Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 216923, 17 pages
doi:10.1155/2012/216923
Research Article
A Numerical Scheme to Solve Fuzzy Linear Volterra
Integral Equations System
A. Jafarian, S. Measoomy Nia, and S. Tavan
Department of Mathematics, Islamic Azad University, Urmia Branch, Urmia 5715944867, Iran
Correspondence should be addressed to A. Jafarian, jafarian5594@yahoo.com
Received 20 February 2012; Revised 14 June 2012; Accepted 16 June 2012
Academic Editor: Said Abbasbandy
Copyright q 2012 A. Jafarian et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The current research attempts to offer a new method for solving fuzzy linear Volterra integral
equations system. This method converts the given fuzzy system into a linear system in crisp case
by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor
coefficients of the solution functions. The proposed method is illustrated by an example and also
results are compared with the exact solution by using computer simulations.
1. Introduction
Many mathematical formulations of physical phenomena contain integral equations. These
equations appear in physics, biological models, and engineering. Since these equations are
usually difficult to solve explicitly, so it is required to obtain approximate solutions. In recent
years, numerous methods have been proposed for solving integral equations. For example,
Tricomi, in his book 1, introduced the classical method of successive approximations for
nonlinear integral equations. Variational iteration method 2 and Adomian decomposition
method 3 were effective and convenient for solving integral equations. Also the Homotopy
analysis method HAM was proposed by Liao 4 and then has been applied in 5.
Moreover, some different valid methods for solving this kind of equations have been
developed. First time, Taylor’s expansion approach was presented for solution of integral
equations by Kanwal and Liu in 6 and then has been extended in 7–9. In addition,
Babolian et al. 10 by using the orthogonal triangular basis functions solved some integral
equation systems. Jafari et al. 11 applied Legendre’s wavelets method to find numerical
solution system of linear integral equations. Also Sorkun and Yalçinbaş 12 approximated a
solution of linear Volterra integral equations system with the help of Taylor’s series.
2
Journal of Applied Mathematics
In this paper, we want to propose a new numerical approach to approximate the
solution of a fuzzy linear Volterra integral equations system. This method converts the given
fuzzy system that supposedly has a unique fuzzy solution, into crisp linear system. For this
scope, first, the Taylor expansions of unknown functions are substituted in parametric form of
the given fuzzy system. Then we differentiate both sides of the resulting integral equations of
the system N times and also approximate the Taylor expansion by a suitable truncation limit.
This work yields a linear system in crisp case, so the solution of the linear system yields the
unknown Taylor coefficients of the solution functions. An interesting feature of this method
is that we can get an approximate of the Taylor expansion in arbitrary point to any desired
degree of accuracy. Here is an outline of the paper. In Section 2, the basic notations and
definitions of the integral equation and the Taylor polynomial method are briefly presented.
Section 3 describes how to find an approximate solution of the given fuzzy Volterra integral
equations system with using proposed approach. Finally in Section 4, we apply the proposed
method by an example to show the simplicity and efficiency of the method.
2. Preliminaries
In this section, the most basic used notations in fuzzy calculus and integral equations are
briefly introduced. We started by defining the fuzzy number.
Definition 2.1. A fuzzy number is a fuzzy set u : R1 → I 0, 1 such that
i u is upper semicontinuous,
ii ux 0 outside some interval a, d,
iii there are real numbers b, c : a ≤ b ≤ c ≤ d, for which:
1 ux is monotonically increasing on a, b,
2 ux is monotonically decreasing on c, d,
3 ux 1, b ≤ x ≤ c.
The set of all fuzzy numbers as given by Definition 2.1 is denoted by E1 13, 14. An
alternative definition which yields the same E1 is given by Kaleva 15 and Ma et al. 16.
Definition 2.2. A fuzzy number v is a pair v, v of functions vr and vr : 0 ≤ r ≤ 1, which
satisfy the following requirements:
i vr is a bounded monotonically increasing left continuous function on 0, 1 and
right continuous at 0,
ii vr is a bounded monotonically decreasing left continuous function on 0, 1 and
right continuous at 0,
iii vr ≤ vr : 0 ≤ r ≤ 1.
Journal of Applied Mathematics
3
A popular fuzzy number is the triangular fuzzy number v vm , vl , vu where vm denotes
the modal value and the real values vl ≥ 0 and vu ≥ 0 represent the left and right fuzziness,
respectively. The membership function of a triangular fuzzy number is defined as follows:
⎧
x − vm
⎪
⎪
⎪
⎪ vl 1,
⎪
⎪
⎨
μv x vm − x 1,
⎪
⎪ vu
⎪
⎪
⎪
⎪
⎩0,
vm − vl ≤ x ≤ vm ,
vm ≤ x ≤ vm vu ,
2.1
otherwise.
Its parametric form is
vr vm vl r − 1,
vr vm vu 1 − r,
0 ≤ r ≤ 1.
2.2
Triangular fuzzy numbers are fuzzy numbers in LR representation where the reference
functions L and R are linear.
2.1. Operation on Fuzzy Numbers
We briefly mentioned fuzzy number operations that have had been defined by the extension
principle 17, 18.
μAB z max μA x ∧ μB y | z x y ,
μfNet z max μA x ∧ μB y | z xy ,
2.3
where A and B are fuzzy numbers, μ∗ · denotes the membership function of each fuzzy
number, ∧ is the minimum operator, and f is a continuous function.
The above operations on fuzzy numbers are numerically performed on level sets i.e.,
α-cuts. For 0 < α ≤ 1, a α-level set of a fuzzy number A is defined as
Aα x | μA x ≥ α, x ∈ R ,
and A0 Aα by
α0,1 A
α
2.4
. Since level sets of fuzzy numbers become closed intervals, we denote
Aα Aαl , Aαu ,
2.5
4
Journal of Applied Mathematics
where Aαl and Aαu are the lower and the upper limits of the α-level set Aα , respectively.
From interval arithmetic 19, the above operations on fuzzy numbers are written for the
α-level sets as follows:
Aα Bα Aαl , Aαu Bαl , Bαu Aαl Bαl , Aαu Bαu ,
fNetα f Netαl , Netαu f Netαl , f Netαu ,
kAα k Aαl , Aαu kAαl , kAαu , if k ≥ 0,
kAα k Aαl , Aαu kAαu , kAαl , if k < 0.
2.6
For arbitrary u u, u and v v, v, we define addition u v and multiplication by k as
13, 14:
u vr ur vr,
u vr ur vr,
kur k · ur,
kvr k · ur,
if k ≥ 0,
kur k · ur,
kvr k · ur,
if k < 0.
2.7
Definition 2.3. For arbitrary fuzzy numbers u, vE1 the quantity
Du, v sup max ur − vr, |ur − vr|
2.8
0≤r≤1
is the distance between u and v. It is shown that E1 , D is a complete metric space 20.
Definition 2.4. Let f : a, b → E1 . For each partition P {t0 , t1 , . . . , tn } of a, b and for
arbitrary ξi ti−1 , ti 1 ≤ i ≤ n, suppose
RP n
fξi ti − ti−1 ,
i1
2.9
Δ : max{|ti − ti−1 |, i 1, . . . , n}.
The definite integral of ft over a, b is
b
a
ftdt lim RP ,
Δ→0
2.10
Journal of Applied Mathematics
5
provided that this limit exists in the metric D. If the fuzzy function ft is continuous in the
metric D, its definite integral exists 13. Also,
b
ft, r dt
a
b
ft, rdt,
a
b
ft, r dt
a
2.11
b
ft, rdt.
a
More details about properties of the fuzzy integral are given in 13, 15.
2.2. System of Integral Equations
The basic definition of integral equation is given in 21.
Definition 2.5. The Fredholm integral equation of the second kind is
Ft ft λkut,
2.12
where
kut b
ks, tFsds,
a ≤ t ≤ b.
2.13
a
In 2.12, ks, t is an arbitrary kernel function over the square a ≤ s, t ≤ b and ft is a
function of t : a ≤ t ≤ b. If the kernel function satisfies ks, t 0, s > t, we obtain the Volterra
integral equation
Ft ft λ
t
ks, tFsds.
2.14
a
In addition, if ft be a crisp function, then the solution of the above equation is crisp as
well. Also if ft be a fuzzy function, we have Fredholm’s fuzzy integral equation of the
second kind which may only process fuzzy solutions. Sufficient conditions for the existence
and uniqueness of the solution of the second kind equation, where ft is a fuzzy function,
are given in 22, 23.
Definition 2.6. The second kind fuzzy linear Volterra integral equations system is in the form
F1 t f1 t t
m λ1j k1j s, tFj sds
j1
..
.
a
6
Journal of Applied Mathematics
Fi t fi t m t
λij
kij s, tFj sds
a
j1
..
.
Fm t fm t m t
λmj
kmj s, tFj sds ,
a
j1
2.15
where a ≤ s ≤ t ≤ b and λij /
0 for i, j 1, . . . , m are real constants. Moreover, in system
2.15, the fuzzy function fi t and kernel ki,j s, t are given and assumed to be sufficiently
differentiable with respect to all their arguments on the interval a ≤ t, s ≤ b. Also we assume
that the kernel function ki,j s, tL2 a, b×a, b and Ft F1 t, . . . , Fm tT is the solution
to be determined.
Now let f t, r, f i t, r and F i t, r, F i t, r 0 ≤ r ≤ 1; a ≤ t ≤ b be parametric
i
form of fi t and Fi t, respectively. To simplify, we assume that λij > 0 for i, j 1, . . . , m.
In order to design a numerical scheme for solving 2.15, we write the parametric form of the
given fuzzy integral equations system as follows:
F 1 t, r f 1 t, r t
m λ1j U1,j s, rds
a
j1
t
m F 1 t, r f t, r λ1j U1,j s, rds
1
a
j1
..
.
2.16
F m t, r f m t, r F m t, r f t, r m
m t
λmj
a
j1
m
j1
Um,j s, rds
b
λmj
a
Um,j s, rds ,
where
Ui,j s, r Ui,j s, r ⎧
⎪
⎨ki,j s, tF j s, r, ki,j s, t ≥ 0,
⎪
⎩k s, tF s, r, k s, t < 0
i,j
i,j
j
⎧
⎪
⎨ki,j s, tF j s, r, ki,j s, t ≥ 0,
⎪
⎩k s, tF s, r, k s, t < 0.
i,j
j
i,j
2.17
Journal of Applied Mathematics
7
2.3. Taylor’s Series
Let us first recall the basic principles of the Taylor polynomial method for solving Fredholm’s
fuzzy integral equations system 2.15. Because these results are the key for our problems,
therefore, we explain them. Without loss of generality, we assume that
λi,j · ki,j s, t ≥ 0,
a ≤ s ≤ ci,j
2.18
λi,j · ki,j s, t < 0,
ci,j ≤ s ≤ t,
With above supposition, the system 2.16 is transformed to the following form:
m
F 1 t, r f 1 t, r λ1j
c
1
k1,j s, tF j s, rds c
j1
t
c1,j
a
j1
m
F 1 t, r f t, r λ1j
1,j
1,j
a
k1,j s, tF j s, rds k1,j s, tF j s, rds
t
k1,j s, tF j s, rds
c1,j
..
.
F m t, r f m t, r F m t, r f t, r m
m
c
λmj
km,j s, tF j s, rds c
λmj
a
j1
t
cm,j
a
j1
m
m,j
m,j
km,j s, tF j s, rds t
km,j s, tF j s, rds
km,j s, tF j s, rds .
cm,j
2.19
Now we want to obtain the solution of the above system in the form of
F j,N t, r N
i0
1 ∂i F j t, r ·
i!
∂ti
· t − z
⎛
,
a ≤ t, z ≤ b, 0 ≤ r ≤ 1,
tz
∂ F j t, r 1
⎝ ·
F j,N t, r i!
∂ti
i0
N
i
2.20
⎞
i
· t − zi ⎠,
tz
a ≤ t, z ≤ b, 0 ≤ r ≤ 1,
8
Journal of Applied Mathematics
for j 1, . . . , m which are the Taylor expansions of degree N at t z for the unknown
functions F j t, r and F j t, r, respectively. For this scope, we calculate pth for p 0, . . . , N
derivative of each equation in the system 2.19 with respect to t and get
m
∂p F 1 t, r ∂p f 1 t, r λ1j
∂tp
∂tp
j1
c
∂p k1,j s, t
· F j s, rds
∂tp
1,j
a
p−1 p−l−1
p − q − 1 ∂q k1,j s, t p−q−1−l
∂tq
l0 q0
⎛
·⎝
p
m
∂p F 1 t, r ∂ f 1 t, r λ1j
p
p
∂t
∂t
j1
c
∂l F j t, r
∂tl
⎞
⎠
t
c1,j
p−q−1−l
st
⎞
∂p k1,j s, t
· F j s, rds⎠
∂tp
∂p k1,j s, t
· F j s, rds
∂tp
1,j
a
p−1 p−l−1
p−q−1 p−q−1−l
l0 q0
·
∂l F j t, r
∂tl
t
c1,j
∂q k1,j s, t ∂tq
p−q−1−l
st
∂p k1,j s, t
· F j s, rds
∂tp
..
.
m
∂p F m t, r ∂p f m t, r λmj
p
p
∂t
∂t
j1
c
m,j
a
∂p km,j s, t
· F j s, rds
∂tp
p−1 p−l−1
p − q − 1 ∂q km,j s, t p−q−1−l
∂tq
l0 q0
⎛
·⎝
p
m
∂p F m t, r ∂ f m t, r λmj
∂tp
∂tp
j1
c
m,j
a
∂l F j t, r
∂tl
⎞
⎠
t
cm,j
·
st
⎞
∂p km,j s, t
· F j s, rds⎠
∂tp
∂p km,j s, t
· F j s, rds
∂tp
p−1 p−l−1
p − q − 1 ∂q km,j s, t p−q−1−l
∂tq
l0 q0
p−q−1−l
∂l F j t, r
∂tl
t
cm,j
p−q−1−l
st
∂p km,j s, t
· F j s, rds .
∂tp
2.21
Journal of Applied Mathematics
9
For brevity, we define symbols as below:
∂p F jN t, r p
F jN z, r :
∂tp
,
p
F jN z, r :
tz
∂p F jN t, r ∂tp
j 1, . . . , m.
,
2.22
tz
p
p
The aim of this study is to determine the coefficients F j z, r and F j z, r, for p 0, . . . , N; j 1, . . . , m in system 2.21. For this intent, we expanded F j s, r and F j s, r in
Taylor’s series at arbitrary point z : a ≤ z ≤ b and substituted its Nth truncation in 2.21.
Now we can write
p
F 1N z, r
∂p f 1 t, r ∂tp
tz
⎞
⎛
p−1
m
N
N
q
1,j
1,j
q
1,j
l
⎝ v
· F z, r wp,q · F jN z, r w p,q · F z, r⎠
j1
p
F 1N z, r
p,l
l0
∂p f t, r 1
∂tp
jN
q0
q0
jN
tz
⎞
⎛
p−1
m
N
N
l
q
1,j
1,j
q
1,j
⎝ v
· F jN z, r wp,q · F z, r w p,q · F jN z, r⎠
j1
l0
p,l
q0
jN
q0
..
.
p
F mN z, r
∂p f m t, r ∂tp
tz
⎞
⎛
p−1
m
N
N
q
m,j
m,j
q
m,j
l
⎝ v
· F z, r wp,q · F jN z, r w p,q · F z, r⎠
j1
p
F mN z, r
l0
∂p f t, r m
∂tp
p,l
jN
q0
q0
jN
tz
⎞
⎛
p−1
m
N
N
l
q
m,j
m,j
q
m,j
⎝ v
· F jN z, r wp,q · F z, r w p,q · F jN z, r⎠,
j1
l0
p,l
q0
jN
q0
2.23
10
Journal of Applied Mathematics
where
λi,j
q!
i,j
wp,q
i,j
w p,q ci,j
λi,j
q!
a
b
ci,j
p−l−1
i,j
vp,l q0
∂p ki,j s, t ∂tp
· s − zq ds,
tz
∂p ki,j s, t ∂tp
p−q−1
p−q−1−l
· s − zq ds,
tz
∂q k1,j s, t ∂tq
p, q 0, . . . , N,
p−q−1−l st
2.24
i, j 1, . . . , m.
,
sz
Consequently, the matrix form of expression 2.23 can be written as follows:
W V Y E,
2.25
where
N
N
Y F 1N a, r, . . . , F 1N a, r, F 1N a, r, . . . , F 1N a, r, . . . ,
N
N
F mN a, r, . . . , F mN a, r, F mN a, r, . . . , F mN a, r
∂N f t, r 1
E ⎣−f a, r, . . . , −
1
∂tN
⎡
ta
∂N f t, r m
−f a, r, . . . , −
m
∂tN
⎡
1,m ⎤
W 1,1 · · · W
⎢ ..
.. ⎥,
..
W ⎣ .
.
. ⎦
m,1
m,m
··· W
W
,
∂N f 1 t, r , −f 1 a, r, . . . , −
∂tN
ta
,...,
ta
∂N f m t, r , −f m a, r, . . . , −
∂tN
⎤
2.26
⎦,
ta
⎡
⎤
V 1,1 · · · V 1,m
⎢
.. ⎥.
..
V ⎣ ...
.
. ⎦
m,1
m,m
··· V
V
Parochial matrices W i,j for i, j 1, ..., m are defined with the following elements:
W i,j
⎤
⎡ i,j
i,j
W1,1 W1,2
⎥
⎢
⎣
⎦,
i,j
i,j
W2,1 W2,2
V i,j
⎡ i,j i,j ⎤
V1,2
V
⎥
⎢ 1,1
⎣
⎦,
i,j
i,j
V2,1 V2,2
i, j 1, . . . , m,
2.27
Journal of Applied Mathematics
11
where
i,j
i,j
i,j
i,j
W1,1 W2,2
W1,2 W2,1
⎡ i,j
⎤
i,j
i,j
i,j
w0,0 − 1 w0,1
···
w0,N−1
w0,N
⎢
⎥
⎢
⎥
⎢
i,j
i,j
i,j
i,j ⎥
⎢ w1,0
⎥
w
−
1
·
·
·
w
w
1,1
1,N−1
1,N ⎥
⎢
⎢
⎥
⎢
⎥
..
..
..
..
..
⎢
⎥,
.
.
.
.
.
⎢
⎥
⎢
⎥
i,j
i,j
i,j
⎢ i,j
⎥
⎢ wN−1,0 wN−1,1 · · · wN−1,N−1 − 1 wN−1,N ⎥
⎢
⎥
⎣
⎦
i,j
i,j
i,j
i,j
wN,0
wN,1
···
wN,N−1
wN,N − 1
⎤
⎡ i,j
i,j
i,j
i,j
w 0,0
w 0,1 · · · w 0,N−1
w 0,N
⎥
⎢
⎥
⎢
⎢ i,j
i,j
i,j
i,j ⎥
⎥
⎢ w 1,0
w
·
·
·
w
w
1,1
1,N−1
1,N ⎥
⎢
⎥
⎢
⎥
⎢ .
..
..
..
..
⎢ ..
⎥,
.
.
.
.
⎥
⎢
⎥
⎢
⎥
⎢ i,j
i,j
i,j
i,j
⎢w N−1,0 w N−1,1 · · · w N−1,N−1 w N−1,N ⎥
⎥
⎢
⎦
⎣
i,j
i,j
i,j
i,j
w N,0 w N,1 · · · w N,N−1 w N,N
⎡
i,j
i,j
V1,2 V2,1
i,j
i,j
V1,1 V2,2
0
0
⎢ i,j
⎢ v
0
⎢ 1,0
⎢
⎢
⎢ .
..
.
⎢
.
⎢ .
⎢
⎢ i,j
i,j
⎢v N−1,0 v N−1,1
⎢
⎣
i,j
i,j
v N,0 v N,1
⎤
⎡
0 0 ··· 0 0
⎥
⎢
⎢0 0 · · · 0 0⎥
⎥
⎢
⎥
⎢
⎥
⎢
⎢ .. .. . . .. .. ⎥
⎢. .
⎥
.
.
.
⎥
⎢
⎥
⎢
⎢0 0 · · · 0 0⎥
⎥
⎢
⎦
⎣
0 0 ··· 0 0
···
0
···
0
..
.
..
.
···
0
i,j
· · · v N,N−1
⎤
0
⎥
0⎥
⎥
⎥
⎥
.. ⎥
.⎥
⎥,
⎥
⎥
0⎥
⎥
⎦
0
2.28
.
N1×N1
3. Convergence Analysis
In this section, we proved that the above numerical method converges to the exact solution
of fuzzy system 2.15.
Theorem 3.1. Let the kernel be bounded and belong to L2 and F j,N t and F j,N t for j 1, . . . , m
be Taylor polynomials of degree N that their coefficients are produced by solving the linear system
12
Journal of Applied Mathematics
2.25. Then these polynomials converge to the exact solution of the fuzzy Volterra integral equations
system 2.15, when N → ∞.
Proof. Consider the system 2.15. Since, the series 2.20 converge to F j t, r and
F j t, r for j 1, . . . , m, respectively, then we conclude that
m
F iN t, r f i t, r λij
j1
m
F iN t, r f t, r λij
i
j1
c
i,j
ki,j s, tF jN s, rds ci,j
a
c
t
i,j
a
ki,j s, tF jN s, rds ki,j s, tF jN s, rds ,
t
3.1
ki,j s, tF jN s, rds ,
ci,j
for i 1, . . . , m and it holds that
F j t, r lim F jN t, r,
F j t lim F jN t, r.
N →∞
N →∞
3.2
We defined the error function eN t, r by subtracting 2.19 and 3.1 as follows:
eN t, r m
ei,N t, r,
3.3
i1
ei,N t, r ei,N t, r ei,N t, r,
where
ci,j
m
! ! eiN t, r F i t, r − F iN t, r λij
Ki,j s, t F j s, r − F jN s, r ds
j1
m
t
λij
j1
ci,j
a
!
Ki,j s, t F j s, r − F jN s, r ds ,
ci,j
m
! eiN t, r F i t, r − F iN t, r λij
Ki,j s, t F j s, r − F jN s, r ds
j1
m
j1
t
λij
ci,j
a
!
Ki,j s, t F j s, r − F jN s, r ds .
3.4
Journal of Applied Mathematics
13
We must prove when N → ∞, the error function eN t becomes zero. Hence, we proceed
as follows:
eN ≤
m
m "
m
" " "
"eiN e " ≤
eiN eiN "eiN "
iN
i1
≤
i1
i1
!" "
"
"
"
" F i t, r − F iN t, r " " F i t, r − F iN t, r "
m "
i1
m
m " "
"!
t" " "
"
"
"
"
λi,j "ki,j " "F j s, r − F jN s, r" "F s, r − F s, r" ds .
j
jN
i1 j1
a
3.5
Since ki,j is bounded, therefore, F j s, r−F jN s, r
→ 0 and F j s, r−F jN s, r
→ 0
imply that eN → 0 and proof is completed.
4. An Example
In this section, we present an example of fuzzy linear Volterra integral equations system and
results will be compared with the exact solution.
Example 4.1. Consider the system of fuzzy linear Volterra integral equations with
2
2
rt2 t2 − 1
t2 r − 2 9 r 3 − 2 t − 1
f 1 t, r 4
10
4
4
4 3
r r 2 t − 1 2t 4t2 6t 3
− tr − 2 ,
10
4
rt2 3 r 3 − 2 t − 1 2t3 4t2 6t 3
−
f t, r rt −
1
4
10
4
2
2
2 2
t t − 1 r − 2
3r r 2 t − 1
−
,
−
10
4
!
! r4t 1t2 − 14
r − 2t 1
3
2
3
− t 3r − 6 t r − 2 f 2 t, r 20
20
2
2 4
rt r 2 2t 1t − 1
,
3
! rt2 r 4 2 4t 1t2 − 14 r − 2 rt 14
5
f t, r t r 2r −
−
−
2
3
20
20
!
− t2 r 3 − 2 2t 1t − 12 ,
4
4.1
14
Journal of Applied Mathematics
kernel functions
!
k1,1 s, t t2 1 − s2 ,
!
k1,2 s, t 1 − t2 1 − s3 ,
!
k2,1 s, t 1 t4 1 − s3 ,
4.2
k2,2 s, t 2t2 1 − s,
0 ≤ s ≤ t ≤ 2,
and a 0, b 2, N 1, λi,j 1 for i, j 1, 2. The exact solution in this case is given by
F 1 t, r t2 − r,
!
F 2 t, r t 6 − 3r 3 ,
F 1 t, r tr,
!
F 2 t, r t r 5 2r .
4.3
In this example, we assume that z 0. Using 2.23–2.25, the coefficients matrix W V is
calculated as follows:
W V # 1,1
W V 1,1
W 1,2 V 1,2
W 2,1 V 2,1
W 2,2 V 2,2
$
,
4.4
where
⎤
3
3
3
3
⎢ 4 10 − 4 − 10 ⎥
⎥
⎢
⎥
⎢
3
5
3 ⎥
⎢ 3
⎥
⎢−
⎢ 2 −5
2
5 ⎥
⎥,
⎢
⎢
3
3
3 ⎥
⎥
⎢ 3
⎥
⎢− −
⎢ 4 10 4 10 ⎥
⎥
⎢
⎣ 5
3
3
3⎦
−
−
2
5
2
5
⎡
W 1,1 V 1,1
⎡
−1
⎢0
⎢
⎣0
0
⎡
W 2,1 V 2,1
0
−1
0
0
0
0
−1
0
⎤
0
0⎥
⎥,
0⎦
−1
⎤
1
1
1
1
⎢ 4 20 − 4 − 20 ⎥
⎢
⎥
⎢
⎥
1⎥
1
⎢
⎢ 1
⎥
0
−
⎢
5
5⎥
⎢
⎥,
⎢
1 1
1 ⎥
⎢ 1
⎥
⎢− −
⎥
⎢ 4 20 4 20 ⎥
⎢
⎥
⎣
1 ⎦
1
1
0 −
5
5
W 1,2 V 1,2
W 2,2 V 2,2
⎤
⎡
−1 0 0 0
⎥
⎢
⎢ 0 −1 0 0 ⎥
⎥
⎢
⎥
⎢
⎥.
⎢
⎢
⎥
0
0
−1
0
⎥
⎢
⎥
⎢
⎦
⎣
0 0 0 −1
4.5
Journal of Applied Mathematics
15
With using above matrices, we can rewrite the linear system 2.25 as follows:
⎡
⎤
3r 5 9r 3 3r 9
−
⎡
⎤ ⎢
10
5
5 ⎥
⎢ 10
⎥
⎢
⎥
F 1,1 0, r
5
3
⎥
3r
9r
11r
18
⎢ ⎥ ⎢
⎢
⎢F 0, r⎥ ⎢−
−
−
⎥
⎢ 1,1
⎥ ⎢ 5
5
5
5⎥
⎥
⎢
⎥ ⎢
⎥
⎢
⎥ ⎢
5
3
⎥
3r
9r
3r
9
⎢F 0, r⎥ ⎢
⎥
⎢ 1,1
⎥ ⎢ −
⎢
⎥ ⎢ 10 − 10 − 5 5 ⎥
⎥
⎢ ⎥ ⎢
⎥
⎢
⎥ ⎢ 5
3
⎢F 1,1 0, r⎥ ⎢ 3r
9r
11r 28 ⎥
⎥
⎢
⎥ ⎢
⎢
⎥ ⎢ 5 5 5 − 5 ⎥
⎥.
W V ⎢
⎥
⎥
⎢F 2,1 0, r⎥ ⎢
⎢
⎥
1
r
⎢
⎥ ⎢
⎥
⎢
⎥ ⎢
−
⎥
⎢ ⎥ ⎢
10
10
⎥
⎢F 2,1 0, r⎥ ⎢
⎥
⎢
⎥ ⎢
⎥
2
8r
⎢
⎥ ⎢
⎥
−
−r 5 −
⎢
⎥ ⎢
⎥
⎢F 2,1 0, r⎥ ⎢
5
5
⎥
⎢
⎥ ⎢
⎥
⎢
⎥ ⎢
r
1
⎥
⎣ ⎦ ⎢
⎥
−
F 2,1 0, r
⎢
⎥
10 10
⎢
⎥
⎣
⎦
28
2r
−
3r 3 −
5
5
4.6
The vector solution of the above linear system is
⎡
⎤
F 1,1 0, r
⎢ ⎥
⎢F 0, r⎥
⎢ 1,1
⎥
⎢
⎥
⎢
⎥ ⎡
⎤
⎢F 0, r⎥
0
⎢ 1,1
⎥
⎢
⎥ ⎢ r ⎥
⎢ ⎥ ⎢
⎥
⎢
⎥ ⎢
⎥
⎢F 1,1 0, r⎥ ⎢ 0 ⎥
⎢
⎥ ⎢ 2−r ⎥
⎢
⎥ ⎢
⎥
⎢
⎥⎢
⎥.
⎢F 2,1 0, r⎥ ⎢ 0 ⎥
⎢
⎥ ⎢ 5
⎥
⎢
⎥ ⎢r 2r ⎥
⎢ ⎥ ⎢
⎥
⎢F 2,1 0, r⎥ ⎣ 0 ⎦
⎢
⎥
⎢
⎥
6 − 3r 3
⎢
⎥
⎢F 2,1 0, r⎥
⎢
⎥
⎢
⎥
⎣ ⎦
F 2,1 0, r
4.7
After propagating this vector solution in 2.20, we have
F 1,1 t, r t2 − r F 1 t, r,
F 1,1 t, r tr F 1 t, r,
!
!
F 2,1 t, r t 6 − 3r 3 F 2 t, r,
F 2,1 t, r t r 5 2r F 2 t, r.
4.8
As shown in Figures 1 and 2, the present method gives the analytical solution for this kind of
fuzzy equations system, if the exact solution be polynomial.
16
Journal of Applied Mathematics
1
0.8
r
0.6
0.4
2
0.2
0
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
t
F 1 (t, r) = F 1,1 (t, r)
F 1 (t, r) = F 1,1 (t, r)
Figure 1: F 1,1 t, r and F 1,1 t, r for Example 4.1.
1
0.8
r
0.6
0.4
2
0.2
0
12
1
10
8
6
0.5
4
2
0
1.5
t
0
F 2 (t, r) = F 2,1 (t, r)
F 2 (t, r) = F 2,1 (t, r)
Figure 2: F 2,1 t, r and F 2,1 t, r for Example 4.1.
5. Conclusions
Fuzzy integral equations systems, which have a very important place in physics and
engineering, are usually difficult to solve analytically. Therefore, it is required to obtain
approximate solutions. In this study, mechanization of solving fuzzy linear Volterra integral
equations system of the second kind by using the Taylor expansion method have proposed.
This Taylor method transforms the given problem to a linear algebraic system in crisp case.
The solution of the resulting system is used to compute unknown Taylor coefficients of the
solution functions. Consider that to get the best approximating solutions of the given fuzzy
equations, the truncation limit N must be chosen large enough. An interesting feature of
this method is finding the analytical solution for given fuzzy system, if the exact solution be
polynomials of degree N or less than N. The results of the example indicate the ability and
reliability of the present method.
Journal of Applied Mathematics
17
References
1 F. G. Tricomi, Integral Equations, Dover Publications, New York, NY, USA, 1982.
2 L. Xu, “Variational iteration method for solving integral equations,” Computers & Mathematics with
Applications, vol. 54, no. 7-8, pp. 1071–1078, 2007.
3 E. Babolian, H. Sadeghi Goghary, and S. Abbasbandy, “Numerical solution of linear Fredholm fuzzy
integral equations of the second kind by Adomian method,” Applied Mathematics and Computation,
vol. 161, no. 3, pp. 733–744, 2005.
4 S. J. Liao, Beyond Perturbation: Introduction to the HomoTopy Analysis Method, Chapman Hall/CRC
Press, Boca Raton, Fla, USA, 2003.
5 S. Abbasbandy, “Numerical solutions of the integral equations: homotopy perturbation method and
Adomian’s decomposition method,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 493–500,
2006.
6 R. P. Kanwal and K. C. Liu, “A Taylor expansion approach for solving integral equations,”
International Journal of Mathematical Education in Science and Technology, vol. 20, pp. 411–414, 1989.
7 K. Maleknejad and N. Aghazadeh, “Numerical solution of Volterra integral equations of the second
kind with convolution kernel by using Taylor-series expansion method,” Applied Mathematics and
Computation, vol. 161, no. 3, pp. 915–922, 2005.
8 S. Nas, S. Yalçınbaş, and M. Sezer, “A Taylor polynomial approach for solving high-order linear
Fredholm integro-differential equations,” International Journal of Mathematical Education in Science and
Technology, vol. 31, no. 2, pp. 213–225, 2000.
9 S. Nas, S. Yalçınbaş, and M. Sezer, “A Taylor polynomial approach for solving high-order linear
Fredholm integro-differential equations,” International Journal of Mathematical Education in Science and
Technology, vol. 31, no. 2, pp. 213–225, 2000.
10 E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, “A direct method for numerically solving
integral equations system using orthogonal triangular functions,” International Journal of Industrial
Mathematics, vol. 2, pp. 135–145, 2009.
11 H. Jafari, H. Hosseinzadeh, and S. Mohamadzadeh, “Numerical solution of system of linear integral
equations by using Legendre wavelets,” International Journal of Open Problems in Computer Science and
Mathematics, vol. 5, pp. 63–71, 2010.
12 H. H. Sorkun and S. Yalçinbaş, “Approximate solutions of linear Volterra integral equation systems
with variable coefficients,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3451–3464, 2010.
13 R. Goetschel and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp.
31–43, 1986.
14 H. T. Nguyen, “A note on the extension principle for fuzzy sets,” Journal of Mathematical Analysis and
Applications, vol. 64, no. 2, pp. 369–380, 1978.
15 O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987.
16 M. Ma, M. Friedman, and A. Kandel, “A new fuzzy arithmetic,” Fuzzy Sets and Systems, vol. 108, no.
1, pp. 83–90, 1999.
17 L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning. III,”
Information Sciences, vol. 9, no. 1, pp. 43–80, 1975.
18 L. A. Zadeh, “Toward a generalized theory of uncertainty GTU an outline,” Information Sciences, vol.
172, no. 1-2, pp. 1–40, 2005.
19 G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, NY,
USA, 1983.
20 M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and
Applications, vol. 114, no. 2, pp. 409–422, 1986.
21 H. Hochstadt, Integral Equations, John Wiley & Sons, New York, NY, USA, 1973.
22 W. Congxin and M. Ming, “On the integrals, series and integral equations of fuzzy set-valued
functions,” Journal of Harbin Institute of Technology, vol. 21, pp. 9–11, 1990.
23 M. Friedman, M. Ma, and A. Kandel, “Numerical solutions of fuzzy differential and integral
equations,” Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35–48, 1999.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014