arXiv:1107.4157v1 [cs.NA] 21 Jul 2011

Linear Differential Equations
with Fuzzy Boundary Values
Nizami Gasilov

Şahin Emrah Amrahov

Afet Golayoğlu Fatullayev

Baskent University,
Eskisehir yolu 20. km, Baglica,
06810 Ankara, Turkey
Email: gasilov@baskent.edu.tr

Ankara University,
Computer Engineering Department,
06100 Ankara, Turkey
Email: emrah@eng.ankara.edu.tr

Baskent University,
Eskisehir yolu 20. km, Baglica,
06810 Ankara, Turkey
Email: afet@baskent.edu.tr

Abstract—In this study, we consider a linear differential
equation with fuzzy boundary values. We express the solution
of the problem in terms of a fuzzy set of crisp real functions.
Each real function from the solution set satisfies differential
equation, and its boundary values belong to intervals, determined
by the corresponding fuzzy numbers. The least possibility among
possibilities of boundary values in corresponding fuzzy sets is
defined as the possibility of the real function in the fuzzy solution.
In order to find the fuzzy solution we propose a method based
on the properties of linear transformations. We show that, if
the corresponding crisp problem has a unique solution then the
fuzzy problem has unique solution too. We also prove that if the
boundary values are triangular fuzzy numbers, then the value of
the solution at any time is also a triangular fuzzy number.
We find that the fuzzy solution determined by our method
is the same as the one that is obtained from solution of crisp
problem by the application of the extension principle.
We present two examples describing the proposed method.

Keywords: fuzzy boundary value problem, fuzzy set, linear
transformation.
I. I NTRODUCTION
Approaches to fuzzy boundary value problems can be of two
types. The first approach assumes that, even if only the boundary values are fuzzy in the handling problem, the solution is
fuzzy function, consequently, the derivative in the differential
equation can be considered as a derivative of fuzzy function.
This derivative can be Hukuhara derivative, or a derivative in
generalized sense. Bede [1] has demonstrated that a large class
of boundary value problems have not a solution, if Hukuhara
derivative is used. To overcome this difficulty, in [2] and [3]
the concept of generalized derivative is developed and fuzzy
differential equations have been investigated using this concept
(see also [4], [5], [6]). Recently, Khastan and Nieto [7] have
found solutions for a large enough class of boundary value
problems with the generalized derivative. However as it is seen
from the examples in mentioned article, these solutions are
difficult to interpret because four different problems, obtained
by using the generalized second derivatives, often does not
reflect the nature of the problem.
The second approach is based on generating the fuzzy
solution from the crisp solution. In particular case, for the
fuzzy initial value problem this approach can be of three

ways. The first one uses the extension principle. In this
way, the initial value is taken as a real constant, and the
resulting crisp problem is solved. Then the real constant in the
solution is replaced with the initial fuzzy value. In the final
solution, arithmetic operations are considered to be operations
on fuzzy numbers ([8], [9]). The second way, offered by
Hüllemerier [10], uses the concept of differential inclusion.
In this way, by taking an alpha-cut of initial value, the given
differential equation is converted to a differential inclusion and
the obtained solution is accepted as the alpha-cut of the fuzzy
solution. Misukoshi et al [11] have proved that, under certain
conditions, the two main ways of the approach are equivalent
for the initial value problem. The third way is offered by
Gasilov et al [12]. In this way the fuzzy problem is considered
to be a set of crisp problems.
In this study, we investigate a differential equation with
fuzzy boundary values. We interpret the problem as a set of
crisp problems. For linear equations, we propose a method
based on the properties of linear transformations. We show
that, if the solution of the corresponding crisp problem exists
and is unique, the fuzzy problem also has unique solution.
Moreover, we prove that if the boundary values are triangular
fuzzy numbers, then the value of the solution is a triangular
fuzzy number at each time. We explain the proposed method
on examples.
II. F UZZY B OUNDARY VALUE P ROBLEM
Below, we use the notation u
e = (uL (r), uR (r)), (0 ≤
r ≤ 1), to indicate a fuzzy number in parametric form. We
denote u = uL (0) and u = uR (0) to indicate the left and the
right limits of u
e, respectively. We represent a triangular fuzzy
number as u
e = (l, m, r), for which we have u = l and u = r.
In this paper we consider a fuzzy boundary value problem
(FBVP) with crisp linear differential equation but with fuzzy
boundary values. For clarity we consider second order differential equation:
 ′′
′
 x + a1 (t)x + a2 (t)x = f (t)
e
x(0) = A
(1)

e
x(T ) = B
We note that the coefficients of the differential equation are
not necessary constant.

e = acr + e
Let us represent the boundary values as A
a and
e
e
B = bcr + b, where acr and bcr are vectors with possibility 1
e and B;
e e
and denote the crisp parts (the vertices) of A
a and eb
denote the uncertain parts with vertices at the origin.
We split the problem (1) to following two problems:
1) Associated crisp problem (which is non-homogeneous):
 ′′
 x + a1 (t)x′ + a2 (t)x = f (t)
x(0) = acr
(2)

x(T ) = bcr
2) Homogeneous problem with fuzzy boundary values:
 ′′
 x + a1 (t)x′ + a2 (t)x = 0
x(0) = e
a
(3)

x(T ) = eb

It is easy to see that, a solution of the given problem
(1) is of the form x(t) = xcr (t) + xun (t) (crisp solution +
uncertainty). Here xcr (t) is a solution of the non-homogeneous
crisp problem (2); while xun (t) is a solution of the homogeneous problem (3) with fuzzy boundary conditions. xcr (t)
can be computed by means of analytical or numerical methods.
Hence, (1) is reduced to solving a homogeneous equation with
fuzzy boundary conditions (3). Therefore, we will investigate
how to solve this problem.
We assume the solution xun of the problem (3) be a fuzzy
e of real functions such as x(t). Each function x(t) must
set X
satisfy the differential equation and must have boundary values
a and b from the sets e
a and eb, respectively. We define the
possibility (membership) of the function x(t) to be equal to
the least possibility of its boundary values.
Mathematically, the fuzzy solution set can be defined as
follows:
e = {x(t) | x′′ + a1 (t)x′ + a2 (t)x = 0;
X
x(0) = a; x(T ) = b; a ∈ e
a; b ∈ eb}

(4)

with membership function


µXe (x(t)) = min µea (a), µeb (b)

(5)

e defined above, can be interpreted as a fuzzy
The solution X,
bunch of functions.
One can also interpret that we consider a FBVP as a set of
crisp BVPs whose boundary values belong to the fuzzy sets e
a
and eb.
A. A matrix representation of the solution in the crisp case

Here we consider crisp BVP for second order homogeneous
linear differential equation:
 ′′
 x + a1 (t)x′ + a2 (t)x = 0
x(0) = a
(6)

x(T ) = b

Let x1 (t) and x2 (t) be linear independent solutions of the
differential equation. Then the general solution is x(t) =

c1 x1 (t) + c2 x2 (t). For c1 and c2 we have the following linear
system

c1 x1 (0) + c2 x2 (0) = a
(7)
c1 x1 (T ) + c2 x2 (T ) = b
Below we obtain a matrix representation for the solution of
the BVP. We rewrite the linear system (7) in matrix form:
M c=u





a
x1 (0) x2 (0)
c1
;u=
.
where M =
;c=
b
c2
x1 (T ) x2 (T )
The solution of the linear system is


c = M −1 u

(8)

We constitute a vector-function of linear independent solutions
s(t) = [x1 (t) x2 (t)]. Then the general solution can be
rewritten in matrix form as


c1
= s(t) c
x(t) = [x1 (t) x2 (t)]
c2
Using (8) we have x(t) = s(t) M −1 u, or,
x(t) = w(t) u =w1 (t) a + w2 (t) b

(9)

w(t) = s(t) M −1

(10)

where
B. The solution method for FBVP
e
Now we show how to find X(t)
(the value of the solution
for the problem (3) at a time t).
Let linear independent solutions of the crisp equation
(3), x1 (t) and x2 (t), be known. Then we can constitute the
vector w (see, formula (10)). According (4) and (9) we have:




a
e
e
X = x(t) = w(t) u | u =
; a∈e
a; b ∈ b
(11)
b
Consider a fixed time t. Put v = w(t). Then from (11) we
have:
n
o
e
X(t)
= v u | u = [a b]T ; a ∈ e
a; b ∈ eb
(12)

e
To determine how is the set X(t)
we consider the transformation T (u) = v u (here v is a fixed vector). One can see that
e is the
T : R2 → R1 is a linear
X(t)
n transformation. Therefore, o
T
e
e
image of the set B = u = [a b] | a ∈ e
a; b ∈ b = (e
a, eb)
under the linear transformation T (u).
We remember some properties of linear transformations
[13]:
1. A linear transformation maps the origin (zero vector) to
the origin (zero vector).
2. Under a linear transformation the images of a pair of
similar figures are also similar.
3. Under a linear transformation the images of nested figures
are also nested.
In addition, we shall reference a property of fuzzy number
vectors.
e = (e
4. The fuzzy set B
a, eb) forms a fuzzy region in the
ab-coordinate plane, vertex of which is located at the origin

and boundary of which is a rectangle. Furthermore, the α-cuts
of the region are rectangles nested within one another.
The facts 1-4 allow us to derive the following conclusion.
e components of which are the boundary values
The vector B,
e
e
a and b, form a fuzzy rectangle in the ab-coordinate plane.
The linear transformation T (u) maps this fuzzy rectangle to a
fuzzy interval in R1 . Therefore, the solution at any time forms
a fuzzy number.
C. Particular case when boundary values are triangular fuzzy
numbers
In particular, if e
a and eb triangular fuzzy numbers, the αe = (e
cuts of the region B
a, eb) are nested rectangles, furthermore, they are similar. According to the discussion above,
their images are intervals that also are nested and similar,
e
consequently, form a triangular fuzzy number X(t).
Therefore,
e
e
X(t)
can be represented in the form X(t)
= (x(t), 0, x(t)).
Now we investigate how to calculate x(t) and x(t).
Let e
a = (a, 0, a), eb = (b, 0, b) and w(t) = (w1 (t), w2 (t)).
Since x(t) is the maximum value among all products
w(t)·u =aw1 (t) + bw2 (t), we have:

x(t) = max {aw1 (t), aw1 (t)} + max bw2 (t), bw2 (t)
x(t) = min {aw1 (t), aw1 (t)} + min bw2 (t), bw2 (t)
e
Note that an α-cut of X(t)
can be determined by similarity:


Xα (t) = xα (t), xα (t) = (1 − α) [x(t), x(t)] = (1 − α)X0 (t)
Formulas for x(t) and x(t) allow us to represent the solution
in a new way:
e
X(t)
= w1 (t) e
a + w2 (t) eb

E. Solution algorithm
Based on the arguments above, we propose the following
algorithm to solve the problem (1):
e = acr + e
e=
1. Represent the boundary values as A
a and B
e
bcr + b.
2. Find linear independent solutions x1 (t) and x2 (t) of
the crisp differential equation x′′ + a1 (t)x′ + a2 (t)x = 0.
Constitute the vector-function s(t) = [x1 (t) x2 (t)], the
matrix M and calculate the vector w(t) = (w1 (t), w2 (t)) by
formula (10).
3. Find the solution xcr (t) of the non-homogeneous crisp
problem (2).
4. The solution of the given problem (1) is
x
e(t) = xcr (t) + w1 (t) e
a + w2 (t) eb

(14)

III. E XAMPLES

Example 1. Solve the FBVP:
 ′′
 x − 3x′ + 2x = 4t − 6
x(0) = (1.5, 2, 3)

x(1) = ( 2, 3, 4)

(15)

Solution: We represent the solution as x
e(t) = xcr (t)+ x
eun (t).
1.
We
solve
crisp
non-homogeneous
problem
 ′′
 x − 3x′ + 2x = 4t − 6
x(0) = 2

x(1) = 3
and find the crisp solution


xcr (t) = 2t + e21−e 2(e2+t − e1+2t ) + (e2t − et ) (the thick
line in Fig.1).

where the operations assumed to be multiplication of real
number with fuzzy one, and addition of fuzzy numbers.
D. General case when boundary values are parametric fuzzy
numbers
In the general case, when e
a and eb are arbitrary fuzzy
numbers,
can be obtained
by using α-cuts.
 the solution



Let
a
=
a
and
b
=
b
.
Then
B
=
a
×
,
a
,
b
,
a
α
α
α
α
α
α
α
α
α


bα , bα . By similar argumentation to the preceding case, for
the α-cut of the solution we obtain the following formulas:


Xα (t) = xα (t), xα (t)
where



xα (t) = max aα w1 (t), aα w1 (t) +max  bα w2 (t), bα w2 (t)
xα (t) = min aα w1 (t), aα w1 (t) + min bα w2 (t), bα w2 (t)

Based on the formulas above we can conclude, that the
representation for solution
e
X(t)
= w1 (t) e
a + w2 (t) eb

(13)

is valid in general. Thus, the solution, defined by formula
(4), becomes the same as the solution obtained from (9) by
extension principle.
Remark: The approach is valid also for the general case,
when nth-order m-point boundary value problem is considered.

Fig. 1. The fuzzy solution and its α = 0.5-cut for Example 1. Dashed and
thick bars represent the values of the fuzzy solution and its α = 0.5-cut at
different times, respectively.

2. We consider fuzzy homogeneous problem

 x′′ − 3x′ + 2x = 0
x(0) = (−0.5, 0, 1)

x(1) = (−1, 0, 1)

x1 (t) = et and x2 (t) = e2t are linear independent
 t
 solutions
2t
for the differential
equation.
Then
s(t)
=
e
e
,

1 1
M=
and
e e2


w = s(t) M −1 = e21−e e2+t − e1+2t e2t − et .
The formula (13) gives the solution of homogeneous problem:
x
eun (t) =

1
2+t
e2 −e ((e

− e1+2t ) (−0.5, 0, 1)+

(e2t − et ) (−1, 0, 1))

(16)

where the arithmetic operations are considered to be fuzzy
operations. We add this solution to the crisp solution and get
the fuzzy solution of the given FBVP (15):
x
e(t) = 2t + e21−e ((e2+t − e1+2t ) (1.5, 2, 3)+
(e2t − et ) (0, 1, 2))

(17)

The fuzzy solution x
e(t) forms a band in the tx-coordinate
plane (Fig. 1). Since w1 (t) > 0 and w2 (t) > 0 for 0 < t < T ,
the upper border of the band, x(t), becomes the solution of
the crisp non-homogeneous problem with the upper boundary
values A = 3 and B = 4, while the lower border x(t)
corresponds to A = 1.5 and B = 2:
x(t)
x(t)

1
((e2+t − e1+2t ) · 3 + (e2t − et ) · 2)
e2 − e
1
((e2+t − e1+2t ) · 1.5 + (e2t − et ) · 0)
= 2t + 2
e −e
= 2t +

We can express theh solution x
e(t)i also via α-cuts, which
are intervals xα (t) = xα (t), xα (t) at any time t. Since the
boundary values are triangular fuzzy numbers, x
eun (t) also is
a triangular fuzzy number, say x
eun (t) = (xun (t), 0, xun (t)).
Consequently, an α-cut of x
eun (t) can be determined by
similarity with coefficient (1 − α), i.e.


xun, α (t) = (1 − α) xun (t), xun (t)
Adding the crisp solution gives the α-cut of the solution x
e(t):
h
i
xα (t), xα (t) = 2t+(1−α)· e21−e ((e2+t −e1+2t ) [1.5, 3] +

Fig. 2. The fuzzy solution and its α = 0.6-cut for Example 2. Dashed and
thick bars represent the values of the fuzzy solution and its α = 0.6-cut at
different times, respectively.




x′′ + 16x = 0
x(0) = ( −1, 0, 0.5)

x(2) = (−0.5, 0, 0.5)
x1 (t) = cos 4t and x2 (t) = sin 4t are linear independent
solutions for the differential equation.
Then 

1
0
s(t) = [cos 4t sin 4t], M =
and
cos 8 sin 8
1
−1
w = s(t) M = sin 8 [sin(8 − 4t) sin 4t].
Using the formula (13) we obtain the solution of homogeneous problem and adding the crisp solution we get the
solution of the given FBVP (18):
x
e(t) = 3 − 0.5t2 +

1
sin 8 (sin(8

− 4t) (−1, 0, 0.5)+

sin 4t (−0.5, 0, 0.5))

(19)

Fuzzy solution generates a band in tx-plane (Fig. 2). Unlike
Example 1, the functions w1 (t) and w2 (t) takes both positive
and negative values in the interval 0 < t < T . Because of
that, in generation of upper and lower borders of the band, a
and a, b and b take charge in alternately.

(e2t − et ) [0, 2])
In Fig. 1 we show the fuzzy solution (dashed bars) and its
α = 0.5-cut (thick bars) at different times.
Example 2. Solve the FBVP:
 ′′
 x + 16x = 47 − 8t2
x(0) = ( 2, 3, 3.5)

x(2) = (0.5, 1, 1.5)

(18)

Solution:
Associated crisp non-homogeneous problem

 x′′ + 16x = 47 − 8t2
x(0) = 3

x(2) = 1
has the solution xcr (t) = 3 − 0.5t2 (thick line in Fig. 2).
To find the uncertain part of the fuzzy solution, x
eun (t), we
solve fuzzy homogeneous problem

IV. C ONCLUSION
In this paper we have investigated the fuzzy boundary value
problem as a set of crisp problems. We have proposed a
solution method based on the properties of linear transformations. For clarity we have explained the proposed method
for second order linear differential equation. We have shown
that the fuzzy solution by our method coincides with the
solution by extension principle. We are planning to make a
comparative analysis between the proposed method and the
method including generalized Hukuhara derivative in future.
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