; then, the

solution of which is

𝑦 (𝑥) = 1. (20)

We plot the absolute error |𝑦(𝑥) − 𝑦(𝑥)| by Matlab 2012ainternal function ezplot. Specially, for 𝑁 = 4, with 𝜔 = 103and 𝛼 = 0.5, solve linear algebraic equations (10); it can befound that the approximate solutions are

𝑦1 (𝑥) = 0.9999999996 − 0.110 × 10−15𝑥 + 0.281 × 10

−9𝑥2

+ 0.323 × 10−15𝑥3+ 0.563 × 10

−10𝑥4,

𝑦2 (𝑥) = 0.460 × 10−14− 0.329 × 10

−9𝑥 + 0.554 × 10

−11𝑥2

+ 0.322 × 10−9𝑥3− 0.555 × 10

−11𝑥4.

(21)

It is easy to see from Figure 1 that the proposed method isconverging and we only need𝑁 = 4 so we could achieve highaccuracy.

Example 2. We consider 𝑦(𝑥) + ∫1−1(𝑒𝑖𝜔(𝑥−𝑡)/|𝑥 − 𝑡|

𝛼)𝑦(𝑡)𝑑𝑡 =

𝑓(𝑥) and set 𝑓(𝑥) = 𝑒𝑥 + ∫1−1(𝑒𝑖𝜔(𝑥−𝑡)/|𝑥 − 𝑡|

𝛼)𝑒𝑡𝑑𝑡; then, the

solution of which is𝑦 (𝑥) = 𝑒

𝑥. (22)

In this case, for 𝑁 = 4, with 𝜔 = 1000 and 𝛼 = 1/3, theapproximate solutions are

𝑦1 (𝑥) = 1.000000001 + 0.9956819990𝑥 + 0.4992866904𝑥2

+ 0.1795192648𝑥3+ 0.04379395375𝑥

4,

𝑦2 (𝑥) = −0.530 × 10−7+ 0.189 × 10

−7𝑥 + 0.426 × 10

−6𝑥2

− 0.246 × 10−7𝑥3− 0.420 × 10

−6𝑥4.

(23)

It is obvious to see from Figure 2 that the proposedmethod is efficient and could achieve high accuracy with littlenumber of collocation points.

Example 3. For a general case, we consider𝑦(𝑥) + ∫

1

−1(𝑒𝑖𝜔(𝑥−𝑡)/|𝑥 − 𝑡|

𝛼)𝑦(𝑡)𝑑𝑡 = 𝑒

𝑥 and set 𝛼 = 0.5.Although the solution of this equation is unknown, we canverify the calculation precision of the method to a certaindegree by comparing the approximate evaluation at meshpoints.

In this case, for 𝑁 = 4, with 𝜔 = 1000 and 𝛼 = 0.5, theapproximate solutions are𝑦1 (𝑥) = 0.9242239091

+ 0.8836218495𝑥 + 0.4292470859𝑥2

+ 0.2458272504𝑥3+ 0.1276741404𝑥

4,



0.5)𝑦(𝑡)𝑑𝑡 = 𝑒


𝑥 cos(𝜋) cos(3𝜋/4) cos(0)𝑦4

10.367419631536920 0.491835233010361 0.997500711679274

𝑦8

10.367419631541538 0.491835233006225 0.997500711679218

𝑦16

10.367419632527633 0.491835233064998 0.997500711678921

𝑦4

20.000458466865252363 0.150404190 × 10

−5−0.219327280 × 10

−5

𝑦8

20.000458466860662437 0.150403636 × 10

−5−0.219326739 × 10

−5

𝑦16

20.000458463698666909 0.150390826 × 10

−5−0.219326736 × 10

−5


0.5)𝑦(𝑡)𝑑𝑡 = 𝑒


𝑥 cos(𝜋/4) 1𝑦4

12.02304108762491 2.71487514654707

𝑦8

12.02304108760947 2.71487514654071

𝑦16

12.02304108780835 2.71487515460854

𝑦4

20.384042582 × 10

−5−0.00339813438891857

𝑦8

20.384040621 × 10

−5−0.00339813431478428

𝑦16

20.3840356665 × 10

−5−0.00339813322237100

𝑦2 (𝑥) = −0.001077278807 + 0.06120358877𝑥

+ 0.05269959347𝑥2− 0.1180433250𝑥

3

− 0.09433340997𝑥4.

(24)

It is also easy to see from Tables 1, 2, 3, and 4 that thepresentedmethod is efficient and accurate, although the exactsolution is unknown.

5. Conclusion

In this paper, we explore quadrature methods for weaklysingular Fredholm integral equation of the second kindwith oscillatory trigonometric kernels and present colloca-tion methods with Chebyshev series for calculation of thesolution. For integral equationwith highly oscillatory kernels,the standard collocation methods with classical quadraturemethods are not suitable for the numerical approximationof the solution of integral equation, since the computationof the highly oscillatory integrals by standard quadraturemethods is exceedingly difficult and the cost steeply increaseswith the frequency. Based on the recursion formula derivedin Section 3, we compute the highly oscillatory integralsoccurring in collocation equation, directly and efficiently.Numerical examples demonstrate the performance of algo-rithm.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This paper was supported by NSF of China (no. 11371376).

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