Download Numerical Solution of fuzzy Volterra Linear Integral equations based

Numerical Solution of fuzzy Volterra Linear Integral equations
based on Sinc method
a
Gh. Kazemi gelian a∗
Department of Mathematics, Shirvan Branch, Islamic Azad University, Shirvan, Iran
Abstract
Some numerical methods for fuzzy integral
equations illustrated by different authors. In [19]
differential transformation method is used to solve
fuzzy integral equations. Also numerical solution of
fuzzy integral equations by polynomials investigated
in [8]. Recently Adomian decomposition method [2]
are used to solve linear fuzzy integral equation of the
second kind.
In [16] fuzzy Fredholm-Volterra integral equation
is solved, where expansion method is applied to
approximate the solution of an unknown function in
the fuzzy Fredholm-Volterra integral equation and
convert this equation to a system of fuzzy linear
equations.
In this paper, numerical solution of fuzzy Volterra
linear integral equations is considered by applying
Sinc method based on double exponential transformation with dual Fuzzy linear system.
For
this purpose, we convert the given fuzzy integral
equation to a fuzzy linear system of equations, the
Sinc collocation method with double exponential
transformation are used. Numerical examples are
provided to verify the validity of proposed method.
Keywords: Fuzzy number, Fuzzy Volterra integral
equations, Sinc method, dual system, double exponential
transformation.
In [17] integral term in the fuzzy Fredholm integral equation is approximated by one of the
Gaussian methods. Fuzzy fredholm integral equation
Fuzzy approximation is a special phenomenon which is transformed to a dual fuzzy linear system that it
explains how complicated functions can be described can be approximated by the method that proposed
by a set of fuzzy rules. In many cases, the way in [19, 20]. In the special case, Chebyshev-Gauss
of designing the concrete set of these rules is left quadrature is applied to approximate the mentioned
unexplained, or rather exhaustive technique based integral.
on neural networks or genetic algorithms is used.
On the other hand, there is a series of theoretical Sinc method is a powerful numerical tool for
papers unified by the key words fuzzy approximation finding fast and accurate solution in various areas of
where the problem of representation of a continu- problems. In [18, 24] a full overview of sinc function
ous function by a so called fuzzy system is considered. and appropriate conditions and theorems have been
1
Introduction
discussed. Recently okayama [23] used sinc method
to singular integral equations and take useful and
interesting results.
Many numerical techniques have been used for
fuzzy integral equation and investigated by many scientists. Dubios, Prade, Goetschel, Voxman, Kaleva,
Matloka [3, 4, 5, 11, 15] and others have important
role in expanding fuzzy integration. Fuzzy integral
equations are used in spread spectral applications
such as fuzzy control, fuzzy financial and economic
systems. In [10] the existence of a unique solution is
established.
∗ Corresponding
Double exponential transformation, abbreviated
as DE was first propose by Takahasi and Mori [28]
in 1974 for one dimensional numerical integration
and it has come to be widely used in applications. It
is known that the double exponential transformation
gives an optimal result for numerical evaluation of
definite integral of an analytic function [25, 26].
However, Sugihara has recently in [27] found that
Author, e-mail: [email protected]
1
Definition 2. [9, 20] An arbitrary fuzzy number in
parametric form is represented by an ordered pair
functions (u(r), ū(r)), 0 ≤ r ≤ 1, which satisfy the
following requirements:
1. u(r) is a bounded left -continuous non - decreasSinc method with double exponential transforma- ing function over [0, 1],
2. ū(r) is a bounded left -continuous non -in creastion is used for various area of problems of integral
equation for more information see [6, 7, 13] and ing function over [0, 1],
3. u(r) ≤ ū(r),
0 ≤ r ≤ 1.
references therin.
the errors in the Sinc numerical methods are
O(exp(−cN/logN )) with some c > 0, which is also
meaningful practically.
A crisp number α is simply represented by
The purpose of this paper is to extend the application of the sinc method for solving the fuzzy u(r) = u(r) = α, 0 ≤ r ≤ 1.
Volterra integral equation of the second kind. For
For arbitrary u = (u(r), ū(r)), v = (v(r), v̄(r))
this perpose we state and verify the definitions
and theorems about sinc approximation for integral and k ∈ R, we define addition and multiplication by
equation then apply sinc collocation method to fuzzy k as
volterra integral equation and convert obtaind dual
(a) u = v if and only if u(r) = v(r) and u(r) = v(r),
fuzzy system to a linear system of equation and
solve it. Also error analysis and measuring some (b) u + v = (u(r) + v(r), u(r) + v(r)),
{
applicable computational remarks such as fuzzy
(ku, ku),
k ≥ 0,
distance and run time of programs is obtained.
(c) ku =
(ku, ku),
k < 0.
In this paper, we propose fuzzy Sinc method based
on double exponential transformation for solving the
fuzzy Volterra integral equations. The rest of this
paper is organized as follow. In section 2, we
present some required preliminaries, theorems for
Sinc method and fuzzy definitions. The proposed
method is drawn in section 3. In section 4, error
analysis is obtained. Numerical examples are given
in section 5.
2
Definition 3.

a11 x1 +



 a21 x1 +
..

.



an1 x1 +
[9, 20] The n × n linear system
a12 x2
a22 x2
+ ···
+ ···
+
+
a1n xn
a2n xn
..
.
=
=
..
.
an2 x2
+ ···
+
ann xn
= yn ,
y1 ,
y2 ,
..
.
where,
the
given
matrix
of
coefficients
A = (aij ), 1 ≤ i, j ≤ n is a real n × n matrix,
the given yi ∈ E, 1 ≤ i ≤ n, with the unknowns
xj ∈ E, 1 ≤ j ≤ n is called a fuzzy linear system
(FLS).
Preliminaries
In this section, we introduce the notations needed in Definition 4. [10, 20] A fuzzy number vector
(x1 , x2 , ..., xn )t given by
the rest of the paper.
xj = (xj (r), xj (r)); 1 ≤ j ≤ n, 0 ≤ r ≤ 1,
2.1
Fuzzy definitions
is called a solution of the fuzzy linear system AX = Y
Definition 1. [9, 20] A fuzzy number is a map u : if
 ∑n
∑n

R −→ I = [0, 1] which satisfies
j=1 aij xj =
j=1 aij xj = y i ,

(i) u is upper semi continuous,

∑n
 ∑n
(ii) u(x) = 0 outside some interval [c, d] ⊂ R,
j=1 aij xj =
j=1 aij xj = y i .
(iii) There exist real numbers a, b such that c ≤
Let for a particular i, aij > 0, for all j, we simply
a ≤ b ≤ d where,
obtain
1. u(x) is monotonic increasing on [c, a]
n
n
∑
∑
2. u(x) is monotonic decreasing on [b, d]
1 ≤ i ≤ n.
aij xj = y i ,
aij xj = y i ,
3. u(x) = 1, a ≤ x ≤ b.
j=1
j=1
The set of all such fuzzy numbers is represented
by E 1 .
In general, an arbitrary equation for either y i or y i
may include a linear combination of xj ’s and xj ’s.
2
Consequently, in order to solve the above system, Definition 6. [10, 20] For arbitrary fuzzy numbers
one must solve a crisp (2n) × (2n) linear system u = (u, u) and v = (v, v), the function
where, the right-hand-side column is the function vector (y 1 , y 2 , ..., y n , −y 1 , −y 2 , ..., −y n )t . Hence, we get dH (u, v) = sup {max(|u(r) − v(r)|, |u(r) − v(r)|)}
0≤r≤1
the following (2n) × (2n) linear system,


















s11 x1 + · · · + s1n xn +
s1,n+1 (−x1 ) + · · · + s1,2n (−xn )
= y1 ,
..
.
sn,1 x1 + · · · + snn xn +
sn,n+1 (−x1 ) + · · · + sn,2n (−xn )
= yn ,
sn+1,1 x1 + · · · + sn+1,n xn +
sn+1,n+1 (−x1 ) + · · · + sn+1,2n (−xn ) = −y 1 ,
..
.















s
x + · · · + s2n,n xn +

 2n,1 1
s2n,n+1 (−x1 ) + · · · + s2n,2n (−xn )
is the distance between u and v. Also is showed by
D(u, v).
2.2
Sinc approximation
Let f be a function defined on R and h > 0 is step
size then the Whittaker cardinal defined by the series
= −y n ,
SX = Y,
where S = (sij ) ≥ 0, 1 ≤ i, j ≤ 2n and
x1
 ..
 .

 xn
X=
 −x1

 .
 ..
−xn




,








Y =




y1
..
.
yn
−y 1
..
.
sin[π(x − jh)/h]
,
π(x − jh)/h
j = 0, ±1, ±2, ...
(2)
where S(j, h)(x) is known as j −th Sinc function evaluated at x. We recall the following definitions that
will become instrumental in establishing our useful
formulas:
Definition 7. [18, 24] Let Dd denotes the infinite
strip domain of width 2d, d > 0 given by
and any sij which is not determined by above condition is zero. Using matrix notation we get,

(1)
whenever this series convergence, and
S(j, h)(x) =
aij ≥ 0 =⇒ sij = aij , si+n,j+n = aij ,
aij < 0 =⇒ si,j+n = −aij , si+n,j = −aij ,

f (jh)S(j, h)(x),
j=−∞
where, sij are determined as follows:

∞
∑
C(f, h)(x) =
Dd = {z ∈ C, z = x + iy, |y| < d}.





.




(3)
Definition 8. [18] Let BFp (Dd ) be the set of fuzzy
valued functions analytic in Dd which satisfies following conditions :
∫
−y n
d
−d
The structure of S implies that sij ≥ 0, 1 ≤ i, j ≤ 2n
and
(
)
B C
S=
,
C B
dH (f (t+iy), 0̃)dy = O(|ta |),
t −→ ±∞, 0 ≤ a < 1.
(4)
and
NFp (f, Dd )
where, B contains the positive entries of A, and C
contains the absolute values of the negative entries
of A, i.e., A = B − C.
∫ ∞
1
=limy→d {(
dpH (f (t + iy), 0̃)dt) p
−∞
∫ ∞
1
dpH (f (t + iy), 0̃)dt) p } < ∞.
+(
−∞
(5)
Definition 5.
[9, 20] The fuzzy system
AX̃ = B X̃ + Y where the coefficient matrix
A = (ai,j ) and B = (bi,j ) are crisp n × n matrices
and X̃, Ỹ are fuzzy number vectors. Above system
is called the daual linear system.
Definition 9. [18] Let B(D) denote the class of functions analytic in D which satisfy for some constant a
with 0 ≤ a < 1 ,
∫
|F (w)dw| = O(|xa |), x → ±∞,
(6)
ψ(x+L)
3
where L = {iy, |y| < d} and γ is a simple closed
contour in D
∫
N (F, D) = limγ→∂D |F (w)dw| < ∞.
(7)
fuzzy function this equation may only possess fuzzy
solution, so we have
∫ x
˜
ũ(x) = f (x) + λ
K(x, t)ũ(t)dt,
a ≤ x, t ≤ b
γ
a
(13)
Definition 10. [21, 22] A Fuzzy function f is said to Application of (8) to the integral term of (13) gives:
be decay double exponentially, if there exist constants
∫ x
N
α and C, such that:
∑
K(x, t)ũ(t)dt = h
K(x, ϕ(jh))ϕ′ (jh)ηh,j (x)ũj
a
dH (f (t), 0̃) ≤ C exp(−α exp |t|), t ∈ (−∞, ∞)
j=−N
+ IEN ,
equivalently, a function g is said to be decay double
exponentially with respect to conformal map ϕ, if
there exist positive constants α and C such that:
(14)
where ũj denotes an approximation value of ũ(xj )
and
IEN denotes error of Sinc integral approximadH (g(ϕ(t))ϕ (t), 0̃) ≤ C exp(−α exp |t|), t ∈ (−∞, ∞).
tion. If we substitute (14) in Eq. (13) we have:
Finally then we have the following formulas for inN
∑
definite integral based on DE transformation.
ũ(x) = f˜(x) + λh
K(x, ϕ(jh))ϕ′ (jh)ηh,j (x)ũj ,
j=−N
∫ x
N
∑
′
(15)
dH (
f (t)dt, h
f (ϕ(jh))ϕ (jh)ηh,j (x))
which
has
2N
+
1
unknowns
ũ
,
j
=
−N..N
.
In
order
a
j
j=−N
(8)
to determine these unknowns, we use Sinc collocation
log N
πdN
≤ O(
exp(−
)),
points:
N
log(πdN/α)
b−a
π
a+b
where
xk = ϕ(kh) =
tanh( sinh(kh)) +
. (16)
2
2
2
π
a+b
b−a
tanh( sinh t) +
,
(9) So we get:
ϕ(t) =
2
2
2
′
ϕ′ (t) =
b−a
π/2 cosh(t)
2 cosh2 (π/2 sinh(t))
∫
Si(t) =
0
t
3
(17)
Finally, following form of the dual fuzzy linear system
is obtained.
AŨ = f˜ + B Ũ
(18)
(11)
where matrices A = (ai,j ), B = (bi,j ), i, j = −N...N
are defined as
sin w
dw,
w
and the mesh size h satisfies h =
1
N
K(xk , ϕ(jh))ϕ′ (jh)ηh,j (xk )ũj .
j=−N
−1
1
1
πϕ (s)
ηh,j (s) = ( + Si(
− jπ)).
2 π
h
Also Si(t) is the Sine integral defined by:
N
∑
ũ(xk ) = f˜(xk )+λh
(10)
A = I(2N +1)(2N +1) ,
log(πdN/α).
bi,j = λhK(xi , ϕ(jh))ϕ′ (jh),
(19)
Also
Sinc approximation to fuzzy
Volterra integral equations
f˜ = [f˜(x−N ), ...f˜(xN )]t
and
Ũ = [ũ−N , ...ũN ]t
In this paper, we consider Fuzzy Volterra integral
are arbitrary fuzzy number vectors.
equations of the form :
∫ x
Similar to definition [4] coefficients of matrices A, B
u(x) = f (x) + λ
K(x, t)u(t)dt,
a ≤ x, t ≤ b
can transfer into 2n × 2n (with n = 2N + 1) crisp
a
(12) linear systems :
where a, b, λ are real constants f (x), K(x, t) are given
SU = F + T U
(20)
functions and u(x) is to be determined. If f (x) is a
4
Step4: Compute S − T and its inverse.
Step5: Solve linear system U = (S − T )−1 F by
newton method.
Step6: Output ũj , j = −N..N and using (25,26)
for approximation and definition 6 for fuzzy distance .
Linear system of Eq. (20) has dimension (4N +
2)(4N + 2) and can be uniquely solved for U , if and
only if the matrix (S − T ) is nonsingular, so
(S − T )U = F,
U = (S − T )−1 F,
(21)
where matrices S, T are defined as definition [4]. Also
we have:
As mentioned above matrices S, T have the following structure:
(
)
C D
S=
,
D C
F = [f (x−N ), ..., f (xN ), −f (x−N ), ..., −f (xN )]t ,
(22)
and
U = [u−N , ..., uN , −u−N , ..., −uN ]t .
(23)
(
and
T =
In order to get an approximate value of ũ = (u, u)
in an arbitrary point we use method in [22]:
E
F
F
E
)
,
where C and E contains the positive entries of A and
B respectively, and D and F the absolute values of
ũN (x) = f˜(x)+hλ
K(x, ϕ(jh))ϕ′ (jh))ηh,j (x)ũj , the negative entries of A and B, i.e. A = C − D and
j=−N
B = E − F so we have :
(24)
(
)
C −E D−E
which can be replaced by the following equations
S−T =
,
D−F C −E
N
∑
Kp (x, tj )uj dt, (25) Theorem 1. [7, 14] The matrix S − T is nonsingular
uN (x, r) = f (x, r) + hλ
if and if the matrix (C + D) − (E + F ) and (C + F ) −
j=−N
(E + D) are both nonsingular.
and
Theorem 2. [9, 20] If (S − T )−1 exists it must have
N
∑
the same structure as S, i.e.
uN (x, r) = f (x, r) + hλ
Kp (x, tj )uj dt, (26)
(
)
j=−N
G H
(S − T )−1 =
,
H G
where Kp = K(x, tj )ϕ′ (jh))ηh,j (x), tj = ϕ(jh)
{
where
Kp uj Kp ≥ 0,
Kp uj =
(27)
Kp uj Kp < 0.
1
G = [((C +D)−(E +F ))−1 +((C +F )−(E +D))−1 ]
Also
2
{
Kp uj Kp ≥ 0,
Kp uj =
(28)
Kp uj Kp < 0.
and
N
∑
Finally, we give the following algorithm to compute numerical solution of Eq. (13):
Algorithm1:
Step1: Input a, b, N, α, f˜, K(x, t), ϕ(x)
Step2: Execute nested loops
for i = −N..N do
xx[i] := ϕ(ih)
for j = −N..N do
A[i + N + 1, j + N + 1] := δij
B[i + N + 1, j + N + 1]
:=
h ∗
K(xx[i], ϕ(jh))ϕ′ (jh)ηh,j (xx[i])
end do
end do
Step3: Convert matrices A, B to transfer matrices
S, T , respectively by definition(4).
H=
1
[((C +D)−(E +F ))−1 −((C +F )−(E +D))−1 ]
2
Theorem 3. [9, 20] The unique solution U of Eq.
(14) is a fuzzy vector for arbitrary Y if and only if
(S − T )−1 is nonnegative, i.e.
((S − T )−1 )ij ≥ 0,
1 ≤ i, j ≤ 2n.
(29)
Definition 11. Let X = {(xi (r), xi (r)), i = 1..n}
denotes the unique solution of linear system AX̃ = Ỹ
then if (y i (r), y i (r)) are linear functions of r, then the
fuzzy number vector U = {(ui (r), ui (r)), i = 1..n}
defined by:
ui (r) = min{xi (r), xi (r), xi (1), xi (1)}
5
ui (r) = min{xi (r), xi (r), xi (1), xi (1)}
π
4.
and a = 0, b =
given by
is called the fuzzy solution. If (xi (r), xi (r)) are all
fuzzy numbers then ui (r) = xi (r), ui (r) = xi (r) and
then U is called a strong fuzzy solution. Otherwise,
U is a weak fuzzy solution.
The exact solution in this case is
u(x, r) = x3 (r5 + 2r)
and
u(x, r) = x3 (6 − 3r3 ).
4
For N = 1 matrices A, B, S, T, (S − T )−1 are listed
blow
Error Analysis
In this section based on sinc approximation
which state in section 2.2, we derive a bound of
dH (ũ(x) − ũN (x)) by the following theorem.


A=
 0
Theorem 4. Let ũN (x) be approximate solution,
there exists constant C such that:
0
5
0
0
log(N )
πdN
exp(−
)
N
log(πdN/α)


0 
,
1
1
0

and
dH (ũ(x) − ũN (x))
≤C
1






S=





(30)
Numerical Examples
In this section, examples are presented to illustrate
effectiveness and importance of proposed method.
Also, in order to show the error of approximation,
the accuracy of approximation are calculated by
three criterion :

0

−0.061

B=
 −0.078 −0.058
−0.021 −0.042
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0.01
0.016
0
0.06
0
−0.005

0.006 

−0.012


0 


0 

,
0 


0 

1
0
0.005



 0
0
0.006 0.07 0.05
0 




 0

0
0
0.02
0.04
0.01


T =

 0.06
0
0.005
0
0.01
0 




 0.07 0.05

0
0
0
0.006


0.02 0.04 0.01
0
0
0
1)Distance between two fuzzy numbers u and v
mentioned by definition(6).
2)Run time of program which is showed by T(s),(s
means second).
3)Condition number of matrix (S − T )−1 based
on infinity norm which is defined as

Cond((S − T )−1 ) = ∥(S − T )−1 ∥∞ ∥(S − T )∥∞ . (31)
We Consider the following two test problems:
(S−T )−1
Example .1 We consider the following fuzzy
Volterra integral equation






=





1.00
0.01 0.00
0.00
1.00 0.00
0.06 0.00 0.00


0.07 0.06 0.00 


0.00 0.00 1.00 0.02 0.04 0.01 


0.06 0.00 0.00 1.00 0.01 0.00 


0.07 0.06 0.00 0.00 1.00 0.00 

0.02 0.04 0.01 0.00 0.00 1.00
f (x, r) = 2x(r5 + 2r)(3 − 3 cos(x) − x2 )
To obtain results, we use Maple 13 software and
take number of Sinc functions as N = 2, 4, 8, 16. Also
in order to have better results we concentrate on 3
criteria in Table 1, Run-Time (column T (s) in second), fuzzy distance (column D(u, v))and Condition
f (x, r) = 6x(2 − r3 )(3 − 3 cos(x) − x2 )
and kernel
K(x, t) = x cos(t − x),
0 ≤ x, t ≤
π
, λ=1
4
6

Figure 1: Comparison between approximate solutions
with N = 3, r = 0.5 and the exact solution for example 1.
Figure 2: Graph of u(x, r) for example 1.
number of matrix (S − T )−1 (column Cond based on
infinity norm). The results are showed in Table 1.
N
2
4
8
16
T(s)
0.84
1.26
1.34
1.56
D(u, v)
3.14E-001
5.71E-004
2.35E-007
1.40E-012
Cond
3.50E+001
3.98E+001
4.00E+001
4.00E+001
Table 1. Results for example 1 by Sinc fuzzy
collocation method.
As shown in Table1, by increasing value of N fuzzy
distance decreased. Based on Sinc method in system
(18) matrices A, B have (2N + 1)(2N + 1) dimension
and transfer matrices S, T have (4N + 2)(4N + 2)
dimension. For example in Table 1 for N = 16 linear
system has 64 unknowns and 64 equations, where in
this case condition number is Cond = 5.00E + 001
which is remarkable. Also we must note that run
time is very short in comparison with size of system.
Figure 1 shows, comparison between approximate
solutions and exact solution for N = 3, r = 0.5 which
is showed by u, uN and u, uN , as shown in figure 1
approximation is very good.
As shown in figures 2,3 and 4,5 exact solution
u(x, r), u(x, r) with x = 0..1, r = 0..1 and approximate solution uN (x, r), uN (x, r), N = 3 are plotted
respect to each others.
Figure 3: Graph of uN (x, r) for example 1.
Example 2 [1, 14] We consider the following fuzzy
Volterra integral equation
Figure 4: Graph of u(x, r) for example 1.
2
4
1
1
1
f (x, r) = rx−x ( rx3 − x3 − rx2 +x2 + r − )
3
3
12
12
2
7
Figure 5: Graph of uN (x, r) for example 1.
2
1
1
1
f (x, r) = (2 − r)x + x2 ( rx3 − rx2 + r − )
3
12
12
and kernel
K(x, t) = x2 (1 − 2t),
0 ≤ x, t ≤
Figure 6: Error plot versus N and D(u, v) for example 2.
1
, λ=1
2
and a = 0, b = 1. The exact solution in this case is
given by
u(x, r) = rx,
u(x, r) = (2 − r)x.
The results for example 2 are listed in Table 2.
N
2
4
8
16
T(s)
1.23
1.47
2.14
3.56
D(u, v)
2.01E-002
8.09E-005
1.84E-006
1.60E-012
Cond
4.30E+001
4.85E+001
5.01E+001
5.05E+001
Table 2. Results for example 2 by Sinc fuzzy
collocation method.
As results present in Table 2, condition number is
invariant under variation of N , however for N = 16
dimension of system is large. Also run time of program is small. Figure 4, shows convergence behavior
of Sinc collocation method in terms of fuzzy distance
Figure 7: Comparison between approximate solutions
versus reciprocal of number of collocation points N .
with N = 3, r = 0.8 and the exact solution for examSimilar to column D(u, v) in table 2, figure 6 shows,
ple 2.
fuzzy distance decreases by increasing the number of
collocation points.
Also figures 7,8 show Comparison between the
8
integral equations of the second kind by Adomian
method, Appl. Math. Comput, 161 (2005) 733-744.
3. Bing Zheng, K. Wang, General fuzzy linear systems,
Appl. Math. Comput, 181 (2006) 1276-1286.
4. W. Congxin and M. Ma, On embedding problem of
fuzzy number spaces, Fuzzy Sets Syst, 44 (1991) 3338.
5. D. Dubois and H. Prade, Towards fuzzy differential
calculus, Fuzzy Sets Syst, 8 (1982) 1-7.
6. M. A. Fariborzi Araghi, Gh.
Kazemi Gelian,
Numerical solution of integro differential equations
based on double exponential transformation in the
sinc-collocation method, App. Math. and Comp.
Intel. 1 (2012) 48-55.
7. M. A. Fariborzi Araghi, Gh. Kazemi Gelian, Numerical solution of nonlinear Hammerstien integral
equations via Sinc collocation method based on Double Exponential Transformation, Mathematical Science. 7:30, doi:10.1186/2251-7456-7-30, 30 (2013).
Figure 8: Comparison between approximate solutions
with N = 3, r = 1 and the exact solution for example
2.
8. M. A. Fariborzi Araghi, N. Parandin, Numerical solution of fuzzy Fredholm integral equation by the
Lagrange interpolation based on extension principle,
Soft Comput, 15 (2011) 2449-2456.
exact solution and approximate solutions with
N = 3 and r = 0.8, r = 1 respectively.
As we see in figure 6 approximation is very well
and note that in this case u = u.
6
9. M. Friedman, M. Ming and A. Kandel, Fuzzy linear
systems, Fuzzy Sets Syst, 96 (1998) 201-209.
10. M. Friedman, M. Ming and A. Kandel, Numerical
solutions of fuzzy differential and integral equations,
Fuzzy Sets Syst, 106 (1999) 35-48.
Conclusion
11. R. Goetschel, W. Voxman, Elementary calculus,
Fuzzy Sets Syst, 18 (1986) 31-43.
We applied the Sinc collocation method based on
double exponential transformation to fuzzy Volterra
integral equations. Sinc collocation method in run
time and condition number have good reliability and
efficiency. Also, we can improve the accuracy of the
solution by selecting the appropriate shape parameters and selecting the large values of N . based on
ontaind results and similar work [6, 7, 13] we conclude that the high accuracy of the proposed method
by taking this view that storing in time and memory, is another useful property in the Sinc method.
In addition this method is portable to other area of
problems and easy to programming.
7
12. S. Haber, Two formulas for numerical indefinite integration, Math. Comp, 60 (2008) 279-296.
13. M. Hadizadeh and Gh. Kazemi Gelian, Error estimate in the Sinc collocation method for VolterraFeredholm integral equations based on DE transformations, ETNA 30, 75-87 (2008).
14. M. Jahantigh, T. Allahviranloo, M. Otadi, Numerical solution of fuzzy integral equations,Appl. Math.
Sci, 2 (2008) 33-46.
15. O. Kaleva, Fuzzy differential equations, Fuzzy Sets
Syst, 24 (1987) 301-317.
16. S. Khezerloo, T. Allahviranloo, S. Haji Ghasemi, S.
Salahshour, M. Khezerloo, M. Khorasan Kiasary,
Expansion method for solving fuzzy FredholmVolterra integral equations, Information processing
and management of uncertainty in knowledge-based
systems. Applications communications in computer
and information science, 81 (2010) 501-511.
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