American Journal of Computational Mathematics, 2011, 1, 134-138
doi:10.4236/ajcm.2011.12014 Published Online June 2011 (http://www.scirp.org/journal/ajcm)
Numerical Solution of Nonlinear Fredholm-Volterra
Integtral Equations via Piecewise Constant Function by
Collocation Method
Ahmad Shahsavaran
Faculty of Science, Islamic Azad University, Borujerd Branch, Borujerd, Iran
Received March 26, 2011; revised April 17, 2011; accepted May 10, 2011
Abstract
In this work, we present a computational method for solving nonlinear Fredholm-Volterra integral equations
of the second kind which is based on replacement of the unknown function by truncated series of well known
Block-Pulse functions (BPfs) expansion. Error analysis is worked out that shows efficiency of the method.
Finally, we also give some numerical examples.
Keywords: Nonlinear Fre dh ol m- Vo lte rr a In te gral Equat io n, Bl ock- Pulse F unction, Error Analysis, Collocation
Points
1. Introduction
The integral equation method is widely used for solving
many problems in mathematical physics and engineering.
This article proposes a computational method for solving
nonlinear Fredholm-Volterra integral equations. Several
numerical methods for approximating the solution of linear
and nonlinear integral equations and specially Fredholm-
Volterra integral equations are known [1-10]. Also, Block-
Pulse functions are studied by many authors and applied
for solving different problems. In presented paper, by using
vector forms of BPfs, the main problem can be easily re-
duced to a nonlinear system of algebraic equations which
can be solved by Newton’s iterative method.
2. Review of Some Related Papers
Some computational methods for approximating the so-
lution of linear and nonlinear integral equations are
known. The classical method of successive approxima-
tion for Fredholm-Hammerstein integral equations was
introduced in [3]. Brunner in [4] applied a collocation
type method and Ordokhani in [8] applied rationalized
Haar function to nonlinear Volterra-Fredholm-Hammer-
stein integral equations. A variation of the Nystrom me-
thod was presented in [5]. A collocation type method
was developed in [6]. The asymptotic error expansion of
a collocation type method for volterra-Hammerstein in-
tegral equations h as been considered in [7]. Yousefi in [9]
applied Legendre wavelets to a special type of nonlinear
Volterra-Fredholm integral equations of the form.
11
0
1
2
0
()()(,)( ( ))d
(,)(())d, 0, 1,
t
utf tKtxFuxx
KtxGux xxt
2






 (1)
where ()
f
t, and 1(, )
K
tx and 2(, )
K
tx are assumed
to be in on the interval
2(LR)0
x
1t, . Yalcinbas in
[10] used Taylor polynomials for solving Equation (1) with
()
p
F
uu

and () q Gu u

. Orthogonal functions and
polynomials receive attention in dealing with various
problems that one of those in integral equation. The main
characteristic of using orthogonal basis is that it reduces
these problems to solving a system of nonlinear algebraic
equations. The aim of this work is to present a numerical
method for approximating the solution of nonlinear Fred-
holm-Volterra integral equation of the form :
11
0
1
2
0
()()(,)( ( ))d
(,)(())d, 0, 1,
tm
n
utf tKtxuxx
Ktxux xxt
2






 (2)
where and are nonnegative integers and 1 m n

and
2

are constants. For this purpose we define a k-set of
BPfs as 1
1,, forall1,2,,
()
0, elsewhere
i
ii
ti
Bt kk

 




k
(3)

A. SHAHSAVARAN ET AL.
135
j
The functions are disjoint and orthogonal. That
is, ()
r
Bt
0,
() ()(),
ji i
i
BtBt Bti j



 (4)
0,
() ()1,
ij
ij
BtBt ij
k







(5)
A function defined over the interval [0, 1) may be
expanded as: ()ut
1
() ()
ii
i
utuB t


. (6)
In practice, only k-term of (6) are considered, where k is
a power of 2, that is,
1
() ()()
k
ki
i
ututuB t


i
t
)
, (7)
with matrix from:
() ()()
t
k
ututuBt
t

, (8)
where, and
12
[, ,,]
k
uu uu
12 ()[(), (),,()]
t
k
tBtBt BtB
m
.
In a similar manner, [( can be approximated in
term of BPfs )]ut
[()] ()
mt
ut t
uB
u
,
that we need to calculate vector whose elements are
nonlinear combination of the elements of the vector u
For this purpose, we can write

() ()ut t
t
uB and [( . )] ()
m
ut t uB
t
So,
() [()]
m
t

uB uB
tt
(9)
now using (4) leads to
1
2
() 0
()
() ()
0(
k
Bt
Bt
tt
Bt







t
BB
also from (3) we get
1
0tk  implies that and
1() 1Bt() 0 i
Bt

for
. 2, ,ik
1
t
kk

2
implies that and
2() 1Bt() 0 i
Bt

for
and . 1, ,ik 2i
11 kt
k
 implies that and () 1
k
Bt() 0
i
Bt

for
. Therefore, simply we obtain
1, ,1ik 
1
() ()dttt
k

IBB
1
0
t
, (10)
where, is the identity matrix of order k. By incorporat-
ing these results we have
I
11
00
() ()d[()d
ttt ttt
ktttk t

 

I
 
uuuBBuB( Bt)
m
t
.
Hence,
1
0
1
1
1
1
1
()] ()d
()]()d
()][()()]d
ttm
i
ktm
k
i
ik
i
ktm t
k
i
ik
kttt
kttt
ktt












t
t
uuBB
uB B
uBu BB
t
[
[
[tt
. (11)
So using (11) leads to
1
1
12
0
1
0
[, ,, ]
0
m
tkk
kuuu
















u.
12
1
2
12 1
10
0
[, ,,]d 0
00
0
1
[, ,,]0
0
k
m
kk
k
uu ut
kuuu





 
















 





.
12
1
1
112
00
1
[, ,,]d 0
00
0
0
[, ,,]
0
1
k
m
kk
k
uu ut
kuuu







 












 




 





.
12
00
0
[, ,,]d
0
01
k
uu ut




 





136 A. SHAHSAVARAN ET AL.
2
111
112 2 1
0
11
112
[0,,0]d[0, , ,0]d
[0,,0, ]d[, ,,]
mm
k
k
k
mmm
kmm m
k
ku utkuut
kuutuu u









m

t Now for evaluating the integral at the
collocation point s 0() ()d
tt
tt
BB
1
2
j
j
tk

, 1, 2,,jk

 , (12)
we may proceed as follows
1/2 1
00 0
2
1
11/2
21
1
2
0
1
() ()() ()d() ()d
()()d
()()d()()d
00
1d0 1d
0
0
00
00
jj
ttt
kk
t
k
k
jj
tt
kk
jj
kk
k
k
k
ttdtttt ttt
ttt
tttttt
tt













 








 




BBBB BB
BB
BB BB
t

1
2
1/2
1
00
0
1d
0
00
00
0
01
1d
0
00
j
k
j
k
j
j
k
j
k
t
tk














































D
(13)
where,
1
[1, 1,,, 0,,0]
2
jkk
Diag   D,
in fact, the diagonal matrix
j
D, is defined
as follows :
1, 2,,j
1,1, 2,1,
1,,
2
0,1,, .
j
mn
mn j
mnj
mn jk












2
Also, may be approxim ated as:
2
(, )[0, 1)Kxt L
kk
11
(, )()()
ij i j
ij
K
xt KBxBt

 ,
or in matrix form
(, )()()
t
K
xtx tBKB, (14)
where 1,
[]
iji jk
K



K and 11
2
00 (, )()
iji j
K
kKxtBx B
.
()ddtxt
3. Solution of the Nonlinear Fredholm-
Volterra Integral Equations
In order to use BPfs for solving nonlinear Fredholm-
Voterra integral equations given in Equation (2), we first
approximate the ,
()ut ()
f
t, , , (())
m
ux (())
n
ux 1(,
K
t
)
x
and 2(, )
K
tx with respect to BPfs
() ()ut tBu
t
t (15)
() ()
f
ttBf
(()) (
mt
ux x uB
(16)
1)
(()) ()
nt
ux x uB
(17)
2
(, )()
t
(18)
1( ) 1
K
tx tBKxB
(, )()()
t
(19)
22
K
txtxBKB (20)
where k-vectors , , 1, 2, and matrices
and are BPfs coefficients of ,
uf u ukk
()ut
1
K2 K()
f
t,
, , 1
(()ux)
m(()ux) n(, )
K
tx and 2(, )
K
tx respectively.
For solving Equation (2), we substitute (15-20) into (2),
therefore
11
22 2
()()()( )( )d
()()()d
tt tt
tt
tttxx
txxx








t
BuBf BKBBu
BKBB u
0
1
0
1
x
k
, (21)
We now collocate Equation (21) at k points ,
i
t
1, 2,,j

 defined by (12) as
11
22 2
()()()() ()d
()()()d
tt tt
jj j
tt
j
ttt xx
txxx








j
t
BuBfBKBB u
BKBB u
0
1
0
1
x
(22)
by using (10) and (13) and the fact that ()
j
j
t

Be
where,
j
e is the j-th column of the identity matrix of
order k, Equati on (2 2) may then be restated as
12
1122 1, 2,,jk
tj t
ji jj
uf
kk

 

eKDueKu , . (23)
Equation (23) gi ves nonlinear equations w hi ch can k
k

A. SHAHSAVARAN ET AL.
137
Table 1.
t Exact A pp roxima t e for 8k

Approximate for 16k

0.1 –1.99 –1.9847 –1.9876
0.2 –1.96 –1.9505 –1.9532
0.3 –1.91 –1.8857 –1.8905
0.4 –1.84 –1.7905 –1.8122
0.5 –1.75 –1.7650 –1.7666
0.6 –1.64 –1.6650 –1.6589
0.7 –1.51 –1.5091 –1.5080
0.8 –1.36 –1.3205 –1.3342
0.9 –1.19 –1.1103 –1.1297
Table 2.
t Exact Approxima t e f o r 8k

Approximate for 16k

0.1 0.0998 0.0625 0.0936
0.2 0.1986 0.1866 0.2070
0.3 0.2955 0.3078 0.2776
0.4 0.3894 0.4242 0.3952
0.5 0.4794 0.5139 0.5067
0.6 0.5646 0.5339 0.5596
0.7 0.6442 0.6353 0.6514
0.8 0.7173 0.7268 0.7243
0.9 0.7833 0.8069 0.7874
be solved for the elements using Newton’s iterative
method. 1
 u
4. Error in BPfs Approximation
Theorem. If a differentiable function with
bounded first derivative on (0, 1) is represented in a se-
ries of BPfs over subinterval
()ut
1
[, )
ii
kk
, we have
1
()( ) et Ok , where . ()() ()
k
etutut
Proof. See [1].
5. Illustrative Examples
Consider the following nonlinear volterra-Fredholm in-
tegral equations.
Example 1.
642
1
2
00
11 55
() 3033 4
()[()]()[()],
t
uttt tt
tx uxdxtx uxdx




0t, 1
x
.
We applied the method presented in this paper for
solving Equation (2) with and .
8k16k
The computational results together with the exact so-
lution are given in Table 1.
2
() 2ut t
Example 2.
2
0
111
() sinsin2[()]d
842
t
uttttux x 
,0t

,1
x

.
The computational results with and 8k16k

together with the exact so lution are given in
Table 2. ()ut sint
6. Conclusions
The aim of present work is to apply a method for solv ing
the nonlinear Volterra-Fredholm integral equations. The
properties of the Block Pulse functions together with the
collocation method are used to reduce the problem to the
solution of nonlinear algebraic equations. Example 1 is
solved in [2] using Chebyshev expansion method (Cem),
comparing the results shows Cem is more accurate than
BPfs method But, it seems the number of calculatio ns of
BPfs method is lower. Also, the benefits of this method
are low cost of setting up the equations due to properties
of BPfs mentioned in Section 2. In addition, the nonlin-
ear system of algebraic equations is sparse. Finally, this
method can be easily extended and applied to nonlinear
Volterra-Fredholm integral equations of the form Equa-
tion (1). Illustrative examples are included to demon-
strate the validity and app licability of the technique.
7. References
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of an Expansion Method for Linear Volterra Integral
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byshev Approximation for Solving Nonlinear Integral
Equations of Hammerstein Type,” Applied Mathematics
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doi:10.1016/j.amc.2006.11.181
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[4] H. Brunner, “Implicity Linear Collocation Method for
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[5] L. J. Lardy, “A Variation of Nysrtom’s Method for
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[7] H. Guoqiang, “Asymptotic Error Expansion Variation of
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A. SHAHSAVARAN ET AL. 138
holm-Hammerstein Integral Equations Via Rationalized
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