Soft Comput (2011) 15:1247–1253
DOI 10.1007/s00500-010-0659-y
ORIGINAL PAPER
Euler method for solving hybrid fuzzy differential equation
T. Allahviranloo • S. Salahshour
Published online: 1 October 2010
Ó Springer-Verlag 2010
Abstract In this paper, we study the numerical method
for solving hybrid fuzzy differential using Euler method
under generalized Hukuhara differentiability. To this end,
we determine the Euler method for both cases of H-differentiability. Also, the convergence of the proposed
method is studied and the characteristic theorem is given
for both cases. Finally, some numerical examples are given
to illustrate the efficiency of the proposed method under
generalized Hukuhara differentiability instead of suing
Hukuhara differentiability.
Keywords Hybrid fuzzy differential equations
Euler method Generalized Hukuhara differentiability
Convergence Characteristic theorem
1 Introduction
The topic of fuzzy differential equations (FDEs) has been
rapidly growing in recent years. The concept of fuzzy
derivative was first introduced by Chang and Zadeh (1972);
it was followed up by Dubois and Prade (1982), who used
the extension principle in their approach. Other methods
have been discussed by Puri and Ralescu (1983) and by
Goetschel and Voxman (1986). The FDE and the initial
value problem (Cauchy problem) were rigorously treated
T. Allahviranloo (&)
Department of Mathematics, Science and Research Branch,
Islamic Azad University, Tehran, Iran
e-mail: tofigh@allahviranloo.com
S. Salahshour
Department of Mathematics, Mobarakeh Branch,
Islamic Azad University, Mobarakeh, Iran
e-mail: soheilsalahshour@yahoo.com
by Kaleva (1987, 1990), Seikkala (1987) and by other
researchers (see Abbasbandy and Allahviranloo 2002;
Abbasbandy et al. 2004; Allahviranloo and Barkhordari
Ahmadi 2010; Allahviranloo et al. 2009, 2007, 2008;
Buckley and Feuring 2003, 2001; Gnana Bhaskar et al.
2004; Kloeden 1991; Friedman et al. 1999a; Georgiou
et al. 2005; Congxin and Shiji 1998; Chalco-Cano and
Roman-Flores 2006; Diamond 1999; Ding et al. 1997).
Strongly generalized differentiability was introduced in
(Bede and Gal 2005) and studied in (Bede et al. 2006;
Stefanini and Bede 2010). The strongly generalized
derivative is defined for a larger class of fuzzy-valued
function than the H-derivative, and fuzzy differential
equations can have solutions which have a decreasing
length of their support. So, we use this differentiability
concept in the present paper.
Also, there are useful research papers for solving fuzzy
differential equations and fuzzy integral equations (Ding
and Kandel 1997; Friedman and Kandel 1992; Friedman
et al. 1996a, b, 1997a, b, 1999b, c).
Hybrid systems are devoted to modeling, design and
validation of interactive systems of computer programs and
continuous systems. That is, control systems that are
capable of controlling complex systems, which have discrete event dynamics as well as continuous time dynamics,
can be modeled by hybrid systems. The differential systems containing fuzzy-valued functions with an interaction
with a discrete time controller are named hybrid fuzzy
differential systems (for analytical results on hybrid fuzzy
differential equations, see Lakshmikantham and Mohapatra
2006; Lakshmikantham and liu 1999; Sambandham 2002).
Pederson et al. (Pederson and Sambandham 2007, 2008)
studied the Euler and Runge-Kutta numerical methods,
respectively, for hybrid fuzzy differential equations. But,
they have been considered as mentioned methods based on
123
1248
the Hukuhara differentiability. However, we will extend
the Euler method based on the generalized Hukuhara
differentiability.
Pederson et al. (Pederson and Sambandham 2009) have
extended Bede’s characterization theorem (2008) to hybrid
fuzzy differential equations. However, they performed such
approaches based on the Hukuhara differentiability. Also,
we extend the Pederson’s characterization theorem to
hybrid fuzzy differential equations under generalized
Hukuhara differentiability and then used this result to
numerically to solve these systems by any suitable method
for ODEs.
This paper is organized as follows. In Sect. 2, we
present the basic notions of fuzzy number, fuzzy-valued
function and generalized Hukuhara differentiability. In
Sect. 3, the hybrid fuzzy differential equations are defined,
and the Euler method using generalized Hukuhara differentiability is investigated. Convergence of the proposed
method is also studied. Besides, we proposed the characterization theorem to solve fuzzy hybrid differential
equations under the mentioned types of H-differentiability.
Some numerical examples are given to illustrate the Euler
method under generalized Hukuhara differentiability in
Sect. 4, and the conclusion is drawn in Sect. 5.
T. Allahviranloo, S. Salahshour
yeðtÞ ¼
1
0
if
if
t¼y
t 6¼ y
Remark 2.1 (see Zimmermann 1991), Let X be Cartesian
product of universes X ¼ X1 Xn ; and A1 ; . . .; An be
n fuzzy numbers in X1 ; . . .;Xn ; respectively. f is a mapping
from X to a universe Y, y ¼ f ðx1 ; . . .; xn Þ: Then the
extension principle allows us to define a fuzzy set B in Y by
B ¼ fðy; uðyÞÞ j y ¼ f ðx1 ; . . .; xn Þ; ðx1 ; . . .; xn Þ 2 Xg
where
uB ðyÞ ¼
supðx1 ;...;xn Þ2f 1 ðyÞ minfuA1 ðx1 Þ; . . .; uAn ðxn ÞÞg;
0
We recall some definitions that are used throughout the
paper. The basic definition of fuzzy numbers is given in
(Xu et al. 2007).
By R; we denote the set of all real numbers. A fuzzy
number is mapping u : R ! ½0; 1 with the following
properties:
(a) u is upper semi-continuous,
(b) u is fuzzy convex, i.e., uðkx þ ð1 kÞyÞ minfuðxÞ;
uðyÞg for all x; y 2 R; k 2 ½0; 1;
(c) u is normal, i.e.,9x0 2R for which uðx0 Þ ¼ 1;
(d) supp u ¼ fx 2 R j uðxÞ [ 0g is the support of the u,
and its closure cl(supp u) is compact.
Let E be the set of all fuzzy number on r. The r-level set
of a fuzzy number u 2 E; 0 r 1; denoted by ½ur , is
defined as
fx 2 R j uðxÞ rg if 0\r 1
½ur ¼
clðsupp uÞ
if r ¼ 0
It is clear that the r-level set of a fuzzy number is a closed
and bounded interval ½uðrÞ; uðrÞ; where uðrÞ denotes the
left-hand endpoint of ½ur and uðrÞ denotes the right-hand
endpoint of ½ur : Since each y 2 R can be regarded as a
fuzzy number ye is defined by
123
f 1 ðyÞ 6¼ 0;
otherwise:
where f 1 is the inverse of f. For n = 1, the extension
principle, of course, reduces to
B ¼ fðy; uB ðyÞÞ j y ¼ f ðxÞ; x 2 Xg
where
uB ðyÞ ¼
supx2f 1 ðyÞ uA ðxÞ;
0
if
if
f 1 ðyÞ 6¼ 0;
otherwise:
According to Zadeh,s extension principle, operation of
addition on E is defined by
ðu vÞðxÞ ¼ sup minfuðyÞ; vðx yÞg;
y2R
2 Preliminaries
if
if
x2R
and scalar multiplication of a fuzzy number is given by
uðx=kÞ; k [ 0;
ðk uÞðxÞ ¼ e
0;
k ¼ 0;
where ~0 2 E:
The Hausdorff distance between fuzzy numbers given
S
by D : E E ! Rþ f0g;
Dðu; vÞ ¼ sup maxfjuðrÞ vðrÞj; juðrÞ vðrÞjg;
r2½0;1
It is easy to see that D is a metric in E and has the following properties (see Puri and Ralescu 1986)
(i) Dðu w; v wÞ ¼ Dðu; vÞ; 8u; v; w 2 E;
(ii) Dðk u; k vÞ ¼ jkjDðu; vÞ; 8k 2 R; u; v 2 E;
(iii) Dðu v;w eÞ Dðu;wÞ þ Dðv; eÞ; 8u; v;w;e 2 E;
(iv) ðD; EÞ is a complete metric space.
Definition 2.1 (see Friedman et al. 1999a), Let f : R !
E be a fuzzy valued function. If for arbitrary fixed t0 2 R
and [ 0; a d [ 0 such that
jt t0 j\d ) Dðf ðtÞ; f ðt0 ÞÞ\;
f is said to be continuous.
It is well known that the H-derivative (differentiability
in the sense of Hukuhara) for fuzzy mappings was initially
Euler method for solving hybrid fuzzy differential equation
introduced by Puri and Ralescu ( 1983) and is based on the
H-difference of sets, as follows.
Definition 2.2 Let x; y 2 E: If there exists z 2 E such that
x ¼ y z; then z is called the H-difference of x and y, and
it is denoted by x y:
In this paper, the sign ‘‘ ’’ always stands for H-difference, and also note that x y 6¼ x þ ð1Þy: In this paper we
consider the following definition for differentiability,
which was introduced by Bede and Gal (Bede and Gal
2005, Bede et al. 2006) (note that we consider only two
cases).
Definition 2.3 Let f : ða; bÞ ! E and x0 2ða; bÞ: We say
that f is strongly generalized differential at x0 : If there
exists an element f 0 ðx0 Þ 2 E; such that
(i)
for all h [ 0 sufficiently small, 9f ðx0 þ hÞ f ðx0 Þ;
9f ðx0 Þ f ðx0 hÞ and the limits (in the metric D)
lim
h&0
(ii)
f ðx0 þ hÞ hf ðx0 Þ
f ðx0 Þ
¼ lim
h&0
h
¼ f 0 ðx0 Þ
f ðx0 hÞ
h
or
for all h [ 0 sufficiently small, 9f ðx0 Þ f ðx0 þ hÞ;
9f ðx0 hÞ f ðx0 Þ and the limits(in the metric D)
lim
h&0
f ðx0 Þ
f ðx0 þ hÞ
f ðx0 hÞ f ðx0 Þ
¼ lim
h&0
h
h
¼ f 0 ðx0 Þ:
For the sake of simplicity, we say that the fuzzy-valued
function f is (i)-differentiable if it satisfies in the
Definition 2.3 case (1) and fuzzy-valued function f is
(ii)-differentiable if it satisfies in the Definition 2.3
case (2).
In the special case when f is a fuzzy-valued function, we
have the following result.
Theorem 2.1 (see e.g. Chalco-Cano and Roman-Flores
2006), Let f : R ! Ebe a function and denote f ðtÞ ¼
ðf ðt; rÞ; f ðt; rÞÞ; for each r2½0; 1. Then
(1)
If f is (i)-differentiable, then f ðt; rÞand f ðt; rÞare
0
(2)
differentiable functions and f 0 ðtÞ ¼ ðf 0 ðt; rÞ; f ðt; rÞÞ
If f is (ii)-differentiable, then f ðt; rÞand f ðt; rÞare
0
differentiable functions and f 0 ðtÞ ¼ ðf ðt; rÞ; f 0 ðt; rÞÞ:
3 Euler method
In this section, we adopt Euler method to solve hybrid
fuzzy differential equations using generalized Hukuhara
1249
differentiability. Consider the hybrid fuzzy differential
equation:
0
x ðtÞ ¼ f ðt; xðtÞ; kk ðxðtk ÞÞÞ; t 2 ½tk ; tkþ1
ð1Þ
xðtk Þ ¼ xk ;
where
0 t0 \t1 \ \tk \. . .; tk ! 1; f 2 C½Rþ
E E; E; kk 2 C½E; E: To be specific, the system will
be as follows:
8 0
x0 ðtÞ ¼ f ðt;x0 ðtÞ;k0 ðx0 ÞÞ; x0 ðt0 Þ ¼ x0 ; t 2 ½t0 ;t1
>
>
>
> x01 ðtÞ ¼ f ðt;x1 ðtÞ;k1 ðx1 ÞÞ; x1 ðt1 Þ ¼ x1 ; t 2 ½t1 ;t2
>
>
>
> x02 ðtÞ ¼ f ðt;x2 ðtÞ;k2 ðx2 ÞÞ; x2 ðt2 Þ ¼ x2 ; t 2 ½t2 ;t3
<
0
x ðtÞ ¼ ..
.
>
>
>
>
x0k ðtÞ ¼ f ðt;xk ðtÞ;kk ðxk ÞÞ; xk ðtk Þ ¼ xk ; t 2 ½tk ; tkþ1
>
>
>
>
.
:.
.
ð2Þ
With respect to the solution of (1), we determine the
following function:
8
x0 ðtÞ; t 2 ½t0 ; t1
>
>
>
>
> x1 ðtÞ; t 2 ½t1 ; t2
>
>
>
< x2 ðtÞ; t 2 ½t2 ; t3
xðt; t0 ; x0 Þ ¼ ..
ð3Þ
.
>
>
>
>
xk ðtÞ; t 2 ½tk ; tkþ1
>
>
>
>
: ..
.
We note that the solutions of (1) are piecewise
differentiable in each interval for t2½tk ; tkþ1 for a fixed
xk 2 E and k ¼ 0; 1; 2; . . .: We may replace (1) by
an equivalent system when xðtÞ is considered as
(i)-differentiable fuzzy-valued function:
0
x ðtÞ ¼ f ðt;x;kk ðxk ÞÞ ¼ Fk ðt; x; x; kk ðxk ÞÞ;xðtk Þ ¼ xk ;
ð4Þ
x0 ðtÞ ¼ f ðt; x; kk ðxk ÞÞ ¼ Gk ðt; x; x;kk ðxk ÞÞ;xðtk Þ ¼ xk
and also (1) is equivalent to the following system when xðtÞ
is considered as (ii)-differentiable fuzzy-valued function:
0
x ðtÞ ¼ f ðt;x;kk ðxk ÞÞ ¼ Fk ðt; x; x; kk ðxk ÞÞ;xðtk Þ ¼ xk ;
ð5Þ
x0 ðtÞ ¼ f ðt; x; kk ðxk ÞÞ ¼ Gk ðt; x; x;kk ðxk ÞÞ;xðtk Þ ¼ xk
That is, for each t, the pair ½xðt;rÞ;xðt;rÞ is a fuzzy number,
where xðt;rÞ;xðt;rÞ are, respectively, the solutions of the
parametric form given by:
0
xk ðt;rÞ ¼ Fk ðt;xðt;rÞ; xðt; rÞ;kk ðxk ÞÞ; xðtk ; rÞ ¼ xk ðrÞ;
ð6Þ
x0 ðt; rÞ ¼ Gk ðt;xðt;rÞ; xðt; rÞ;kk ðxk ÞÞ;xðtk ;rÞ ¼ xk ðrÞ
in the sense of (i)—differentiability and
0
xk ðt;rÞ ¼ Gk ðt; xðt; rÞ;xðt;rÞ;kk ðxk ÞÞ;xðtk ;rÞ ¼ xk ðrÞ;
x0 ðt;rÞ ¼ Fk ðt;xðt;rÞ; xðt; rÞ;kk ðxk ÞÞ;xðtk ;rÞ ¼ xk ðrÞ
ð7Þ
in the sense of (ii)-differentiability and for each 0 r 1:
For a fixed r, to integrate the system (6) and (7) in
½t0 ;t1 ; ½t1 ; t2 ;. ..;½tk ;tkþ1 ;. ..; we replace each interval by a
123
1250
T. Allahviranloo, S. Salahshour
set of Nkþ1 discrete equally spaced grid points (including
the endpoints) at which the exact solution ½xðt;rÞ;xðt;rÞ
is approximated by some ½yk ðt;rÞ;yk ðt;rÞ: For the chosen
k
;
grid points on ½tk ; tkþ1 at tk;n ¼ tk þ nhk ; hk ¼ tkþ1Nt
k
Let
½Y k ðt;rÞ;Y k ðt;rÞ ½xðt;rÞ; xðt; rÞ:
h
i
Moreover, Y k ðt; rÞ;Y k ðt;rÞ and yk ðt;rÞ; yk ðt;rÞ may
be denoted respectively by Y k;n ðt;rÞ;Y k;n ðt; rÞ and
h
i
yk;n ðt;rÞ;yk;n ðt;rÞ : We allow the Nk0 s to vary over the
0 n Nk ;
0
h0k s
½tk ;tkþ1 s so that
may be comparable. The Euler
method under generalized Hukuhara differentiability is the
0
first-order approximation of Y 0k ðt; rÞ and Y k ðt; rÞ; which can
be written for case (i)-differentiability as:
(
Y k;nþ1 ðrÞ Y k;n ðrÞ þ hk Fk tk ; Y k;n ðtÞ; Y k;n ðrÞ ;
ð8Þ
Y k;nþ1 ðrÞ Y k;n ðrÞ þ hk Gk tk ; Y k;n ðtÞ; Y k;n ðrÞ
and in the sense of (ii)-differentiability we get the
following:
(
Y k;nþ1 ðrÞ Y k;n ðrÞ þ hk Gk tk ; Y k;n ðtÞ; Y k;n ðrÞ ;
ð9Þ
Y k;nþ1 ðrÞ Y k;n ðrÞ þ hk Fk tk ; Y k;n ðtÞ; Y k;n ðrÞ
Now, we define the Euler method based on the (8) and (9),
respectively, as follows:
h
i
8
<y
ðrÞ
¼
y
ðrÞ
þ
h
F
t
;
y
ðtÞ;
y
ðrÞ
;
k
k
k
k;n
k;nþ1
k;n
k;n
h
i
ð10Þ
:y
ðrÞ
¼
y
ðrÞ
þ
h
G
t
;
y
ðtÞ;
y
ðrÞ
k
k
k
k;nþ1
k;n
k;n
k;n
and
h
i
8
<y
ðrÞ
¼
y
ðrÞ
þ
h
G
t
;
y
ðtÞ;
y
ðrÞ
;
k
k
k
k;n
k;nþ1
k;n
k;n
h
i
:y
k;nþ1 ðrÞ ¼ yk;n ðrÞ þ hk Fk tk ; yk;n ðtÞ; yk;n ðrÞ
ð11Þ
However, (10) and (11) will use y0;0 ðrÞ ¼ x0 ðrÞ; y0;0 ðrÞ ¼
x0 ðrÞ and yk;0 ðrÞ ¼ yk1;N ðrÞ; yk;0 ðrÞ ¼ yk1;Nk1 ðrÞ if
k1
k 1: Then (10) and (11) represent the approximations of
Y k ðt; rÞ and Y k ðt; rÞ for each of intervals ½t0 ; t1 ; ½t1 ; t2 ;
. . .; ½tk ; tkþ1 ; . . .: For a prefixed k and r 2 ½0; 1 proof of
convergence of the approximations in (10) and (11) is
ðrÞ ¼ xðtkþ1 ; rÞ;
lim
y
lim
yk;Nk ðrÞ ¼ xðtkþ1 ; rÞ
h0 ;h1 ;...;hk !0 k;Nk
h0 ;h1 ;...;hk !0
is an application of Theorem 1 in (Bede 2008) and Lemma
3.1 below.
Lemma 3.1 Suppose i 2 Zþ ; [ 0; r 2 ½0; 1 and hi \1
i
are fixed. Let fZi;n gNn¼0
be the Euler approximation with
N ¼ Ni to the fuzzy IVP:
123
x0 ðtÞ ¼ f ðt; xðtÞ; ki ðxðti ÞÞÞ; t 2 ½ti ; tiþ1
xðti Þ ¼ xi ;
Ni
If yi;n ðrÞ n¼0
denotes the results of (10) or (11) for some
yi;0 ðrÞ; then there exists a di [ 0 such that
j zi;0 ðrÞ yi;0 ðrÞ j \di ;
j zi;0 ðrÞ yi;0 ðrÞ j \di
implies
j zi;Ni ðrÞ yi;N ðrÞ j \i ;
i
j zi;Ni ðrÞ yi;Ni ðrÞ j \i
Proof It is proved for Eq. (10) in (Pederson and
Sambandham 2007). Proof of Eq. (11) is completely
similar to the previous one.
Theorem 3.1 Consider the system (4) and (10) or system
(5) and (11), for fixed k 2 Zþ and r2½0; 1
lim
y
ðrÞ ¼ xðtkþ1 ; rÞ;
ð12Þ
lim
yk;Nk ðrÞ ¼ xðtkþ1 ; rÞ;
ð13Þ
h0 ;h1 ;...;hk !0 k;Nk
h0 ;h1 ;...;hk !0
Proof If system (4) and (10) applied, the proof is given in
Pederson and Sambandham (2007) and for system (5) and
(11), proof is completely similar.
Also, the contribution of this paper is to extend Pederson’s
characterization theorem (Pederson and Sambandham 2009)
to hybrid fuzzy differential equations under generalized
Hukuhara differentiability.
Theorem 3.2 Consider the hybrid fuzzy differential
equations (1) expanded as (2) where for k ¼ 0; 1; 2; . . .;
each fk : ½tk ; tkþ1 E ! E is such that
(i) fk ðt; x; rÞ ¼ ½Fk ðt; xðt; rÞ; xðt; rÞÞ; Gk ðt; xðt; rÞ; xðt; rÞÞ
(ii) Fk ðt; xðt; rÞ; xðt; rÞÞ and Gk ðt; xðt; rÞ; xðt; rÞÞ are equicontinuous(that is, for and [ 0 there is a dk ðÞ such
that jFk ðt; x; yÞ Gk ðt1 ; x1 ; y1 Þj\ for all r 2 ½0; 1;
whenever ðt; x; yÞ; ðt1 ; x1 ; y1 Þ 2 ½tk ; tkþ1 R2 and
kðt; x; yÞ ðt1 ; x1 ; y1 Þ\dk ðÞk) and uniformly bounded
on any bounded set.
(iii) There exists Lk [ 0 such that
jFk ðt; x2 ; y2 Þ Gk ðt1 ; x1 ; y1 Þj
\Lk maxfjx2 x1 j; jy2 y1 jg
Then, Eq. (1) and the hybrid system ODEs
8 0
x ðt; rÞ ¼ Fk ðt; xk ðt; rÞ; xk ðt; rÞÞ
>
>
< k0
xk ðt; rÞ ¼ Gk ðt; xk ðt; rÞ; xk ðt; rÞÞ
x ðt ; rÞ ¼ xk1 ðtk ; rÞ if k [ 0; x0 ðt0 ; rÞ ¼ x0 ðrÞ
>
>
: k k
xk ðtk ; rÞ ¼ xk1 ðtk ; rÞ if k [ 0; x0 ðt0 ; rÞ ¼ x0 ðrÞ
ð14Þ
are equivalent when x(t) is (i)-differentiable and, Eq. (1)
and the hybrid system of ODEs
Euler method for solving hybrid fuzzy differential equation
1251
8 0
x ðt; rÞ ¼ Fk ðt; xk ðt; rÞ; xk ðt; rÞÞ
>
>
< k0
xk ðt; rÞ ¼ Gk ðt; xk ðt; rÞ; xk ðt; rÞÞ
> xk ðtk ; rÞ ¼ xk1 ðtk ; rÞ if k [ 0; x0 ðt0 ; rÞ ¼ x0 ðrÞ
>
:
xk ðtk ; rÞ ¼ xk1 ðtk ; rÞ if k [ 0; x0 ðt0 ; rÞ ¼ x0 ðrÞ
yðiiÞEuler : Clearly, the exact solution of Eq. (14) is as
follows:
yExact ð1; rÞ ¼ ð0:75 þ 0:25 rÞe1 ;ð1:125 0:125 rÞe1
ð15Þ
are equivalent when x(t) is (ii)-differentiable.
Proof It is completely similar to Pederson’s paper
(Pederson and Sambandham 2009).
4 Numerical examples
In this section, we first take an illustrative example to show
the defect of using Hukuhara differentiability (i.e., only
using (i)-differentiability in Bede et al. 2006). In addition,
we are going to solve the hybrid fuzzy systems by Euler
method under generalized Hukuhara differentiability.
Moreover, the characteristic theorem is given for obtaining
the solutions of hybrid fuzzy differential equations.
4.1 Solution via Euler method
Example 4.1
equation
First, we solve the simple fuzzy differential
x0 ðtÞ ¼ xðtÞ; xð0; rÞ
¼ ½0:75 þ 0:25 r; 1:125 0:125 r;
t 2 ½0; 1; r 2 ½0; 1
ð16Þ
If we apply the method that is proposed in Friedman et al.
1999a, we get the solution as follows:
1 eð1=2Þ
2
1 eð1=2Þ
2
þ ð1=2Þ xð0Þ þ
ð1=2Þ xð0Þ;
yð1; rÞ ¼
2
2
2
2
e
e
ð17Þ
ð1=2Þ
ð1=2Þ
1 e
2
1 e
2
ð1=2Þ xð0Þ þ
þ ð1=2Þ xð0Þ
yð1; rÞ ¼
2
2
2
2
e
e
ð18Þ
For the sake of simplicity, we named the above solution as
(i)-Euler solution with notation yðiÞEuler : Also, by using
our proposed method to obtain the solution of Eq. (14) in
the sense of (ii)-differentiability, we get the following:
"
1 10
yð1; rÞ ¼ ð0:75 þ 0:25 rÞ 1
;
10
#
1 10
1:125 0:125 rÞ 1
ð19Þ
10
Similarly, we named such solution in the sense of
(ii) - differentiability as (ii)-Euler solution with notation
It is easy to see that yExact ð1; 1Þ ¼ 0:3679; yðiiÞEuler ð1; 1Þ ¼
0:3487 and yðiÞEuler ð1; 1Þ ¼ 0:8244:
Example 4.2 Consider the following hybrid fuzzy differential equation
0
x ðtÞ ¼ xðtÞ þ mðtÞkk ðxðtk ÞÞ; tk ¼ k;k ¼ 0;1;2; ...
xð0; rÞ ¼ ½0:75 þ :25 r; 1:125 0:125 r;0 r 1
ð20Þ
where
mðtÞ ¼
and
kk ðlÞ ¼
2ðtðmod1ÞÞ;
2ð1 tðmod1ÞÞ;
b
0; if
l; if
if
if
tðmod1Þ 0:5
tðmod1Þ [ ; 0:5
k ¼ 0;
k 2 f1; 2; . . .g
The hybrid fuzzy differential Eq. (19) is equivalent to
the following system of fuzzy differential equations:
8 0
¼ x0 ðtÞ; t 2 ½0; 1;
< x0 ðtÞ
x0 ð0; rÞ ¼ ½0:75 þ 0:25 r; 1:125 0:125 r; 0 r 1;
: 0
xi ðtÞ
¼ xi ðtÞ þ mðtÞxi ðti ÞÞ; t 2 ½ti ; tiþ1 ; xi ðti Þ ¼ xi1 ðti Þ; i ¼ 1; 2; . . .
ð21Þ
By applying Example 4.1, we have
"
1 10
;
y1;0 ðrÞ ¼ ð0:75 þ 0:25 rÞ 1
10
#
1 10
ð1:125 0:125 rÞ 1
; 0 r 1:
10
ð22Þ
Then, for i ¼ 1; . . .; 6; we get:
1
i1
i1
;r þ
y1;i ðrÞ ¼ 1
y1;i1 1
y1;0 ð1; r Þ;
10
10
50
ð23Þ
1
i1
i1
;r þ
y1;i ðrÞ ¼ 1
y1;i1 1
y1;0 ð1; r Þ
10
10
50
ð24Þ
and for i ¼ 7; 8; 9; 10; we get the following:
1
i1
;r
y1;i ðrÞ ¼ 1
y1;i1 1
10
10
11 i
þ
y1;0 ð1; r Þ;
50
ð25Þ
123
1252
y1;i ðrÞ ¼
T. Allahviranloo, S. Salahshour
1
i1
11 i
;r þ
1
y1;i1 1
y1;0 ð1; r Þ
10
10
50
ð26Þ
For t 2 ½1; 2Þ; the exact solution is as follows:
(
xðt; rÞ ¼ xð1; rÞð3e1t 2tÞ; t 2 ½1; 1:5; 0 r 1;
xðt; rÞ ¼ xð1; rÞ 2t 2 þ e1:5t p3ffiffie 4 ; t 2 ½1:5; 2; 0 r 1;
ð27Þ
So, the exact solution xð2; 1Þ ¼ 0:2492 and the Euler
approximation in sense of (ii)-differentiability is obtained
y1;10 ð1Þ ¼ yðiiÞEuler ð2; 1Þ ¼ :6841
1
1 10
10
¼ 0:2385:
Remark 4.1 Please notice that one can easily obtain the
solution under the case of (i)-differentiability, which is as
follows:
8 0
>
< x ðt; rÞ ¼ xðt; rÞ þ mðtÞxð1; rÞ;
xð1; rÞ ¼ ð0:75 þ 0:25 r Þð0:9Þ10 ; ;
>
:
xð1; rÞ ¼ ð1:125 0:125 r Þð0:9Þ10 ;
8 0
>
< x ðt; rÞ ¼ xðt; rÞ þ mðtÞxð1; rÞ;
xð1; rÞ ¼ ð0:75 þ 0:25 r Þð0:9Þ10
>
:
xð1; rÞ ¼ ð1:125 0:125 r Þð0:9Þ10
By the Euler method for ODEs with N = 10, we get the
solution yðiiÞEuler ð2; 1Þ ¼ y1;10 as following:
y1;10 ¼ ðð0:9Þ10 þ ð0:9Þ8 0:02 þ ð0:9Þ7 0:04 þ ð0:9Þ6 0:06
þ ð0:9Þ5 0:08 þ ð0:9Þ4 0:1 þ ð0:9Þ3 0:08
þ ð0:9Þ2 0:06 þ ð0:9Þ0:04 þ 0:02Þy1;0 :
Clearly, the obtained solution agrees with the Euler
method with N = 10 in Example 4.2.
yðiÞEuler ð2; 1Þ ¼ 0:8582:
5 Conclusion
4.2 Solution via characterization theorem
Now, we will obtain the solution of hybrid fuzzy differential equations under generalized Hukuhara differentiability by applying characterization theorem. Suppose that
k = 0 and t2½t0 ; t1 ; then Eq. (19) is equivalent to the
following:
0
x ðt; rÞ ¼ xðt; rÞ;
xð0; rÞ ¼ ½0:75 þ 0:25 r; 1:125 0:125 r ; 0 r 1;
This is equivalent to the systems of ODEs
0
x ðt; rÞ ¼ xðt; rÞ;
xð0; rÞ ¼ 0:75 þ 0:25 r; 0 r 1;
x0 ðt; rÞ ¼ xðt; rÞ;
xð0; rÞ ¼ 1:125 0:125 r; 0 r 1;
So, by applying Euler method for ODEs with N ¼ 10, we
get the following:
xðt0;n ; rÞ y0;n ðrÞ
h
i
¼ ð0:75 þ 0:25 r Þð0:9Þ10 ; ð1:125 0:125 r Þð0:9Þ10
n
; n ¼ 0; 1; 2; . . .; 10: Indeed, this agrees
where, t0;n ¼ 10
with the obtained solution in the direct sense in Example
1. Now, let us consider k ¼ 1 and t2½t1 ; t2 : Then (20) is
equivalent to the following:
x0 ðt; rÞ ¼ xðt; rÞ þ mðtÞxð1Þ;
xð1; rÞ ¼ ½ð0:75 þ 0:25 rÞe1 ; ð1:125 0:125 rÞe1 ; 0 r 1;
By using Theorem 3.1 and the method proposed in Sect. 4,
we get the following:
123
Recently, many works have been performed for solving
hybrid fuzzy systems by several authors (Pederson and
Sambandham 2007, 2008, 2009). According to Bede et al.
(2008) ‘‘The importance of converting a hybrid fuzzy differential equation to a hybrid system of ODEs is that then
any suitable numerical method for ODEs may be implemented’’. In this paper, we solved hybrid fuzzy differential
equations under generalized Hukuhara differentiability by
applying the Euler method. To this end, we applied the
Euler method for two cases (i)- or (ii)-differentiability.
Also, the convergence in the sense of (i)- differentiability
was established by Pederson and Sambandham (2007);
however, Example 4.1 shows that the case of (i)-differentiability does not have sufficient efficiency to be applied.
So, we extended the Euler method based on the generalized
Hukuhara differentiability and discussed on the convergence of Euler method for two cases.
For future research, we will try to extend the fuzzy
Laplace transformation method (Allahviranloo and
Barkhordari Ahmadi 2010) to solve such hybrid fuzzy
differential equations. Clearly, Euler method has a big
propagation error, so we will apply Runge-Kutta or predictor–corrector method for solving fuzzy hybrid differential equations.
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Tofigh Allahviranloo