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Australian Journal of Basic and Applied Sciences, 5(3): 38-43, 2011

ISSN 1991-8178

Corresponding Author:Elayaraja Aruchunan, Department of Engineering and Science, School of Foundation and

Continuous StudiesCurtin University of Technology, Sarawak Malaysia.

Email: elayarajah@curtin.edu.my

38

Half-sweep Conjugate Gradient Method for Solving First Order Linear Fredholm

Integro-differential Equations

Elayaraja Aruchunan, Jumat Sulaiman

Department of Engineering and Science, School of Foundation and Continuous Studies

Curtin University of Technology, Sarawak Malaysia.

2School of Scince and Technology, Universiti Malaysia Sabah, Malaysia.

Abstract: The main purpose of this paper is to examine the effectiveness of Half-Sweep Conjugate

Gradient (HSCG) method. The trapezoidal and central difference scheme will be used to discretize

linear Fredholm integro-differential equations of the first order to formulate system linear equation.

The basic formulation and implementation of the proposed method is also presented. Some numerical

tests were carried out to show the effectiveness of the proposed method compared to the Full-Sweep

Conjugate Gradient (FSGS) iterative method.

Key words: Integro-differential equations, trapezoidal, central difference and Half-Sweep conjugate

gradient

INTRODUCTION

In the recent years, the studies of integro-differential equations (IDE) are developed very rapidly and

intensively. IDE is an equation that the unknown function appears under the sign of integration and it also

contains the derivatives and functional arguments of the unknown function. It can be classified into Fredholm

equations and Volterra equations. The upper bound of the region for integral part of Volterra type is variable,

while it is a fixed number for that of Fredholm type (1). However, in this paper, we focus on Fredholm

integro-differential. Generally, first-order linear Fredholm integro-differential equations can be defined as

follows

, . (1)

'() ()() (,)() ()

b

a

y

xqx

y

xKxt

y

tdt

f

x

,,

x

tab

a

ya y

where the functions f (x), q (x) and the kernel K (x,t) are known and y(x) is the solution to be determined.

Habitually, Fredholm IDE of the first order cannot be solved analytically for y(x). In many application

areas, it is necessary to use the numerical approach to obtain an approximation solution for the problem (1).

To solve a linear integro-differential equation numerically, discretization of integral equation to the solution

of system of linear algebraic equations is the basic concept used by researchers to solve integro-differential

problems. By taking into consideration numerical techniques, there are many methods can be used to discretize

problem (1) such as compact finite difference (Zhao, 2006) Wavelet-Galerkin (Avudainayagam,2000)

rationalized Haar functions (Maleknejad, 2004) Lagrange interpolation (Rashed, 2003) Tau (Hosseini, 2003)

quadrature-difference (Fedotov, 2008) and Generalised minimal residual (GMRES) method (Aruchunan, 2009).

The idea of the half-sweep iteration method has been introduced by Abdullah (1991) via the Explicit

Decoupled Group (EDG) iterative method to solve two-dimensional Poisson equation. Half-sweep iteration is

also known as the complexity reduction approach (Hasan, 2007). Following to that, an application of the half-

sweep iteration concept with the iterative methods have been extensively studied by many researchers; see

(Yousif, 1995; Abdullah, 1996; Othman, 2000; Sulaiman, 2004; Sulaiman, 2008; Abdullah, 2006).

The purpose of this paper is to examine Half-Sweep Conjugate Gradient (HSCG) iterative methods for

solving linear algebraic equations produced by the discretization of the first-order linear Fredholm integro-

differential equations by using repeated trapezoidal (RT) and central difference (CD) scheme. The integral term

is discretized by RT scheme and the differential term is approximated by CD scheme. The standard CG method

Aust. J. Basic & Appl. Sci., 5(3): 38-43, 2011

39

also can be called as Full Sweep Conjugate Gradient (FSCG).

In Section 2 of this paper, the formulation of the RT and CD schemes are elaborated. The latter section

of this paper discussed the formulations of the HSCG iterative methods in solving dense linear systems

generated from discretization of the Eq. (1). Meanwhile, some numerical outcome are shown in fourth section

to emphasize the efficiency of the proposed method and concluding remarks are given in Section 5.

II. Full- and Half-sweep Approximation Equations:

As afore-mentioned, a discretization scheme based on method of quadrature and finite difference were used

to construct approximation equations for problem (1) Generally, quadrature method can be defined as follows

(2)

0

n

b

jj n

aj

y

tdt Ayt y

where tj (j = 0,1,2,.....n) is the abscissas of the partition points of the integration interval [a,b], Aj (j =

0,1,2,....,n) is numerical coefficients that do not depend on the function y (t) and gn (y) is the truncation error

of Eq. (3). To ease in formulating the full- and half-sweep quadrature approximation equations for problem

(1), further discussion will be restricted onto repeated trapezoidal (RT) scheme, which is based on linear

interpolation formula with equally spaced data.

(4)

1,0,

2,,2,,

j

ph j n

A

p

hjppnp

where the constant step size, h is defined as

(5)

ba

hn

and n is the number of subintervals in the interval [a,b]. Meanwhile, the value of p, which corresponds to

1 and 2, represents the full and half-sweep cases respectively.

In this paper, central difference approximation as follow:

(6)

'( ) 2

ip ip

i

yy

yx

p

h

where is approximated by the gradient of the line passing '( )

i

yx

,'()

ii

xy

x

Based on Fig. 1, the full and half-sweep iterative methods will compute approximate values onto node

points of type

Conly until the convergence criterion is reached. Then, other approximate solutions at remaining points

(points of the different type,) are computed directly as discussed in (Hasan, 2007; Yousif, 1995; Abdullah,

1996; Othman, 2000; Sulaiman, 2004; Sulaiman, 2007; Sulaiman, 2008). By applying Eq. (2) and Eq. (6)

into Eq. (1) and neglecting the error, a system of linear equations can be formed for approximation values

of. The following linear system generated using RT and CD schemes can be easily shown in matrix form

as follows

My = f (4)

Conjugate Gradient Method (CG):

As mentioned in the previous section, the CG method will be used to solve a system of linear equations.

CG method is proposed by Hestenes and Stiefel (Hestenes, 1952) and was originally developed as a direct

Aust. J. Basic & Appl. Sci., 5(3): 38-43, 2011

40

method designed to solve positive definite linear system. Following is algorithm of general form of CG

method.

Algorithm: CG Method:

Computer

0000

:,:.rfMypr

For until convergence Do:Computer For until convergence Do:

0,1, ,j

aj : = (rj, rj)/(Mpj, pj)

yj+1 : = yj + aj p

j

rj+1 : = rj - ajMpj

βj : = (rj+1,rj+1)/(rj,rj+1)

Pj+1 : = rj+1 + βj p

j

End do

The vectors pj are multiples of the pj 's of algorithm.

Discussion:

Numerical comparison parameters are considered such as number of iterations, execution time and aximum

absolute error. As comparisons, the Gauss-Seidel (GS) method acts as the control of comparison of numerical

results. Throughout the simulations, the convergence test considered the tolerance error of the g = 10-16. In

order to compare the performances of the methods described in the previous section, several experiments were

carried out on the following problems.

Example 1 (Darania, 2007):

(7)

1

0

'( ) ( )

xx

y x xe e x xy t dt

(0) 0y

with exact solution given as.

Results of numerical experiments, which were obtained from implementations of the FSGS, FSGM and

HSGM methods for Example 1, have been recorded in Table 1. Figs.2 and 3 show number of iterations and

execution time versus mesh size respectively for Example 1.Results of example 1 have been recorded in Tables

1.

Example 2 [19]

1

20

111

'( ) ( ) ( ) ln(1 )

121

ln

x

ux utdt ux x x

tx

(0) 1u

with exact solution is given by () ln( 1)ux x

V. Conclusion:

Through the results obtained for Example 1(in Table 1) shows that number of iterations of FSCG and

HSCG methods have decreased approximately 23.81%-28.57% and 27.27%-28.57% respectively compared to

GS method.

Aust. J. Basic & Appl. Sci., 5(3): 38-43, 2011

41

In terms of execution time, both the FSCG and HSCG methods are much faster than the GS method about

12.13% - 40.74% and 49.27%-78.57% respectively. Number of iterations for FSCG and HSCG iterative

methods for Example 2 as shown in Table 2 decreased approximately 37.20% - 39.03% and 37.20% - 41.87%

compared with GS method. Through the observation in Table 2 and Fig. 5, show that execution time for FSCG

and HSCG methods decreased about 22.17% - 31.30% and 41.17% - 83.51% respectively compared to the GS

method.

Generally, the numerical results have shown that the HSCG method is more superior in term of number

of iterations and the execution time than FSCG and GS method.

Fig. 1: a and b show distribution of uniform node points for the full- and half-sweep cases respectively.

Fig. 2: Number of Iterations Versus Mesh Size of the GS, FSCG and HSCG Methods for Example 1.

Fig. 3: The Execution Time (Seconds) Versus Mesh Size of the GS, FSCG and HSCG Methods for Example

1

Fig. 2: Number of Iterations Versus Mesh Size of the GS, FSCG and HSCG Methods for Example 2.

Aust. J. Basic & Appl. Sci., 5(3): 38-43, 2011

42

Fig. 3: The Execution Time (Seconds) Versus Mesh Size of the GS, FSCG and HSCG Methods for Example

2.

Table 1: Comparison of a Number of Iterations, Execution Time and Maximum Absolute Error for the Iterative Methods of Example 1

Methods Number of iteration

----------------------------------------------------------------------------------------------------------------------------

Mesh Size

----------------------------------------------------------------------------------------------------------------------------

1024 2048 4096 8192

GS 21 21 22 22

FSCG 15 16 16 16

HSCG 15 15 16 16

Methods Execution time (seconds)

----------------------------------------------------------------------------------------------------------------------------

Mesh Size

1024 2048 4096 8192

GS 0.27 1.12 4.70 17.23

FSCG 0.16 0.97 3.71 15.14

HSCG 0.07 0.24 1.69 8.74

Methods Maximum absolute error

Mesh Size

1024 2048 4096 8192

GS 2.67E-03 1.33E-03 6.67E-04 3.34E-04

FSCG 2.47E-03 1.23E-03 6.51E-04 3.20E-04

HSCG 5.01E-03 2.47E-03 1.23E-03 6.51E-04

Table 2: Comparison of a Number of Iterations, Execution Time and Maximum Absolute Error for the Iterative Methods of Example 2

Methods Number of iteration

Mesh Size

1024 2048 4096 8192

GS 40 41 43 43

FSCG 25 25 27 27

HSCG 25 25 25 27

Methods Execution time (seconds)

Mesh Size

1024 2048 4096 8192

GS 1.02 5.88 19.57 83.65

FSCG 0.74 4.04 15.23 60.76

HSCG 0.60 0.97 5.11 25.41

Methods Maximum absolute error

Mesh Size

1024 2048 4096 8192

GS 4.56E-04 2.85E-04 9.08E-05 8.56E-05

FSCG 3.91E-04 2.06E-04 8.85E-05 7.76E-05

HSCG 5.65E-04 3.19E-04 2.06E-04 8.85E-05

Aust. J. Basic & Appl. Sci., 5(3): 38-43, 2011

43

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