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International Journal of Applied Mathematics and ComputationVolume 2(1),pp 3249, 2010http://ijamc.psit.in

A set of multi-dimensional orthogonal basis functions andits application to solve integral equations

Esmail Babolian 1, Zahra Masouri 2, Saeed Hatamzadeh-Varmazyar 3

1 Department of Mathematics, Science and Research Branch,Islamic Azad University (IAU), Tehran, Iran Email: babolian@tmu.ac.ir

2 Department of Mathematics, Khorramabad Branch,Islamic Azad University (IAU), Khorramabad, Iran Email: nmasouri@yahoo.com

3 Department of Electrical Engineering, Science and Research Branch,Islamic Azad University (IAU), Tehran, Iran Email: s.hatamzadeh@yahoo.com

Abstract :A new set of multi-dimensional orthogonal basis functions and some of their prop-

erties are introduced. These functions are extension of triangular functions (TFs)in n dimensions. Expansion of multi-variable functions with respect to these func-tions is presented. Also, the relation of these new functions to the block-pulse func-tions (BPFs) in n dimensions will be investigated.

Many applied problems are often discussed in n dimensions, consequently themulti-dimensional moment method using the current orthogonal basis functions willbe used to solve two-variable integral equations. The obtained results are comparedwith those of the multi-dimensional moment method using BPFs. These comparisonsshow efficiency and accuracy of the new orthogonal basis functions applied to solvemulti-dimensional integral equations. Finally, a study of the representational errorwill be made to estimate the mean integral squared error for the TF approximationof a function f (s, t ) of Lebesgue measure.

Keywords: Multi-dimensional functions; Orthogonal basis functions; Triangularfunctions; Moment method; Multi-variable integral equations.

1 Introduction

Many physical and engineering problems can be formulated to the mathematical models.These models are often of the forms integral and integro-differential equations, differential

Corresponding author: Z. Masouri

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equations, dynamic systems, control systems, etc. There are many approaches to solvethese problems.

In recent years, numerical methods using orthogonal basis functions have been recom-mended by many authors [7, 8, 10, 19, 20]. Some authors have used Sine-Cosine func-tions and orthogonal polynomials such as Chebyshev, Legendre, and Lagrange polyno-mials [2, 11, 12]. Piecewise constant or linear orthogonal functions such as block-pulsefunctions, Walsh functions, Haar functions, triangular functions [36, 9, 1317, 23], alsovarious wavelets [1, 11, 12, 13, 18] have been considered by many researchers for model-ing the physical or engineering problems and to solve different functional equations, oftenproducing very good accuracy.

Whereas many problems are occurred in n dimensions, therefore introducing new setsof multi-dimensional orthogonal basis functions and consequently illustrating numericalmethods based on them can be necessary.

In this article, rst of all, some characteristics of TFs are briey described. Then, thenew set of multi-dimensional orthogonal basis functions as a generalization of triangular

functions is introduced. The characteristics of these new functions and the expansion of multi-variable functions with respect to them are presented. Also, the relation of presentedfunctions to multi-dimensional BPFs is shown.

In section 4, multi-dimensional moment method using the new basis functions will beproposed. This method is applied to solve some two-variable integral equations. Theobtained results are compared with those of the moment method using two-dimensionalBPFs.

Finally, an estimation of the mean integral squared error will be presented.

2 Review of Triangular Functions

A special representation of one-dimensional triangular functions have been presented byDeb et al. [9] and studied and used by Babolian et al. [4,5].

Two m-sets of triangular functions (TFs) are dened over the interval [0 , T ) as [9]

T 1i (t) =1 t ihh , ih t < (i + 1) h,0, otherwise ,

T 2i (t) =t ih

h , ih t < (i + 1) h,0, otherwise ,

(2.1)

where i = 0 , 1, . . . , m 1, with a positive integer value for m. Also, consider h = T/m , andT 1i as the ith left-handed triangular function and T 2i as the ith right-handed triangularfunction.

Assume that T = 1, so TFs are dened over [0 , 1), and h = 1 /m .From the denition of TFs, it is clear that triangular functions are disjoint, orthogonal,

and complete [9]. Therefore, we can write

1

0T 1i (t)T 1 j (t)dt =

1

0T 2i (t)T 2 j (t)dt =

h3 , i = j,0, i = j.

(2.2)

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Also,

i (t) = T 1i (t) + T 2i (t), i = 0 , 1, . . . , m 1, (2.3)

where i (t) is the ith block-pulse function dened as

i (t) =1, ih t < (i + 1) h,0, otherwise ,

(2.4)

where i = 0 , 1,...,m 1.Consider the rst m terms of left-handed triangular functions and the rst m terms of

right-handed triangular functions and write them concisely as m-vectors

T1 (t) = [T 10(t), T 11(t),...,T 1m 1(t)]T

,

T2 (t) = [T 20(t), T 21(t),...,T 2m 1(t)]T ,(2.5)

where T1 (t) and T2 (t) are called left-handed triangular functions (LHTF) vector andright-handed triangular functions (RHTF) vector, respectively.

The expansion of a function f (t) over [0, 1) with respect to TFs, may be compactlywritten as

f (t) m 1

i=0

ci T 1i (t) +m 1

i=0

di T 2i (t)

= c T T1 (t) + d T T2 (t),

(2.6)

where we may put ci = f (ih ) and di = f (( i+1) h) for i = 0 , 1, . . . , m 1. So, approximatinga known function by TFs needs no integration to evaluate the coefficients.

In the next section, TFs will be generalized to n dimensions and a new set of multi-dimensional orthogonal basis functions will be introduced.

3 Multi-dimensional Triangular Functions

Here, TFs are generalized to n dimensions. We rstly dene two-dimensional TFs, thenthese functions will be extended to n dimensions. Also, the expansion of multi-variablefunctions with respect to these new basis functions is presented and the relation of multi-dimensional BPFs to these functions will be shown.

3.1 Denition and some characteristics

We dene a set of two-dimensional triangular functions as T 1i,j (s, t ) and T 2i,j (s, t ) over aregion [0, T 1) [0, T 2), as follows:

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0hT=2h0 h T=2h

0

0.5

1

st

T 1

0 , 0

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 1

1 , 0

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 1

0 , 1

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 1

1 , 1

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 2

0 , 0

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 2

1 , 0

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 2

0 , 1

( s , t

)

0h

T=2h0 h T=2h

0

0.5

1

st

T 2

1 , 1

( s , t

)

Figure 1: A set of two-dimensional LHTFs T 1i,j (s, t ) and RHTFs T 2i,j (s, t ), for i, j = 0 , 1.

T 1i,j (s, t ) =1 s+ t

( ih 1 + jh 2 )h 1 + h 2 ,

ih 1 s < (i + 1) h1, jh 2 t < ( j + 1) h2,

0, otherwise ,

T 2i,j (s, t ) =s + t ( ih 1 + jh 2 )

h 1 + h 2 ,ih 1 s < (i + 1) h1, jh 2 t < ( j + 1) h2,

0, otherwise ,

(3.1)

where i = 0 , 1, . . . , m 1 1 and j = 0 , 1, . . . , m 2 1, with positive integer values for m1, m 2.Also, consider h1 = T 1/m 1 and h2 = T 2/m 2, and T 1i,j as the i, j th left-handed triangularfunction (LHTF) and T 2i,j as the i, j th right-handed triangular function (RHTF).

Figure 1 shows the set of two-dimensional orthogonal triangular basis functions, wherefor convenience, m1 = m2 has been arbitrarily chosen as 2 and T 1 = T 2 = T , so h1 = h2 =h.

Now, consider the rst m1m2 terms of left-handed triangular functions and the rstm1m2 terms of right-handed triangular functions and write them concisely as m1m2-

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vectors

T1 (s, t ) = [T 10,0(s, t ), T 10,1(s, t ), . . . , T 10,m 2 1(s, t ), . . . ,

T 1m 1 1,0(s, t ), . . . , T 1m 1 1,m 2 1(s, t )]T ,

T2 (s, t ) = [T 20,0(s, t ), T 20,1(s, t ), . . . , T 20,m 2 1(s, t ), . . . ,

T 2m 1 1,0(s, t ), . . . , T 2m 1 1,m 2 1(s, t )]T ,

(3.2)

where T1 (s, t ) and T2 (s, t ) are called left-handed triangular functions vector and right-handed triangular functions vector, respectively.

From denition (3.1), it is clear that these functions are disjoint, complete, and orthog-onal; so that for s [0, T 1) and t [0, T 2), we can write

T 1

0

T 2

0T 1i,j (s, t )T 1 p,q (s, t )dsdt

=h 1 h 2

6(h 1 + h 2 )2 (2h21 + 3 h1h2 + 2 h22), i = p, j = q,

0, otherwise ,

T 1

0 T 2

0T 2i,j (s, t )T 2 p,q (s, t )dsdt

=h 1 h 2

6(h 1 + h 2 )2 (2h21 + 3 h1h2 + 2 h22), i = p, j = q,

0, otherwise ,

(3.3)

and

T 1

0 T 2

0T 1i,j (s, t )T 2 p,q (s, t )dsdt

=h 1 h 2

6(h 1 + h 2 )2 (h21 + 3 h1h2 + h22), i = p, j = q,

0, otherwise .

(3.4)

As a simple case, consider T 1 = T 2 = T and m1 = m2. Hence, h1 = h2 = h and theEqs. (3.3) and (3.4) may be read as

T

0

T

0T 1i,j (s, t )T 1 p,q (s, t )dsdt =

7h 224 , i = p, j = q,

0, otherwise ,

T

0 T

0T 2i,j (s, t )T 2 p,q (s, t )dsdt =

7h 224 , i = p, j = q,

0, otherwise ,

(3.5)

and

T

0 T

0T 1i,j (s, t )T 2 p,q (s, t )dsdt =

5h 224 , i = p, j = q,

0, otherwise .(3.6)

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Now, we extend the new set of orthogonal basis functions dened by Eqs. (3.1) todimension n.

A system of n-dimensional TFs is dened as

T 1i 1 ,i 2 ,...,i n (t1, t 2, . . . , t n )

=1

nk =1 t k

nk =1 i k h k

nk =1 h k

, ik hk tk < (ik + 1) hk ,

0, otherwise ,

T 2i 1 ,i 2 ,...,i n (t1, t 2, . . . , t n )

=

nk =1 t k

nk =1 i k h k

nk =1 h k

, ik hk tk < (ik + 1) hk ,

0, otherwise ,

(3.7)

where tk [0, T k ), hk = T k /m k , and ik = 0 , 1, 2, . . . , m k 1, for k = 1 , 2, . . . , n . Also,

m k denotes the number of segments on [0 , T k ) along the coordinate axis of the Cartesiandirection tk .Note that in the case of n-dimensional, left-handed triangular functions vector and

right-handed triangular functions vector can be dened as in Eqs. (3.2). But it shouldbe noted that these vectors have m1m2 . . . , m n components. Ultimately, it can be easilyshown that these functions are disjoint and orthogonal.

3.2 Expansion of multi-variable functions

Initially, the expansion of a two-dimensional function f (s, t ) with respect to the new setof TFs dened by Eqs. (3.1) is presented. This expansion may be compactly written as

f (s, t ) m 1 1

i=0

m 2 1

j =0

ci,j T 1i,j (s, t ) +m 1 1

i=0

m 2 1

j =0

di,j T 2i,j (s, t )

= c T T1 (s, t ) + d T T2 (s, t ),

(3.8)

where ci,j and di,j are constant coefficients for i = 0 , 1, . . . , m 1 1 and j = 0 , 1, . . . , m 2 1,respectively and the m1m2-vectors c , d can be written as

c T = [c0,0, c0,1, . . . , c0,m 2 1, c1,0, . . . , cm 1 1,0, . . . , c m 1 1,m 2 1]T ,

d T = [d0,0, d0,1, . . . , d 0,m 2 1, d1,0, . . . , d m 1 1,0, . . . , d m 1 1,m 2 1]T .

(3.9)

The coefficients ci,j and di,j , for i = 0 , 1, . . . , m 1 1 and j = 0 , 1, . . . , m 2 1 can be setto

ci,j = f (ih 1, jh 2),di,j = ci+1 ,j +1 = f (( i + 1) h1, ( j + 1) h2).

(3.10)

So, approximating a known function by two-dimensional TFs may need no integrationto evaluate the coefficients. It can be done only by sampling of two-variable function f at(m1 + 1)( m2 + 1) specic points on the region [0 , T 1] [0, T 2].

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The component dm 1 1,m 2 1 = f (m1h1, m 2h2) = f (T 1, T 2) of vector d can be computedif the function f (s, t ) is dened over [0, T 1] [0, T 2]. If this function is dened over [0 , T 1) [0, T 2) and be a periodic function, we assume f (T 1, T 2) = f (0, 0).

Also, it should be noted that the above computations of coefficients are not an optimal

case. If we consider an optimal representation of f (s, t ) leading to least mean-absoluteerror, the coefficients ci,j and di,j , for i = 0 , 1, . . . , m 1 1 and j = 0 , 1, . . . , m 2 1 are tobe determined from the following two equations:

( i+1) h 1

ih 1 ( j +1) h 2

jh 2f (s, t )T 1i,j (s, t ) dsdt

= ci,j ( i+1) h 1

ih 1 ( j +1) h 2

jh 2[T 1i,j (s, t )]2 dsdt

+ di,j ( i+1) h 1

ih 1 ( j +1) h 2

jh 2T 1i,j (s, t )T 2i,j (s, t ) dsdt,

( i+1) h 1

ih 1 ( j +1) h 2

jh 2f (s, t )T 2i,j (s, t ) dsdt

= ci,j ( i+1) h 1

ih 1 ( j +1) h 2

jh 2T 1i,j (s, t )T 2i,j (s, t ) dsdt

+ di,j ( i+1) h 1

ih 1 ( j +1) h 2

jh 2[T 2i,j (s, t )]2 dsdt.

(3.11)

Using orthogonality of two-dimensional TFs from Eqs. (3.3) and (3.4) we obtain

A = h1h26(h1 + h2)2(2h21 + 3 h1h2 + 2 h22)ci,j

+h1h2

6(h1 + h2)2(h21 + 3 h1h2 + h

22)di,j ,

B =h1h2

6(h1 + h2)2(h21 + 3 h1h2 + h

22)ci,j

+h1h2

6(h1 + h2)2(2h21 + 3 h1h2 + 2 h

22)di,j ,

(3.12)

where,

A = ( i+1) h 1

ih 1 ( j +1) h 2

jh 2f (s, t )T 1i,j (s, t ) ds dt,

B = ( i+1) h 1

ih 1 ( j +1) h 2

jh 2f (s, t )T 2i,j (s, t ) ds dt,

(3.13)

where s [ih 1, (i + 1) h1) and t [ jh 2, ( j + 1) h2).

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To compare these two kinds of representations, an example will be illustrated in sub-section 3.3.

Now, let f (t1, t 2, . . . , t n ) be an n-variable function. The expansion of f with respect ton-dimensional TFs has the following form:

f (t1, t 2, . . . , t n ) m 1 1

i 1 =0

m 2 1

i 2 =0

m n 1

i n =0

ci1 ,i 2 ,...,i n T 1i1 ,i 2 ,...,i n (t1, t 2, . . . , t n )

+m 1 1

i1 =0

m 2 1

i2 =0

m n 1

i n =0

di 1 ,i 2 ,...,i n T 2i1 ,i 2 ,...,i n (t1, t 2, . . . , t n )

= c T T1 (t1, t 2, . . . , t n ) + d T T2 (t1, t 2, . . . , t n ),

(3.14)

where ci1 ,i 2 ,...,i n and di1 ,i 2 ,...,i n , are constant coefficients for ik = 0 , 1, . . . , m k 1, k =0, 1, . . . , n , respectively. These coefficients can be set as follows:

ci1

,i2

,...,in

= f (i1h

1, i

2h

2, . . . , i n hn ),

di 1 ,i 2 ,...,i n = ci 1 +1 ,i 2 +1 ,...,i n +1 = f (( i1 + 1) h1, (i2 + 1) h2, . . . , (in + 1) hn ).(3.15)

Corresponding to the two-dimensional case, it is clear that c and d are m1m2 . . . m n -vectors and f (T 1, T 2, . . . , T n ) can be set to a suitable value like the two-dimensional case.Also, it is clear that the optimal coefficients ci 1 ,i 2 ,...,i n and di1 ,i 2 ,...,i n , can be easily computedusing orthogonality of n-dimensional TFs like the two-dimensional case.

3.3 Multi-dimensional BPFs and comparing them with multi-dimensionalTFs

In this section, the n-dimensional BPFs are breiy described, and their relation to multi-dimensional TFs is shown.

A system of n-dimensional BPFs over a region tk [0, T k ), for k = 1 , 2, . . . , n , isdened as i 1 ,i 2 ,...,i n (t1, t 2, . . . , t n ) to form an orthogonal basis for approximating a multi-dimensional squar-integrable function f (t1, t 2, . . . , t n ). For each set of integers i1, i2, . . . , i n ,the n-dimensional BPF is dened as follows [21]:

i 1 ,i 2 ,...,i n (t1, t 2, . . . , t n ) =1, ik hk t < (ik + 1) hk ,0, otherwise ,

(3.16)

where mk denotes the number of segments on [0 , T k ) along the coordinate axis of theCartesian direction tk and ik = 0 , 1, . . . , m k 1, and hk = T k /m k , for k = 1 , 2, . . . , n . It isclear that these functions are complete, disjoint and orthogonal.

The set of BPFs may be written as a vector ( t1, t 2, . . . , t n ) of dimension nk=1 mk .For instance, when n = 2:

(t1, t 2) = [0,0(t1, t 2), 0,1(t1, t 2), . . . , 0,m 2 1(t1, t 2), . . . ,

m 1 1,0(t1, t 2), . . . , m 1 1,m 2 1(t1, t 2)]T .

(3.17)

A multi-dimensional square-integrable function in the region tk [0, T k ), for k =1, 2, . . . , n , may be approximated as

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where s, t [0, 1) and i, j = 0 , 1, . . . , m 1.Now, let f (s, t ) = exp( 12 (s + t)). For m = 2, the BPFs representation of f using

Eqs. (3.18) and (3.19) is

f BP F (s, t ) = [1 .29073, 1.65733, 1.65733, 2.12805](s, t ). (3.22)

Similarly, optimal TFs representation of f from Eqs. (3.8), (3.12) and (3.13) is givenby

f T F 1(s, t ) = [0 .96838, 1.24343, 1.24343, 1.59659]T1 (s, t )+ [1 .61307, 2.07123, 2.07123, 2.65951]T2 (s, t ),

(3.23)

and using Eqs. (3.8) and (3.10), non-optimal TFs representation of f is given by

f T F 2(s, t ) = [1 , 1.28403, 1.28403, 1.64872]T1 (s, t )

+ [1 .64872, 2.11700, 2.11700, 2.71828]T2 (s, t ).

(3.24)

To better evaluation of the above representations, the mean-absolute error at points(s, t ), are computed as follows:

E (m )n =1n

n

i=1

|f (s i , t i ) f m (s i , t i )|, (3.25)

where f (s, t ) is the exact solution and f m (s, t ) is the approximate solution.These mean-absolute errors at one hundred points of the forms ( s i , t j ), for s i = 0 .1i

and t j = 0 .1 j , for i, j = 0 , 1, . . . , 9 in [0, 1) [0, 1), are 1 .5E 1, 8.8E 3, and 4 .2E 2,

for BPFs, optimal TFs, and non-optimal TFs representations of f (s, t ), respectively.Also, Figs. 2(a), 2(b), 2(c), and 2(d) show the exact, BPFs, optimal TFs, and non-

optimal TFs representations of f (s, t ), respectively. Hence, it is apparent that TFs repre-sentation is superior to BPFs representation.

4 Multi-dimensional Moment Method

In this section, by using the results obtained in the previous section about multi-dimensionaltriangular functions as the orthogonal basis functions, an effective and very accuratemethod for solving multi-variable integral equations is presented.

For convenience, we consider two-variable Fredholm integral equations of the secondkind. Note that the method presented here can be easily extended and applied to anymulti-variable linear Fredholm and Volterra integral equations of the rst and second kind.

As it was mentioned in previous section, two-dimensional TFs can be extended overany region such as ( s, t ) [a, b) [c, d) in the plane R 2. Therefore, consider the followingFredholm integral equation of the second kind with two variables:

x(s, t ) + b

a d

ck(s,t ,s , t )x(s , t )ds dt = y(s, t ), (4.1)

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Figure 2: (a) The exact representation of f (s, t ). (b) The BPF representation of f (s, t ). (c) The optimalTF representation of f (s, t ). (d) The non-optimal TF representation of f (s, t ).

where ( s, t ) [a, b) [c, d) and the functions k(s,t ,s , t ), y(s, t ), and the parameter areknown but x(s, t ) is the unknown function to be determined.

Now, the moment method is presented for Eq. (4.1). Approximating the function x(s, t )with respect to the two-dimensional triangular functions by Eq. (3.8) gives

x(s, t ) c T T1 (s, t ) + d T T2 (s, t ), (4.2)

where the m1m2-vectors c and d are TFs coefficients of x(s, t ) that should be determined.

Substituting Eq. (4.2) into (4.1) givesm 1 1

i=0

m 2 1

j =0

ci,j T 1i,j (s, t ) + b

a d

ck(s,t ,s , t )T 1i,j (s

, t )ds dt

+ di,j T 2i,j (s, t ) + b

a d

ck(s,t ,s , t )T 2i,j (s

, t )ds dt

y(s, t ).

(4.3)

For computing the unknown m1m2-vectors c and d , we must consider 2 m1m2 appro-priate points ( s p, t q), for s p [a, b), and tq [c, d). Substituting these points in Eq. (4.3)

gives

m 1 1

i=0

m 2 1

j =0

ci,j T 1i,j (s p, tq) + ( i+1) h 1

ih 1 ( j +1) h 2

jh 2k(s p, t q , s

, t )T 1i,j (s , t )ds dt

+ di,j T 2i,j (s p, t q) + ( i+1) h 1

ih 1 ( j +1) h 2

jh 2k(s p, t q , s

, t )T 2i,j (s , t )ds dt

y(s p, t q).

(4.4)

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Replacing with = yields a linear system of 2 m1m2 algebraic equations for 2 m1m2 un-known components c0,0, . . . , c0,m 2 1, . . . , cm 1 1,m 2 1 and d0,0, . . . , d0,m 2 1, . . . , dm 1 1,m 2 1.Therefore, an approximate solution x(s, t ) c T T1 (s, t ) + d T T2 (s, t ) is obtained forEq. (4.1).

It is clear that one can apply this method for two-variable Fredholm integral equationsof the rst kind. Also, this method can be easily used to solve linear Volterra integralequations.

Finally, note that the above moment method uses optimal case of TF approximation.Obviously, one can consider non-optimal case of TF approximation. Therefore, there are(m1 + 1)( m2 + 1) unknowns instead of 2 m1m2 unknowns because di,j = ci+1 ,j +1 . So, weneed (m1 + 1)( m2 + 1) collocation points over [ a, b) [c, d).

5 Numerical Examples

The moment method presented in this article, is applied to some examples. This approach

uses both the new set of two-dimensional orthogonal basis TFs and two-dimensional BPFs,so the results obtained via two sets of basis functions can be compared. Additionally, theseexamples have been presented in [11] too, hence the numerical results obtained here canbe compared with the results of another method.

The computations associated with the examples were performed using Matlab 7 on aP ersonal C omputer.

Example 5.1. Consider the following two-variable Fredholm integral equation of thesecond kind [11]:

x(s, t )

1

0 1

0

(ss 2 + t2t )x(s , t ) ds dt = y(s, t ), (5.1)

with the exact solution x(s, t ) = exp(st ), and suitable right hand side. For m1, m 2 = 10,Fig. 3 shows the exact and approximate solutions. Also, the numerical results at tenspecic points in [0 , 1) [0, 1) are presented in Table 1.

The mean-absolute errors from Eq. (3.25), at one hundred points of the forms ( s i , t j ),for s i = 0 .1i and t j = 0 .1 j , for i, j = 0 , 1, . . . , 9 over [0, 1) [0, 1), are 6.5E 2 and 5.6E 3,for BPFs and TFs approximate solutions, respectively.

Example 5.2. For the following two-variable linear Fredholm integral equation of thesecond kind [11]:

x(s, t ) 1 1 1 1 cos(ss )cos( tt )x(s , t ) ds dt = y(s, t ), (5.2)

where y(s, t ) is chosen so that Eq. (5.2) has the exact solution x(s, t ) = sin (s + t), form1, m 2 = 8, Fig. 4 shows the exact and approximate solutions. Table 2 shows the numer-ical results at some specic points in [ 1, 1) [ 1, 1) too.

The mean-absolute errors from Eq. (3.25), at one hundred points of the forms ( s i , t j ),for s i = 1 + 0 .2i and t j = 1 + 0 .2 j , for i, j = 0 , 1, . . . , 9 over [ 1, 1) [ 1, 1), are6.3E 2 and 3.6E 3, for BPFs and TFs approximate solutions, respectively.

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Table 1: Numerical results for Example 5.1

(s,t) Exact solutionApproximate solution

using TFsApproximate solution

using BPFs(0,0) 1 0.997492 1.002428

(0.1,0.8) 1.083287 1.078875 1.134592(0.2,0.4) 1.083287 1.079786 1.118388(0.3,0.6) 1.197217 1.192452 1.254273(0.4,0.3) 1.127497 1.123622 1.169745(0.5,0.7) 1.419068 1.412062 1.508894(0.6,0.9) 1.716007 1.705436 1.851894(0.7,0.2) 1.150274 1.145733 1.205069(0.8,0.1) 1.083287 1.078990 1.134748(0.9,0.5) 1.568312 1.559442 1.684402

0

0.5

1

0

0.5

11

1.5

2

2.5

3

s

(a)

t

x T F

( s , t )

0

0.5

1

0

0.5

11

1.5

2

2.5

3

s

(c)

t

x ( s

, t )

0

0.5

1

0

0.5

11

1.5

2

2.5

3

s

(b)

t

x B P F

( s , t )

Figure 3: (a) The TF approximate solution. (b) The BPF approximate solution. (c) The exact solution.

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Table 2: Numerical results for Example 5.2

(s,t) Exact solutionApproximate solution

using TFsApproximate solution

using BPFs( 1, 1) 0.909297 0.939424 0.983986

( 0.8,0.4) 0.389418 0.391667 0.479426( 0.6, 0.2) 0.717356 0.718223 0.681639( 0.4,0.6) 0.198669 0.198958 0.247404( 0.2,0.8) 0.564642 0.571885 0.681639

(0,0) 0 0.005176 0.247404(0.2, 0.8) 0.564642 0.571885 0.681639(0.4,0.2) 0.564642 0.567184 0.479426

(0.6, 0.6) 0 0 0(0.8, 0.4) 0.389418 0.391667 0.479426

1

0

1

1

0

11

0.5

0

0.5

1

s

(a)

t

x T F

( s , t

)

1

0

1

1

0

11

0.5

0

0.5

1

s

(c)

t

x ( s

, t )

1

0

1

1

0

11

0.5

0

0.5

1

s

(b)

t

x B P F

( s , t

)

Figure 4: (a) The TF approximate solution. (b) The BPF approximate solution. (c) The exact solution.

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6 Error Analysis

The error analysis for BPF expansion of any square integrable function has been investi-gated by Rao [21]. Also, Deb et al. [9] have carried out an error analysis in the triangularfunction domain. Here, we present an error analysis for two-dimensional TF expansion of any square integrable function of Lebesgue measure.

Let us consider m1 = m2 = m and h1 = h2 = h. The mean integral squared error inthe i, j th region is given by

[E i,j ]2 = ( i+1) h

ih ( j +1) h

jhf (s, t ) f (s, t )

2dsdt, (6.1)

so that s [ih, (i + 1) h), t [ jh, ( j + 1) h), and f (s, t ) is the approximation of f (s, t ) inthe i, j th region and is dened as

f (s, t ) = ci,j T 1i,j + di,j T 2i,j , (6.2)

where ci,j = f (ih,jh ), di,j = f (( i + 1) h, ( j + 1) h). So,

f (s, t ) = f (ih,jh ) + m i,j (s + t) m i,j (i + j )h, (6.3)

in which,

m i,j =f (( i + 1) h, ( j + 1) h) f (ih,jh )

2h. (6.4)

Let the function f (s, t ) be extended by rst order Taylor series in the i, j th regionaround the point ( i , j ). Hence,

f (s, t ) f (i , j ) + ( s i ) f s (i , j ) + ( t j ) f t (i , j ), (6.5)

where f s (i , j ) = f (s,t )s s = i ,t = jand f t (i , j ) = f (s,t )t s = i ,t = j

. Also, i [ih, (i + 1) h],

j [ jh, ( j + 1) h]. Substituting Eqs. (6.3) and (6.5) into Eq. (6.1) gives

[E i,j ]2 = ( i+1) h

ih ( j +1) h

jhf (ih,jh ) + m i,j (s + t) m i,j (i + j )h

f (i , j ) (s i ) f s (i , j ) (t j ) f t (i , j ) dsdt.(6.6)

Now consider

A = f (ih,jh ) f (i , j ) + i f s (i , j ) + j f t (i , j ) m i,j (i + j )h,

B = m i,j f s (i , j ),

C = m i,j f t (i , j ).

(6.7)

Therefore, Eq. (6.6) can be written as

[E i,j ]2 = ( i+1) h

ih ( j +1) h

jh[A + Bs + Ct]2 dsdt. (6.8)

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The Eq. (6.8) may be simplied to

[E i,j ]2 = A2h2 +3i2 + 3 i + 1

3B 2h4 + (2 i + 1) ABh 3 + (2 j + 1) ACh 3

+(2i + 1)(2 j + 1)

2 BCh4

+3 j 2 + 3 j + 1

3 C 2h

4.

(6.9)

Then, the mean integral squared error over m2 rectangles is given by

E 2 =m 1

i=0

m 1

j =0

[E i,j ]2. (6.10)

Let us now consider some special cases that can be derived from Eq.(6.9).

Case I: Assume that the function f (s, t ) is a constant function. Then, f s (s, t ) =f t (s, t ) = 0 and m i,j = 0. Hence, A = B = C = 0 and Eq. (6.9) follows [ E i,j ]2 = 0. So,

the mean integral squared error for constant functions is zero.

Case II: Let f (s, t ) = as + bt + c, where a, b, c are constant. Since f s (s, t ) = a,f t (s, t ) = b, and m i,j = a + b2 , from Eqs. (6.7) we get A =

h2 (a b)( i j ) and B = C =

b a2 .

So, from Eq. (6.9), the mean integral squared error is

[E i,j ]2 =h4

24(a b)2. (6.11)

In this case, [ E i,j ]2 = O(h2), proportional to h2. Also, it will be negligible if |a b|approaches to zero. It is clear that if a = b, then [E i,j ]2 = 0. Therefore, the mean integralsquared error for any function of the form f (s, t ) = a(s + t) + c is zero.

Here, the mean integral squared error has been obtained for two-dimensional functions.However, the presented procedure can be easily extended to obtain this error for arbitrarydimension.

7 Conclusion

This article introduced a new set of multi-dimensional orthogonal basis functions and itscharacteristics. Its effectiveness for representation of functions was established for optimaland non-optimal cases of multi-dimensional TFs. Also, their relation to the well-knownBPFs was shown.

Then, the multi-dimensional moment method based on the new set of orthogonal basisfunctions was proposed. This approach, transforms a multi-variable integral equation toa set of algebraic equations. Its applicability and accuracy was checked on two examples.In these examples the approximate solution was briey compared with exact and BPFapproximate solutions only at ten specic points. These points have been selected over aregion in R 2 at random. But the mean-absolute errors have been computed for one hundredpoints in the related region. It follows from the gures, tables and mean-absolute errorsthat the accuracy of the obtained solutions using multi-dimensional TFs is quite good incomparison with BPF approximate solutions. Increasing the number of TFs decreases the

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error of the solution rapidly. To show the convergence and stability of this approach, thecurrent method can be run with increasing m1 and m2 until the computed results haveappropriate accuracy.

[11] proposes a computational meshless method to solve Examples 5.1 and 5.2. Com-paring gures obtained by current method using multi-dimensional TFs with gures in[11], it seems that the presented method is more accurate than the proposed approachin [11].

Finally, the mean integral squared error estimated in section 6 shows that for somecases the representational error is zero.

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[8] H. Brunner, Collocation Method for Volterra Integral and Relation Functional Equations, CambridgeUniversity Press, Cambridge, 2004.

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[10] L.M. Delves and J.L. Mohamed, Computational Methods for Integral Equations, Cambridge Univer-sity Press, Cambridge, 1985.

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[16] S. Hatamzadeh-Varmazyar and M. Naser-Moghadasi, New numerical method for determining thescattered electromagnetic elds from thin wires, Progress In Electromagnetics Research B , 3 (2008),207218.

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