Journal of the Association of Arab Universities for Basic and Applied Sciences (2012) 12, 65–73

University of Bahrain

Journal of the Association of Arab Universities for
Basic and Applied Sciences
www.elsevier.com/locate/jaaubas
www.sciencedirect.com

ORIGINAL ARTICLE

Approximate solutions for mixed nonlinear
Volterra–Fredholm type integral equations
via modified block-pulse functions
Farshid Mirzaee *, Elham Hadadiyan
Department of Mathematics, Faculty of Science, Malayer University, Malayer 65719-95863, Iran
Available online 10 July 2012

KEYWORDS
Mixed nonlinear Volterra–
Fredholm type integral
equations;
Block-pulse functions;
Operational matrix

Abstract In this article a robust approach for solving mixed nonlinear Volterra–Fredholm type
integral equations of the first kind is investigated. By using the modified two-dimensional blockpulse functions (M2D-BFs) and their operational matrix of integration, first kind mixed nonlinear
Volterra–Fredholm type integral equations can by reduced to a nonlinear system of equations. The
coefficients matrix of this system is a block matrix with lower triangular blocks. Some theorems are
included to show the convergence and advantage of this method. Numerical results show that the
approximate solutions have a good degree of accuracy.

ª 2012 University of Bahrain. Production and hosting by Elsevier B.V. All rights reserved.

1. Introduction
In this paper we applied the direct method for solving mixed
nonlinear Volterra–Fredholm type integral equations of the
first
Z Zkind of the form:
x

Gx; y; s; t; us; tịịdtds ẳ fx; yị;

0

x; yị 2 ẵ0; 1ị X;

X

1ị
* Corresponding author. Tel./fax: +98 8513339944.
E-mail addresses: ,
(F. Mirzaee).
1815-3852 ª 2012 University of Bahrain. Production and hosting by
Elsevier B.V. All rights reserved.
Peer review under responsibility of University of Bahrain.
/>
Production and hosting by Elsevier

where u(s,t) is an unknown function, f(x,y) and G(x,y,s,t,u(s,t))
are analytical function on [0,1) · X and [0,1) · X4, respectively,
where X is a close subset on Rd d ẳ 1; 2; 3ị. Existence and

uniqueness results for Eq. (1) may be found in (Diekmann,
1978; Pachpatte, 1978; Thieme, 1977).
Equation of type (1) often arise from the mathematical
modeling of the spreading, in space and time, of some contagious disease in a population living in a habitat X (Diekmann,
1978; Thieme, 1977), in the theory of nonlinear parabolic
boundary value problems (Pachpatte, 1978), and in many
physical and biological models.
The literature on numerical methods for solving Eq. (1)
mainly consists of projection methods, collocation methods,
the trapezoidal Nystroăm method, Adomain decomposition
method, He’s homotopy perturbation method and the twodimensional block-pulse functions (Adomian, 1990, 1994;
Adomian and Rach, 1992; Biazar et al., 2011; Brunner, 1990;
Cardone et al., 2006; Cherruault et al., 1992; Guoqiang,
1995; Hacia, 1996; Kauthen, 1989; Maleknejad and Fadaei
Yami, 2006; Maleknejad and Hadizadeh, 1999; Maleknejad
and Mahdiani, 2011; Wazwaz, 2006; Yee, 1993).


66

F. Mirzaee, E. Hadadiyan
Assume now that:

Gðx; y; s; t; uðs; tịị ẳ kx; y; s; tịẵus; tịp ;

2ị

where p is a positive integer. In the present paper, we apply a
modification of block-pulse functions (Maleknejad and
Rahimi, 2011), to solve the mixed nonlinear Volterra–

Fredholm type integral Eq. (1) with Eq. (2).
2. M2D-BFs and their properties
Definition 1. An (m + 1)2-set of M2D-BFs consists of
(m + 1)2 functions which are defined over district
D = [0,1) Ã [0,1) as follows:
&
/i1 ;i2 x; yị ẳ

1 ðx; yÞ 2 Di1 ;i2 ;
0 otherwise:

Now suppose that X be a (m + 1)2-vector. Hence by using Eq.
(9) we obtain:
e m;e x; yị;
Um;e x; yịUTm;e x; yịX ẳ XU

10ị

e ¼ diagðXÞ is a (m + 1) · (m + 1) diagonal matrix.
where X
2

2

2.2. M2D-BFs expansions
A function f(x,y) defined over district L2(D) may be expanded
by the M2D-BFs as:
m X
m
X

fðx; yÞ fm;e x; yị ẳ
fi1 ;i2 /i1 ;i2 x; yị
i1 ¼0 i2 ¼0

¼

¼ UTm;e ðx; yÞFm;e ;

FTm;e Um;e ðx; yÞ

ð11Þ

2

i1 ; i2 ẳ 01ịm;

3ị

where Fm,e is an (m + 1) Ã 1 vector given by
Fm;e ẳ ẵf0;0 ; . . . ; f0;m ; . . . ; fm;0 ; . . . ; fm;m ŠT ;

where Di1 ;i2 ¼ fðx; yÞjx 2 Ii1 ;e ; y 2 Ii2 ;e g, and
8
a ẳ 0;
>
< ẵ0; h eị
Ia;e ẳ ẵah e; a ỵ 1ịh eị a ẳ 11ịm;
>
:
ẵ1 e; 1ị

a ẳ m:

4ị
1
.
m

where m is an arbitrary positive integer, and h ¼
Since, each M2D-BF takes only one value in its subregion,
the M2D-BFs can be expressed by the two modied onedimensional block-pulse functions (M1D-BFs):
/i1 ;i2 x; yị ẳ /i1 ðxÞ/i2 ðyÞ;
ð5Þ
where /i1 ðxÞ and /i2 ðyÞ are the M1D-BFs related to variables x
and y, respectively. The M2D-BFs are disjointed with each
other:
&
/i1 ;i2 x; yị i1 ẳ j1 ; i2 ¼ j2 ;
ð6Þ
/i1 ;i2 ðx; yÞ/j1 ;j2 ðx; yÞ ¼
0
otherwise:
and are orthogonal with each other:
Z 1Z 1
/i1 ;i2 ðx; yÞ/j1 ;j2 x; yịdydx
0
&0
MIi1 ;e ịMIi2 ;e ị i1 ẳ j1 ; i2 ẳ j2 ;
ẳ
0
otherwise:


12ị

and Um,e(x,y) is dened in Eq. (8), and fi1 ;i2 , are obtained as:
Z Z
1
fðx; yÞdydx:
ð13Þ
fi1 ;i2 ẳ
MIi1 ;e ịMIi2 ;e ị Ii1 ;e Ii2 ;e
Similarly a function of four variables, k(x,y,s,t), on district
L2(D · D) may be approximated with respect to M2D-BFs
such as:
kðx; y; s; tÞ ’ UTm;e ðx; yÞKm;e Um;e ðs; tÞ;

ð14Þ

where Um,e(x,y) and Um,e(s,t) are M2D-BFs vector of dimension (m + 1)2, and Km,e is the (m + 1)2 · (m + 1)2 M2D-BFs
coefficients matrix.
3. Convergence analysis
In this section, we show that the given method in the previous
À1Á
sections, is convergent and its order of convergence is O km
.
For our purposes we will need the following theorems.
Theorem 1. Let

7ị

fm;e x; yị ẳ


m X
m
X
fi1 ;i2 /i1 ;i2 x; yị;
i1 ẳ0 i2 ẳ0

where (x,y) 2 D, i1,i2,j1,j2 = 0(1)m and MðIi1 ;e Þ and MðIi2 ;e Þ are
length of intervals Ii1 ;e and Ii2 ;e , respectively.

and
fi1 ;i2 ẳ

2.1. Vector forms

1
MIi1 ;e ịMIi2 ;e ị

Z

1
0

Z
0

1

fx; yị/i1 ;i2 x; yịdxdy;


i1 ; i2 ẳ 01ịmị:
2

Consider the rst (m + 1) terms of M2D-BFs and write them
concisely as (m + 1)2-vector:
Um;e x; yị ẳ ẵ/0;0 x; yị; . . . ; /0;m ðx; yÞ; . . . ; /m;0 ðx; yÞ; . . . ;
T

/m;m ðx; yފ ; ðx; yị 2 D:

8ị

B
B
B
T
Um;e x; yịUm;e x; yị ẳ B
B
@

0

15ị

0

achieves its minimum value and also we have

Whence Eqs. (6) and (8) implies that:
0


Then the following equation
Z 1Z 1
ðfðx; yÞ À fm;e ðx; yÞÞ2 dxdy;

/0;0 ðx; yÞ

0

...

0

0
..
.

/0;1 ðx; yÞ
..
.

...
..
.

0
..
.

0


0

. . . /m;m x;yị

1

Z

C
C
C
C
C
A

:

0

mỵ1ị2 mỵ1ị2

9ị

1

Z

1


f2 x; yịdxdy ẳ
0

1 X
1
X
f2i1 ;i2 k/i1 ;i2 x; yịk2 :

16ị

i1 ẳ0 i2 ẳ0

Proof. It is an immediate consequence of theorem which was
proved by Jiang and Schaufelberger (1992). h


Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations
Theorem 2. Assume f(x,y) is continuous and is differentiable
over district [Àh,1 + h] · [Àh,1 + h], and fm;ei ðx; yÞ; ei ¼ ihk ;
for i = 0(1)(k À 1), are correspondingly M2D-BFs(e0) = 2DBFs, M2D-BFs(e1), . . ., M2D-BFs(ekÀ1) expansions of f(x, y)
based on (m + 1)2 M2D-BFs over district D and
1
fm;k x; yị ẳ
k

iẳ0

lim fm x; yị ẳ lim fm;e0 x; yị ẳ fx; yị:
m!ỵ1


Proof. See (Maleknejad et al., 2010).

@f
@f
Proof.
consider
x; yị and @y
x; yị in the district
@x
i1 iỵ1We i1

iỵ1

which
are
approximately
equal to con;
;
m
m
m
m
stants n1 and n2, respectively, where m is so large. Also, we
use
n1xÁ+ n2y + b instead of f(x,y) in the district
i1 page
z=

iỵ1
i1 iỵ1

i i
; ỵ e1
m m
im ;i m ÂÁ m ; m . Now in the district
; ỵ e1 we have:
m m




k1
k1 & 
X
1X
1
i lh
i lh
n1

ỵ n2

fm;el ẳ
k lẳ0 4
m k
m k
lẳ0





i lh
i ỵ 1 lh
ỵ b ỵ n1

ỵ n2

m k
m
k




i ỵ 1 lh
i lh

ỵ n2

ỵ b ỵ n1
m
k
m k




'
i ỵ 1 lh
i ỵ 1 lh


ỵ n2

ỵb
ỵ b ỵ n1
m
k
m
k
 i iỵ1
ỵ
n
ỵ
n
ịhk
1ị
1
2
; 17ị
ẳ n1 ỵ n2 ị m m ỵ b
2k
2

1
fm;k x; yị ẳ
k

Theorems 2 and 3 conclude
that error estimation for M2DÀ1Á
.
BFs is keðx; yịk ẳ O km

If we assume E1 and E2 are errors between f(x,y) and its
2D-BFs and M2D-BFs expansions, respectively, from
Theorem 2 we have Ep2ffiffi 6 k1 E1 , and from (Maleknejad et al.,
2010) we have E1 6 m2M, where M is bounded of iDf(x,y)i
and m shows number of 2D-BFs.
So, we have
p
2M
E2 ẳ kex; yịk 6
;
21ị
km
where k is times of modications of the M2D-BFs series.
Assume now that f(x,y) is approximated by
fm;ei x; yị ẳ

m X
m
X
fi1 ;i2 /i1 ;i2 x; yị;
i1 ẳ0 i2 ¼0

whereas, fi1 ;i2 are the approximation of fi1 ;i2 and
fm;ei x; yị ẳ

m X
m
X
fi1 ;i2 /i1 ;i2 x; yị;
i1 ¼0 i2 ¼0


ð18Þ

kfi1 ;i2 /i1 ;i2 ðx; yÞ À fðx; yịk ẳ kfi1 ;i2 /i1 ;i2 x; yị fx; yị
fi1 ;i2 /i1 ;i2 x; yị
ỵ fi1 ;i2 /i1 ;i2 x; yịk

In other words:
max

x;y2ẵmi ;mi ỵe1 ị



6 kfi1 ;i2 /i1 ;i2 x; yị fx; yịk
ỵ kfi ;i / ðx; yÞ

jfðx; yÞ À fm;k ðx; yÞj

1 2

i
i
max jn1 x þ n2 y þ b À fm;k ðx; yÞj K jn1 ỵ n2
i
i
m
m
x;y2ẵm;mỵe1 ị


ỵ b fm;k x; yịj

n1 ỵ n2 ịh
;
2k
so, we have:
ẳ

max jfx; yị
max kfx; yị fm;ei ðx; yÞk1 P
ei
ei
ðx; yÞ 2 D
ðx; yÞ 2 D0
i
i
À fm;ei x; yịj jn1 ỵ n2 ỵ b
m
m
&


1
i
i
i
i
n1 ỵ n2 þ b þ n1 þ n2
þh
À

4
m
m
m
m




i
i
i
þ h þ n2 þ b ỵ n1
ỵh
ỵ bỵn1
m
m
m


'
 n1 ỵ n2 ịh
i
ỵ n2
ỵ h þ b  ¼
;
m
2
Â
Á

Â
Á
where D0 ¼ mi ; mi þ h mi ; mi ỵ h .
By using Eqs. (19) and (20) the proof is completed.

h

then for ðx; yÞ 2 Di1 ;i2 we have

ẳ mi ỵ h and Eq. (17) can be reformulated as:

i
n1 ỵ n2 ịh
:
fm;k x; yị ẳ n1 ỵ n2 ị ỵ b ỵ
m
2k

ex; yị ẳ fx; yị fm x; yị:

m!ỵ1

1
kfx; yị fm;k x; yịk1 K maxkfx; yị fm;ei x; yịk1 :
k ei

iỵ1
m

Theorem 3. Let the representation error between f(x,y) and its

two-dimensional block-pulse functions, fm x; yị ẳ fm;e0 x; yị
(M2D-BFse0 ị ẳ 2D À BFsÞ, over the district D, as follows:

Then keðx; yịk ẳ Om1 ị and

k1
X
fm;ei x; yị;

then for sufcient large m we have:

but

67

i1 ;i2

À fi1 ;i2 /i1 ;i2 ðx; yÞk:

ð22Þ

We have
ð19Þ

kfi1 ;i2 /i1 ;i2 ðx; yÞ À fi1 ;i2 /i1 ;i2 x; yịk
ẳ

!12
2


fi1 ;i2 /i1 ;i2 x; yị fi1 ;i2 /i1 ;i2 x; yịị dydx

Z

Z
Ii1 ;ei

Ii2 ;ei

ẳ jfi1 ;i2 fi1 ;i2 j

!12

Z

Z

dydx
Ii1 ;ei

Ii2 ;ei

ẳ MIi1 ;ei ịMIi2 ;ei Þjfi1 ;i2 À fi1 ;i2 j
6 MðIi ;e ÞMðIi ;e Þkfm À fk :
1 i

ð20Þ

2 i


1

ð23Þ

Consequently by using Eqs. (21)–(23), the following error
bound is obtained:
p
2M
ỵ MIi1 ;ei ịMIi2 ;ei ịkfm À fk1 : ð24Þ
kfi1 ;i2 /i1 ;i2 À fðx; yÞk 6
km
Moreover Eq. (24) implies that:

h

lim fm;ei x; yị ẳ fx; yị:

m!ỵ1

25ị


68

F. Mirzaee, E. Hadadiyan
ẵux; yịpỵ1 ẳ ux; yịẵux; yịp

4. Method of solution

ẳ UTm;e Um;e x; yịUTm;e x; yịUm;e;p

In this section, we solve mixed nonlinear Volterra–Fredholm
type integral equations of the first kind of the form Eq. (1) with
Eq. (2) by using M2D-BFs.
We now approximate functions u(x,y),f(x,y),[u(x,y)]p and
k(x,y,s,t) with respect to M2D-BFs by manipulation as
Section 2:
8
uðx; yÞ ’ UTm;e ðx; yÞUm;e ;
>
>
>
>
< fðx; yÞ ’ UT ðx; yÞFm;e ;
m;e
> ðuðx; yÞÞp ’ UTm;e ðx; yÞUm;e;p ;
>
>
>
:
kðx; y; s; tÞ ’ UTm;e ðx; yịKm;e Um;e s; tị;

e m;e;p Um;e x; yị:
ẳ UTm;e U

29ị

Now by using Eq. (28) we obtain
h
iT
pỵ1

pỵ1
pỵ1
e m;e;p ẳ upỵ1
;
UTm;e U
0;0 ; . . . ; u0;m ; . . . ; um;0 ; . . . ; um;m

ð30Þ

therefore Eq. (28) holds for (p + 1), and the lemma is
established. h
To approximate the integral part in Eq. (1) with Eq. (2),
from Eq. (26) we get
Z xZ 1
kx; y; s; tịẵus; tịp dtds

ð26Þ

where Um,e(x,y) is defined in Eq. (8), the vectors Um,e, Fm,e,
Um,e,p, and matrix Km,e are M2D-BFs coefficients of u(x,y),f(x,y), [u(x,y)]p and k(x,y,s,t) respectively.

0

’

0

Lemma 1. Let (m + 1)2-vectors Um,e and Um,e,p be M2D-BFs
coefficients of u(x,y) and [u(x,y)]p, respectively. If
Um;e ¼ ½u0;0 ; . . . ; u0;m ; . . . ; um;0 ; . . . ; um;m ŠT ;


0

Z

ẳ

x

Z

1

UTm;e x; yịKm;e Um;e s; tịUTm;e s; tịUm;e;p dtds

0

UTm;e x; yịKm;e

Z

x

0

Z



1


Um;e s; tịUTm;e s; tịdtds

Um;e;p :

0

31ị

27ị

then we have:
h
iT
Um;e;p ẳ up0;0 ; . . . ; up0;m ; . . . ; upm;0 ; . . . ; upm;m ;

Now by using Eqs. (5) and (9), denoting Rj for the (j + 1)th
row of the conventional integration operational matrix Pm,e
((Pm,e)(m+1)·(m+1) is operational matrix of 1D-BFs defined
overR [0,1), see Maleknejad and Mahdiani, 2011) and consider1
ing 0 /i tịdt ẳ MIi;e ị follows:

28ị

where p P 1, is a positive integer.
Proof. (By induction) When p = 1, Eq. (28) follows at once
from [u(x,y)]p = u(x,y). Suppose that Eq. (28) holds for p,

Z


x
0

Z

1

Um;e ðs; tÞUTm;e s; tịdtds

0

0Rx R1

B
B
B
B
B
ẳB
B
B
B
@

0

0

/0 sị/0 tịdtds
..

.
0
..
.
0

0
B
B
B
B
B
B
B
B
B
ẳB
B
B
B
B
B
B
B
@

h eịR0 Um;e xị
0
..
.

0
..
.
0
..
.
0

...
..
.
...
..
.

0
..
.
Rx R1
/
sị/
0
m tịdtds
0
0
...

...
..
.

...
..
.

...

0

...

0

...

hR0 Um;e xị . . .
..
..
.
.
0
...
..
..
.
.
0
...
..
..
.

.
0

...

0

0

0

C
C
C
C
C
C
C
C
C
A

/m sị/m tịdtds

mỵ1ị2 mỵ1ị2

...

0


1

0
...
..
..
.
.
0
...
..
..
.
.
h eÞRm Um;e ðxÞ . . .
..
...
.

0
..
.
0
..
.
0
..
.

C

C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A

0

...

0
...
..
..
.
.
eR0 Um;e ðxÞ . . .
..

..
.
.
0
...
..
..
.
.
0

Rx R1

1

0
..
.
0
..
.

...

we shall deduce it for (p + 1). Since [u(x,y)]p+1 = u(x,y)
[u(x,y)]p, from Eqs. (26) and (10) it follows that

0

...


eRm Um;e xị

32ị

:

mỵ1ị2 mỵ1ị2

Also by using Eq. (5), Eq. (8) can be reformulated as:


Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations

0

/0 ðxÞ . . .
B .
..
B ..
.
B
B
B 0
...
B
B .
..
B
Um;e x; yị ẳ B ..

.
B
B 0
.
.
.
B
B .
..
B .
@ .
.
0
...

0
..
.
/0 ðxÞ
..
.
0
..
.
0

...
..
.
...

..
.
...
..
.
...

0
..
.
0
..
.
/m ðxÞ
..
.
0

...
..
.
...
..
.
...
..
.
...

69


1
0
.. C
. C
C
C
0 C
C
.. C
C
: ẵ/0 yị; . . . ; /m yị; . . . ; /0 ðyÞ; . . . ; /m yịTmỵ1ị2 1 :
. C
C
0 C
C
.. C
C
. A
/m xị mỵ1ị2 mỵ1ị2

33ị

So, we have

UTm;e x; yịKm;e ẳ ẵ/0 yị; . . . ; /m ðyÞ; . . . ; /0 ðyÞ; . . . ; /m yịmỵ1ị2 1
0
k1;1 /0 xị
...
k1;mỵ1ị /0 xị

...
k1;mmỵ1ị /0 xị
B
..
.
.
.
.
B
..
..
..
..
.
B
B
B kmỵ1ị;1 / xị . . . kmỵ1ị;mỵ1ị / xị . . . kmỵ1ị;mmỵ1ị / xị
0
0
0
B
B
B
.
.
.
.
.
..
..

..
..
..
B
B
B
B kmmỵ1ị;1 /m xị . . . kmmỵ1ị;mỵ1ị /m xị . . . kmmỵ1ị;mmỵ1ị /m xị
B
B
..
..
..
..
..
B
.
.
.
.
.
@
kmỵ1ị2 ;1 /m xị . . . kmỵ1ị2 ;mỵ1ị /m xị . . . kmỵ1ị2 ;mmỵ1ị /m xị

k1;mỵ1ị2 /0 xị

...
..

1


C
C
C
C
kmỵ1ị;mỵ1ị2 /0 xị C
C
C
C
..
:
C
.
C
C
kmmỵ1ị;mỵ1ị2 /m xị C
C
C
..
C
.
A
kmỵ1ị2 ;mỵ1ị2 /m xị mỵ1ị2 mỵ1ị2
..
.

.

...
..


.
...
..

.
...

34ị

Also, we have:

8 heị
>
< 2 /0 xị ỵ h eị/1 xị ỵ ỵ h eị/m xị;
Ri Uxị ẳ h2 /i xị ỵ h/iỵ1 xị ỵ ỵ h/m xị;
>
:e
/ xị;
2 m
and
&
/i xị; i ẳ j
:
/i xị/j xị ẳ
0;
otherwise

iẳ0
i ẳ 11ịm 1ị ;
iẳm


35ị

By using Eqs. (32), (34) and (35), Eq. (31) can be reformulated
as:
ẵ/0 yị; . . . ; /m yị; . . . ; /0 ðyÞ; . . . ; /m yịmỵ1ị2 1
0
1
A00
0
0
...
0
B
C
0
...
0 C
B A10 A11
B
C
36ị
B
0 C
:B A20 A21 A22 . . .
:Um;e;p ;
C
B .
C
..

B .
... C
...
...
.
@ .
A
Am0 Am1 Am2 . . . Amm mỵ1ị2 mỵ1ị2
where
Ai;j ẳ

8 MI ị
j;e
>
< 2 MðIr;e Þklz /i ðxÞ;
>
:

MðIj;e ÞMðIr;e Þklz /i ðxÞ;

and 0 is a zero matrix. Also
0
1
A00 0
0 ... 0
B
C
B A10 A11 0 . . . 0 C
B
C

B A20 A21 A22 . . . 0 C
B
C
B .
.. C
.. . .
..
B .
C
.
.
.
.
.
@
A
Am0 Am1 Am2 . . . Amm mỵ1ị2 mỵ1ị2
0
/0 xị . . .
0
...
0
...
B .
..
..
..
B ..
...
...

.
.
.
B
B
B 0
. . . /0 ðxÞ . . .
0
...
B
B .
.
.
.
.
.
B
..
..
..
ẳ B ..
..
..
B
B 0
...
0
. . . /m xị . . .
B
B .

..
..
..
..
..
B .
@ .
.
.
.
.
.
0

...

0

0
..
.
. . . /m xị

0

...

0
..
.

0
..
.

1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A

:Q;

mỵ1ị2 mỵ1ị2

38ị
iẳj
;
otherwise


where
l ẳ m ỵ 1ịi ỵ 1ị1ịm ỵ 1ịi ỵ 1ịị;
z ẳ m ỵ 1ịj ỵ 1ị1ịm ỵ 1ịj ỵ 1ịị;
!
z
;
r ẳ z m ỵ 1ị
m ỵ 1ị

37ị

where
0

Q00
BQ
B 10
B
B
Q ẳ B Q20
B .
B .
@ .

0
Q11
Q21
..
.


0
0
Q22
..
.

...
...
...
..
.

0
0
0
..
.

Qm0

Qm1

Qm2

...

Qmm

1
C

C
C
C
C
C
C
A
mỵ1ị2 mỵ1ị2

;

39ị


70

F. Mirzaee, E. Hadadiyan

Table 1

Numerical results of Example 1 with M2D-BFs.

Table 2

Nodes x; yị

Error for m ẳ 8

x; yị ẳ 2l


k=1

k=2

k=3

l=1
l=2
l=3
l=4

0.03748936
0.05090571
0.02574872
0.04112879

0.02837304
0.03943282
0.01813532
0.02553757

0.02522153
0.03190341
0.01391462
0.02222506

MIj;e ị
MIr;e ịklz ;
2


MIj;e ịMIr;e ịklz ;

iẳj
:
otherwise

Nodes x; yị

Present method

x; yị ẳ 2l

m = 8 and k = 2

Method of Maleknejad
and Mahdiani (2011)
m = 16

l=1
l=2
l=3
l=4

0.02837304
0.03943282
0.01813532
0.02553757

0.0288649
0.0398778

0.0310669
0.0277814

So, we have :
Z xZ 1
kðx; y; s; yịẵus; tịp dtds UTm;e x; yịQUm;e;p :

40ị

0

0

Error(x,y)

0.15
0.1
0.05
0
1

0.8

0.6

y

0.4

0.2


0

0.2

0

0.4

0.6

0.8

1

x

(a) m = 8 and k = 1

0.12
0.09
Error(x,y)

Qi;j ¼

0.06
0.03
0
1


0.8

0.6
y

0.4

0.2

0

0

0.2

0.6

0.4

0.8

1

x

(b) m = 8 and k = 2

0.06

Error(x,y)


(

Error results for Example 1.

0.04

0.02

0
1
0.8
0.6

y

0.4
0.2
0

0

0.2

0.4

0.6

x


(c) m = 8 and k = 3
Figure 1

Absolute value of error, Example 1 with m = 8 and k = 1,2,3.

0.8

1

ð41Þ


Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations
um;e ðx; yị ẳ UTm;e Um;e x; yị:

Numerical results of Example 2 with M2D-BFs.

Table 3

Nodes x; yị
x; yị ẳ 2

k=1

k=2

k=3

l=1
l=2

l=3
l=4

0.04289052
0.06470576
0.03803418
0.03592263

0.03148406
0.04585369
0.02437943
0.02836157

0.02743874
0.04071913
0.02249845
0.02045372

ux; yị um;k x; yị ẳ

ẳ

UTm;e x; yịQUm;e;p

) Fm;e ẳ QUm;e;p :

k1
1X
um;ei x; yị;
k iẳ0


44ị

where ei ẳ ihk ; i ẳ 01ịk À 1Þ is the estimation of the solution
of mixed nonlinear Volterra–Fredholm type integral equation
of the first kind.

Substituting Eqs. (26) and (41) into Eq. (1) with Eq. (2) gives:
UTm;e ðx; yÞFm;e

ð43Þ

Then

Error for m = 8

Àl

71

5. Numerical examples

ð42Þ
In this section to demonstrate the effectiveness of our approach several examples are presented. All results are computed by using a program written in the Matlab. The

After solving the above nonlinear system by using Newton–
Raphson method, we can find Um,e and then

Error(x,y)


0.15
0.1
0.05
0
1

0.8

0.6
y

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

x


(a) m=8 and k = 1

0.08

Error(x,y)

0.06
0.04
0.02
0
1

0.8

0.6

0.4

y

0.2

0

0

0.2

0.4


0.6

0.8

1

x

(b) m = 8 and k = 2

0.05

Error(x,y)

0.04
0.03
0.02
0.01
0
1

0.8

0.6
y

0.4

0.2


0

0

0.2

0.4

0.6
x

(c) m = 8 and k = 3
Figure 2

Absolute value of error, Example 2 with m = 8 and k = 1,2,3.

0.8

1


72

F. Mirzaee, E. Hadadiyan

Table 4

Error results for Example 2.


Nodes ðx; yị

Present method

x; yị ẳ 2l

m = 8 and k = 2

Method of Maleknejad
and Mahdiani (2011)
m = 16

l=1
l=2
l=3
l=4

0.03148406
0.04585369
0.02437943
0.02836157

0.04289006
0.04589757
0.04034072
0.04312157

answer is the coefficients of M2D-BFs expansion of the solution of mixed nonlinear Volterra–Fredholm type integral equation. Also, we have shown that
À 1 our
Á approach is convergent and

its order of convergence is O km
. This method can be easily extended and applied to mixed nonlinear Volterra–Fredholm
type integral equations of the second kind and nonlinear system
of the mixed Volterra–Fredholm type integral equations.
References

numerical experiments are carried our for the selected grid
point which are proposed as (2Àl; l = 1,2,3,4) and m terms
and k times of modifications of the M2D-BFs series. The following problems have been tested.
Example 1. Consider the following mixed linear Volterra–
Fredholm type integral equation (Maleknejad and Mahdiani,
2011):
Z

x

0

Z

1

cosy tịesx us; tịdtds ẳ fx; yị;

x; yị 2 ẵ0; 1ị X;

0

45ị
where

1
fx; yị ẳ xex 2 cosyị ỵ sin2 yị ỵ sinyịị:
4

46ị

The exact solution is u(x,y) = eÀxcos(y). Table 1 and Fig. 1
illustrate the numerical results for this example.
The error results for proposed method besides the error for
method of Maleknejad and Mahdiani (2011) are tabulated in
Table 2.
Example 2. Consider the following mixed nonlinear Volterra–
Fredholm type integral equation (Maleknejad and Mahdiani,
2011):
Z
0

x

Z

1

t ỵ yịe2sx u2 s; tịdtds ẳ fx; yị;

x; yị 2 ẵ0; 1ị X;

0

47ị

where
1
1
1
3
fx; yị ẳ xyex ỵ xex xyex2 xex2 :
2
4
2
4

48ị

The exact solution is u(x,y) = eÀxÀy. Table 3 and Fig. 2 illustrate the numerical results for this example.
The error results for proposed method besides the error for
method of Maleknejad and Mahdiani (2011) are tabulated in
Table 4.
6. Conclusion
In this paper a computational method for approximate solution
of mixed nonlinear Volterra–Fredholm type integral equations
of the first kind, based on the expansion of the solution as series
of M2D-BFs was presented. This method converts a mixed
nonlinear Volterra–Fredholm type integral equation whose

Adomian, G., 1990. A review of the decomposition method and some
recent results for nonlinear equation. Mathematical and Computer
Modelling 13 (7), 17–43.
Adomian, G., 1994. Solving Frontire Problems of Physics-the
Decomposition Method. Kluwer, Dordrecht.
Adomian, G., Rach, R., 1992. Noise terms in decomposition series

solution. Computer and Mathematics with Applications 24 (11),
61–64.
Biazar, J., Ghanbari, B., Gholami Porshokouhi, M., Gholami Porshokouhi, M., 2011. He’s homotopy perturbation method: a
strongly promising method for solving non-linear systems of the
mixed Volterra–Fredholm integral equations. Computer and
Mathematics with Applications 61, 1016–1023.
Brunner, H., 1990. On the numerical solution of nonlinear Volterra–
Fredholm integral equation by collocation methods. SIAM Journal
on Numerical Analysis 27 (4), 978–1000.
Cardone, A., Messina, E., Russo, E., 2006. A fast iterative method for
discretized Volterra–Fredholm integral equations. Journal of
Computational and Applied Mathematics 189, 568–579.
Cherruault, Y., Saccomandi, G., Some, B., 1992. New results for
convergence of Adomian’s method applied to integral equations.
Mathematical and Computer Modelling 16 (2), 85.
Diekmann, O., 1978. Thresholds and travelling waves for the
geographical spread of infection. Journal Mathematical Biology
6, 109–130.
Guoqiang, H., 1995. Asymptotic error expansion for the Nystrom
method for a nonlinear Volterra–Fredholm integral equations.
Computer and Mathematics with Applications 59, 49–59.
Hacia, L., 1996. On approximate solution for integral equations of
mixed type. Zeitschrift fuăr Angewandte Mathematik 76, 415
416.
Jiang, Z.H., Schaufelberger, W., 1992. Block Pulse functions and their
applications in control systems. Spriger-Verlag, Berlin.
Kauthen, P.G., 1989. Continuous time collocation methods for
Volterra–Fredholm integral equations. Numerische Mathematik
56, 409–424.
Maleknejad, K., Fadaei Yami, M.R., 2006. A computational method

for system of Volterra–Fredholm integral equations. Applied
Mathematics and Computation 183, 589–595.
Maleknejad, K., Hadizadeh, M., 1999. A new computational method
for Volterra–Fredholm integral equations. Computer and Mathematics with Applications 37, 1–8.
Maleknejad, K., Mahdiani, K., 2011. Solving nonlinear mixed
Volterra–Fredholm integral equations with two dimensional
block-pulse functions using direct method. Communications in
Nonlinear Science and Numerical Simulation 16, 3512–3519.
Maleknejad, K., Rahimi, B., 2011. Modification of block pulse
functions and their application to solve numerically Volterra
integral equation of the first kind. Communications in Nonlinear
Science and Numerical Simulation 16, 2469–2477.
Maleknejad, K., Sohrabi, S., Baranji, B., 2010. Application of 2DBPFs to nonlinear integral equations. Communications in Nonlinear Science and Numerical Simulation 15, 527–535.
Pachpatte, B.G., 1978. On mixed Volterra–Fredholm type integral
equations. Indian Journal of Pure and Applied Mathematics 17,
488–496.


Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations
Thieme, H.R., 1977. A model for spatial spread of an epidemic.
Journal of Mathematical Biology 4, 337–351.
Wazwaz, A.M., 2006. A reliable treatment for mix Volterra–Fredholm
integral equations. Applied Mathematics and Computation 189,
405–414.

73

Yee, E., 1993. Application of the decomposition method to the solve of
the reaction–convection–diffusion equation. Applied Mathematics
and Computation 56, 1–14.