Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 27-38
This paper is available online at

FUZZY SOLUTIONS FOR GENERAL HYPERBOLIC PARTIAL
DIFFERENTIAL EQUATIONS WITH LOCAL INITIAL CONDITIONS

Nguyen Thi My Ha1 , Nguyen Thi Kim Son2 and Ha Thi Thanh Tam3
1

2

Faculty of Mathematics, Hai Phong University
Faculty of Mathematics, Hanoi National University of Education
3
Diem Dien High School, Thai Binh

Abstract. In this paper, we study the existence and uniqueness of fuzzy solutions
for general hyperbolic partial differential equations with local conditions making
use of the Banach fixed point theorem. Some examples are presented to illustrate
our results.
Keywords: Hyperbolic differential equations, fuzzy solution, local conditions, fixed
point.

1. Introduction
Fuzzy set theory was first introduced by Zadeh [15]. The ambition of fuzzy
set theory is to provide a formal setting for incomplete, inexact, vague and uncertain
information. Today, after its conception, fuzzy set theory has become a fashionable theory
used in many branches of real life such as dynamics, computer technology, biological
phenomena and financial forecasting, etc. The concepts of fuzzy sets, fuzzy numbers,
fuzzy metric spaces, fuzzy valued functions and the necessary calculus of fuzzy functions

have been investigated in papers [3, 7-10]. The fuzzy derivative was first introduced by
Chang and Zadeh in [5]. The study of differential equations was considerd in [12-14].
The recent results on fuzzy differential equations and inclusion was presented in the
monograph of Lakshmikantham and Mohapatra [11].
Nowadays, many fields of science can be presented using mathematical models,
especially partial differential equations. When databases that are transformed from real
life into mathematical models are incomplete or vague, we often use fuzzy partial
differential equations. Hence, more and more authors have studied solutions for fuzzy
partial differential equations. In [4], Buckley and Feuring found the existence of B-F
Received January 15, 2013. Accepted May 24, 2013.
Contact Nguyen Thi Kim Son, e-mail address:

27


Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam

solutions and Seikkala solutions for fuzzy partial differential equations by using crisp
solution and the extension principle. Some other efforts have been recently made to find
the numerical solutions for fuzzy partial differential equations by Allahviranloo [1]. With
regards to the fuzzy hyperbolic partial differential equations with local and nonlocal
initial conditions, Arara et. al. [2] used the Banach fixed point theorem to investigate
the existence and uniqueness of fuzzy solutions. However, their results depended on the
form and the size of the domain Ja × Jb . Meanwhile, it is absolutely not necessary. In
this paper, by using the ideas of a new metric in a complete metric space, we show that
fuzzy solutions of more general hyperbolic partial differential equations exist without any
condition on the domain.

2. Preliminaries
In this section, we give some basic notations, necessary concepts and results which

will be used later.
We denote the set consisting of all nonempty compact, convex subsets of Rn by KCn .
Let A and B be two nonempty bounded subsets of KCn . Denote by ||.|| a norm in Rn . The
distance between A and B is defined by the Hausdorff metric
{
}
dH (A, B) = max sup inf ||a − b|| , sup inf ||a − b||
a∈A b∈B

b∈B a∈A

and (KCn , dH ) is a complete space [11].
Let E n be the space of functions u: Rn → [0, 1] satisfying:
i) there exists a x0 ∈ Rn such that u(x0 ) = 1;
ii) u is fuzzy convex, that is for x, z ∈ Rn and 0 < λ ≤ 1,
u(λx + (1 − λ)z) ≥ min[u(x), u(z)];
iii) u is semi-continuous;
iv) [u]0 = {x ∈ Rn : u(x) > 0} is a compact set in Rn .
If u ∈ E n , u is called a fuzzy set and the α-level of u is defined by
[u]α = {x ∈ Rn : u(x) ≥ α} for each 0 < α ≤ 1.
Then from (i) to (iv), it follows that [u]α is in KCn .
The fuzzy sets u ∈ E 1 is called fuzzy numbers. The triangular fuzzy numbers are
those fuzzy sets in E 1 for which the sendograph is a triangle. A triangular fuzzy number
u is defined by three numbers a1 < a2 < a3 such that [u]0 = [a1 , a3 ], u1 = a2 . We write
u > 0 if a1 > 0, u ≥ 0 if a1 ≥ 0, u < 0 if a3 < 0 and u ≤ 0 if a3 ≤ 0. The α-level set
of a fuzzy number is presented by an ordered pair of function [u1 (α), u2 (α)], 0 < α < 1
which satisfies the following requirements:
28

Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

i) u1 (α) is a bounded left continuous non-decreasing function of α,
ii) u2 (α) is a bounded left continuous non-increasing function of α,
iii) u1 (1) ≤ u2 (1).
If g : Rn × Rn → Rn is a function, then, according to Zadeh’s extension principle
we can extend g to E n × E n → E n by the function defined by
g(u, u)(z) = sup min{u(x), u(z)}.
z=g(x,z)

If g is continuous then
[g (u, u)]α = g ([u]α , [u]α )
for all u, u ∈ Rn , 0 ≤ α ≤ 1.
Especially, we will define addition and scalar multiplication of fuzzy sets in E n
levelsetwises, that is, for all u, u ∈ Rn , 0 ≤ α ≤ 1, k ∈ R, k ̸= 0
[u + u]α = [u]α + [u]α
and
[ky]α = k [u]α ,
where
(u + u)(x) =

sup min{u(x1 ), u(x2 )}
x=x1 +x2

and
ku(x) = u(x/k).
Supremum metric is the most commonly used metric on E n defined by the
Hausdorff metric distance between the level sets of the fuzzy sets
d∞ (u, u) = sup Hd ([u]α , [u]α )
0<α≤1

for all u, u ∈ E n . It is obvious that (E n , d∞ ) is a complete metric space [11]. From the
properties of Hausdorff metric, we have:
i) d∞ (cu, cv) = |c|.d∞ (u, v),
ii) d∞ (u + u′ , v + v ′ ) ≤ d∞ (u, v) + d∞ (u′ , v ′ ),
iii) d∞ (u + w, v + w) = d∞ (u, v)
for all u, v, u′ , v ′ , w ∈ E n and c ∈ R.
29


Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam

Definition 2.1. Let Ja = [0, a], Jb = [0, b], Jab = [0, a] × [0, b]. A map f : Jab → E n
is called continuous at (t0 , s0 ) ∈ Jab if the multi-valued map fα (t, s) = [f (t, s)]α is
continuous at (t0 , s0 ) with respect to the Hausdorff metric dH for all α ∈ [0, 1].
In this paper, we denote C(Jab , E n ) be a space of all continuous functions f : J →
E with the supremum weighted metric H1 defined by
n

H1 (f, g) = sup d∞ (f (t, s) , g (t, s))e−λ(t+s) .
(s,t)∈J

Since (E n , d∞ ) is a complete metric space, it can shown that (C(Jab , E n ), H1 ) is
also a complete metric space (see [11]).
Definition 2.2. A map f : Jab × E n → E n is called continuous at point (t0 , s0 , x0 ) ∈
Jab × E n provided, for any fixed α ∈ [0, 1] and arbitrary ϵ > 0, there exists δ(ϵ, α) > 0
such that
dH ([f (t, s, x)]α , [f (t0 , s0 , x0 )]α ) < ϵ
whenever max |t − t0 | , |s − s0 | < δ(ϵ, α) and dH ([x]α , [x0 ]α ) < δ(ϵ, α) for all (t, s, x) ∈
Jab × E n .

Definition 2.3. A function f : Jab → E n is called integrably bounded if there exists an
integrable function h ∈ L1 (J, Rn ) such that || y || ≤ h (s, t) for all y ∈ f0 (s, t).
∫ a∫ b
Definition 2.4. Let f : Jab → E n . The integral of f over Jab , denoted by 0 0 f (t, s) dsdt
is defined by
)α ∫ a ∫ b
(∫ a ∫ b
f (t, s) dsdt
=
fα (t, s) dsdt
0
0
0
0
∫ a∫ b
={
v (t, s) dsdt|v : Jab → Rn is a measurable
0

0

selection for fα }
for all α ∈ (0, 1] (see [3]). A function f : Jab → E n is integrable on Jab if
∫ a∫ b
f (t, s) dsdt is in E n .
0 0
Let f, g : Jab → E n be integrable and λ ∈ R. The intergral has the elementary
properties as follows
∫ a∫ b
∫ a∫ b

∫ a∫ b
i) 0 0 [f (t, s) + g(t, s)]dsdt = 0 0 f (t, s)dsdt + 0 0 g(t, s)dsdt,
∫ a∫ b
∫ a∫ b
ii) 0 0 λf (t, s)dsdt = λ 0 0 f (t, s)dsdt,
∫ a∫ b
∫ a∫ b
∫ a∫ b
ii) d∞ ( 0 0 f (t, s)dsdt, 0 0 (t, s)dsdt) ≤ 0 0 d∞ (f (t, s), g(t, s))dsdt.
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Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

Definition 2.5. Let x, y ∈ E n . If there exists z ∈ E n such that x = y + z then we call z
the Hukuhara-difference of x and y, denoted x − y (see [11]).
The definition of the fuzzy partial derivative is one of the most important concepts
for fuzzy partial differential equation.
Definition 2.6. Let f : Jab → E n . The fuzzy partial derivative of f with respect to x at
∂f (x0 , y0 )
the point (x0 , y0 ) ∈ J is a fuzzy set
∈ E n which is defined by
∂x
∂f (x0 , y0 )
f (x0 + h, y0 ) − f (x0 , y0 )
= lim
.
h→0
∂x
h

Here the limit is taken in the metric space (E n , d∞ ) and u − v is the Hukuhara-difference
of u and v in E n . The fuzzy partial derivative of f with respect to y and the higher order
of fuzzy partial derivative of f at the point (x0 , y0 ) ∈ Jab are defined similarly (see [6]).

3. Main result
The aim of this section is to consider the existence and uniqueness of the fuzzy
solutions for the general hyperbolic partial differential equation
∂ 2 u(x, y) ∂(p1 (x, y)u(x, y)) ∂(p2 (x, y)u(x, y))
+
+
+ c(x, y)u(x, y) = f (x, y, u(x, y))
∂x∂y
∂x
∂y
(3.1)
for (x, y) ∈ Jab . The local initial conditions are
u(0, 0) = u0 , u(x, 0) = η1 (x), u(0, y) = η2 (y), (x, y) ∈ Jab ,

(3.2)

where pi ∈ C(Jab , R); i = 1, 2, c ∈ C(Jab , R), η1 ∈ C(Ja , E n ), η2 ∈ C(Jb , E n ) are
given functions; u0 ∈ E n and f : Jab × C(Jab , E n ) → E n which satisfies the following
hypothesis:
Hypothesis H
There exists K > 0 such that
d∞ (f (x, y, u(x, y)), f (x, y, u(x, y))) ≤ Kd∞ (u(x, y), u(x, y))
holds for all u, u ∈ E n , (x, y) ∈ Jab .
Definition 3.1. A function u ∈ C(Jab , E n ) is called a solution of the problem (3.1), (3.2)
if it satisfies
∫ y

∫ x
u(x, y) =q1 (x, y) −
p1 (x, s)u(x, s)ds −
p2 (t, y)u(t, y)dt
0
0
∫ x∫ y
∫ x∫ y
−
c(t, s)u(t, s)dsdt +
f (t, s, u(t, s)) dsdt,
0

0

0

0

31


Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam

where

∫
q1 (x, y) = η1 (x) + η2 (y) − u0 +

∫

y

p1 (0, s)η2 (s)ds +
0

x

p2 (t, 0)η1 (t)dt
0

for all (x, y) ∈ Jab .
By using the new weighted metric H1 in the space C(Jab , E n ), we receive the
following result about the existence and uniqueness of solutions of the problem.
Theorem 3.1. Assume that hypothesis H is satisfied. Then the problem (3.1), (3.2) has a
unique solution in C(Jab , E n ) .
Proof. Let p1 = sup(t,s)∈Ja ×Jb |p1 (t, s)|, p2 = sup(t,s)∈Ja ×Jb |p2 (t, s)|, c =
sup(t,s)∈Ja ×Jb |c(t, s)|. From Definition 3.1 for a fuzzy solution, we relize that the fuzzy
solution of the problem (3.1), (3.2) (if it exists) is a fixed point of the operator N :
C(Jab , E n ) → C(Jab , E n ) defined as follows:
∫ y
∫ x
N (u)(x, y) =q1 (x, y) −
p1 (x, s)u(x, s)ds −
p2 (t, y)u(t, y)dt
0
0
∫ x∫ y
∫ x∫ y
−

c(t, s)u(t, s)dsdt +
f (t, s, u(t, s)) dsdt.
0

0

0

0

We will show that N is a contraction operator. Indeed, if u, u ∈ C(Jab , E n ) and α ∈ (0, 1]
then
∫ y
∫ x
N (u(x, y)) =q1 (x, y) −
p1 (x, t)u(x, t)dt −
p2 (s, y)u(s, y)dx
0
0
∫ x∫ y
∫ x∫ y
−
c(t, s)u(t, s)dsdt +
f (t, s, u(t, s)) dsdt
0

0

0

and

∫

0

∫

y

x

N (u(x, y)) =q1 (x, y) −
p1 (x, t)u(x, y)dt −
p2 (s, y)u(s, y)ds
0
0
∫ x∫ y
∫ x∫ y
−
c(t, s)u(t, s)dsdt +
f (t, s, u(t, s)) dsdt.
0

0

0

0

From the properties of supremum metric, we have:
∫ y
∫ y
d∞ (N (u(x, y)),N (u(x, y))) ≤ d∞ (
p1 (x, s)u(x, s)ds,
p1 (x, s)u(x, s)ds)
0
0
∫ x
∫ x
+ d∞ (
p2 (t, y)u(t, y)dt,
p2 (t, y)u(t, y)dt)
0
0
∫ x∫ y
∫ x∫ y
c(t, s)u(t, s)dsdt)
c(t, s)u(t, s)dsdt,
+ d∞ (
0
0
0
0
∫ x∫ y
∫ x∫ y
+ d∞ (
f (t, s, u(t, s))dtds,
f (t, s, u(t, s))dsdt).
0

32

0

0

0


Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

Moreover

∫
d∞ (

∫

y

∫ 0y

0

≤

y

p1 (x, s)u(x, s)ds,

sup

(t,s)∈Ja ×Jb
∫ y

≤ p1

|p1 (t, s)|d∞ (

p1 (x, s)u(x, s)ds)
∫ y
u(x, s)ds,
u(x, s)ds)

0

0

d∞ (u(x, s), u(x, s))ds.
0

Hence for each (x, y) ∈ Ja × Jb , one gets
∫ y
∫ y
−λ(x+y)
e
d∞ (
p1 (x, s)u(x, s)ds,
p1 (x, s)u(x, s)ds)
0

0
∫ y
−λ(x+y)
≤ p1 e
d∞ (u(x, s), u(x, s))e−λ(x+s) eλ(x+s) ds
0
∫ y
−λ(x+y)
eλ(x+s) ds
≤ p1 H1 (u, u)e
0

p1
≤ H1 (u, u).
λ
Similarly, we obtain:
e

−λ(x+y)

∫
d∞ (

∫

x
0

Nevertheless

∫
d∞ (

x

x

p2 (t, y)u(t, y)dt) ≤

p2 (t, y)u(t, y)dt,
0

∫

∫

y

x

∫

p2
H1 (u, u).
λ

y

c(s, t)u(s, t)dsdt,
c(s, t)u(s, t)dsdt)

0
0
0
0
∫ x∫ y
≤ sup |c(t, s)|
d∞ (u(s, t), u(s, t))dsdt
(t,s)∈Ja ×Jb
0
0
∫ x∫ y
≤c
d∞ (u(s, t), u(s, t))dsdt.
0

Hence

0

∫ x∫ y
∫ x∫ y
c(t, s)u(t, s)dsdt)
c(t, s)u(t, s)dtds,
e
d∞ (
0
0
∫0 x ∫0 y
d∞ (u(s, t), u(s, t))e−λ(t+s) eλ(t+s) dsdt
≤ ce−λ(x+y)

0
0
∫ x∫ y
−λ(x+y)
eλ(t+s) dsdt
≤ cH2 (u, u)e
−λ(x+y)

c
≤ 2 H1 (u, u).
λ

0

0

33


Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam

Moreover, one gets
∫
d∞ (

x

∫

∫

y

x

∫

y

f (t, s, u(t, s))dtds,
0
x∫ y

∫

0

≤K

f (t, s, u(t, s))dsdt)
0

0

d∞ (u(t, s), u(t, s))dsdt.
0

0

It implies that

e

−λ(x+y)

∫

x

∫

∫

y

x

∫

∫

0

0

0

d∞ (u(x, y), u(x, y))e−λ(t+s) eλ(t+s) dsdt
0
0
∫ x∫ y

−λ(x+y)
≤ KH1 (u, u)e
eλ(t+s) dsdt
≤ Ke

−λ(x+y)

0
x∫ y

y

f (t, s, u(t, s))dsdt)

f (t, s, u(t, s))dxdy,

d∞ (

0

0

K
≤ 2 H1 (u, u).
λ
That shows
H1 (N (u(x, y)), N (u(x, y))) ≤ [

p1 + p2 c + K
+

]H1 (u, u).
λ
λ2

Since we can choose λ > 0 satisfying
p1 + p2 c + K
+
< 1,
λ
λ2
we receive N which is a contraction operator and by the Bannach fixed point theorem,
N has a unique fixed point, that is a solution of the problem (3.1) - (3.2). The proof is
completed.

4. Examples
Example 4.1. The hyperbolic equation has the form
∂ 2 u(x, y)
= −C1 , (x, y) ∈ [0, 1] × [0, 1]
∂x∂y

(4.1)

with the local conditions
u(x, 0) = u(0, y) = u(0, 0) = C2 ,

(4.2)

where C1 , C2 are triangular fuzzy numbers in [0, M ], M > 0 with the following level sets
α
α

α
α
[C1 ]α = [C11
, C12
], [C2 ]α = [C21
, C22
]

34


Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

for α ∈ [0, 1] and (x, y) ∈ [0, 1] × [0, 1].
In this problem, if f (x, y, u(x, y)) = −C1 then condition (H) is satisfied with K = 2.
Therefore, from Theorem 3.1 there exists a solution to this problem.
Next, we find this fuzzy solution. Assume that solution u has level sets [u]α =
[u1 (x, y)α , u2 (x, y)α ] for α ∈ [0, 1] and (x, y) ∈ [0, 1] × [0, 1]. We also have
[

∂ 2 u(x, y) α
∂ 2 uα1 (x, y) ∂ 2 uα1 (x, y)
] =[
,
].
∂x∂y
∂x∂y
∂x∂y

Applying the extension principle, the fuzzy number −C1 has level sets

α
α
α
α
[−C1 ]α = [min {−C11
, −C12
} , max {−C11
, −C12
}]
α
α
= [−C12 , −C11 ]

for α ∈ [0, 1] and (x, y) ∈ [0, 1]×[0, 1]. Thus the equation (4.1) is equivalent to the system
2 α
∂ 2 uα1 (x, y)
α ∂ u2 (x, y)
α
= −C12
,
= −C11
.
∂x∂y
∂x∂y

(4.3)

The local conditions (4.2) is equivalent to the following system
α
uα1 (x, 0) = uα1 (0, y) = uα1 (0, 0) = C21

,

(4.4)

α
uα2 (x, 0) = uα2 (0, y) = uα2 (0, 0) = C22
.

(4.5)

The solutions of system (4.3) with conditions (4.4), (4.5) are
α
α
α
α
uα1 (x, y) = −C12
xy + C21
, uα2 (x, y) = −C11
xy + C22
.

Hence, the solution of problem (4.1), (4.2) has level sets
α
α
α
α
[u]α = [−C12
xy + C21
, −C11
xy + C22

]

for α ∈ [0, 1] and (x, y) ∈ [0, 1] × [0, 1]. We can write u(x, y) = −C1 xy + C2 .
Example 4.2. Consider the fuzzy hyperbolic equation
∂ 2 u(x, y) ∂u(x, y) ∂u(x, y)
+
+
+ u(x, y) = 4Cex+y , (x, y) ∈ [0, 2] × [0, 2],
∂x∂y
∂x
∂y

(4.6)

with the local conditions
u(x, 0) = Cex , u(0, y) = Cey , u(0, 0) = C,

(4.7)

where C is a fuzzy triangular number in [0, M ], M > 0. C has level sets [C]α = [C1α , C2α ]
for α ∈ [0, 1].
We have f (x, y, u(x, y)) = 4Cex+y then f satisfies condition (H) with K = 1. Hence, the
35


Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam

condition of Theorem 3.1 holds. Therefore, there exists a fuzzy solution of this problem.
Next, we will give a clear solution. Suppose that solution u has level sets [u]α =
[u1 (x, y)α , u2 (x, y)α ] for α ∈ [0, 1] and (x, y) ∈ [0, 2] × [0, 2]. Define

φ(Dx , Dy )U (x, y) =

∂ 2 u(x, y) ∂u(x, y) ∂u(x, y)
+
+
+ u(x, y)
∂x∂y
∂x
∂y

then φ(Dx , Dy )U (x, y) also has level sets
∂ 2 uα1 (x, y) ∂uα1 (x, y) ∂uα1 (x, y)
+
+
+ uα1 (x, y),
∂x∂y
∂x
∂y
∂ 2 uα2 (x, y) ∂uα2 (x, y) ∂uα2 (x, y)
+
+
+ uα2 (x, y)]
∂x∂y
∂x
∂y

[φ(Dx , Dy )U (x, y)]α =[

for all α ∈ [0, 1] and (x, y) ∈ [0, 2] × [0, 2]. By the extension principle, we have
[4Cex+y ]α = [min{4C1α ex+y , 4C2α ex+y }, max{4C1α ex+y , 4C2α ex+y }]

= [4C1α ex+y , 4C2α ex+y ]
Similarly
[Cex ]α = [C1α ex , C2α ex ], [Cey ]α = [C1α ey , C2α ey ]
for all α ∈ [0, 1] and (x, y) ∈ [0, 2] × [0, 2]. Hence, the equation (4.6) is equivalent to the
system
∂ 2 uα1 (x, y) ∂uα1 (x, y) ∂uα1 (x, y)
+
+
+ uα1 (x, y) = 4C1α ex+y ,
(4.8)
∂x∂y
∂x
∂y
∂ 2 uα2 (x, y) ∂uα2 (x, y) ∂uα2 (x, y)
+
+
+ uα2 (x, y) = 4C2α ex+y .
∂x∂y
∂x
∂y

(4.9)

The local conditions (4.7) are equivalent to the following
uα1 (x, 0) = 4C1α ex , uα1 (0, y) = 4C1α ex , uα1 (0, 0) = 4C1α ,

(4.10)

uα2 (x, 0) = 4C2α ex , uα1 (0, y) = 4C2α ex , uα2 (0, 0) = 4C2α .

(4.11)

Solving the problems (4.8), (4.10) and (4.9), (4.11) we have the solutions:
uα1 (x, y) = 4C1α ex+y , uα2 (x, y) = 4C2α ex+y
for all α ∈ [0, 1] and (x, y) ∈ [0, 2] × [0, 2].
Thus, the solution of problem (4.6), (4.7) is a fuzzy function u, which has level sets
[u]α = [4C1α ex+y , 4C2α ex+y ]
and we can write u = 4Cex+y .
36


Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

Example 4.3. We study the following hyperbolic equation:
∂ 2 u(x, y) ∂u(x, y)
1
1
−
= 2C1 − 2C1 (x + y), (x, y) ∈ [0, ] × [0, ],
∂x∂y
∂x
8
8

(4.12)

u(x, 0) = C1 x2 + C3 , u(0, y) = C1 y 2 + C2 siny + C3 , u(0, 0) = C3 ,

(4.13)

with (x, y) ∈ [0, 18 ] × [0, 18 ] and Ci being triangular fuzzy numbers having level sets
α
α
[Ci ]α = [Ci1
, Ci2
] for i = 1, 3, α ∈ [0, 1] and (x, y) ∈ [0, 81 ] × [0, 81 ].
We have f (x, y, u) = C1 y 2 + C2 siny + C3 . It follows that the condition in Theorem 3.1 is
satisfied with K = 1. Therefore there exists a solution of this problem.
Suppose that solution u has level sets [u]α = [u1 (x, y)α , u2 (x, y)α ] for α ∈ [0, 1] and
(x, y) ∈ [0, 18 ] × [0, 18 ]. Using the extension principle, we have
[

∂ 2 u(x, y) ∂u(x, y) α
∂ 2 uα1 (x, y) ∂uα2 (x, y) ∂ 2 uα2 (x, y) ∂uα1 (x, y)
−
] =[
−
,
−
]
∂x∂y
∂x
∂x∂y
∂x
∂x∂y
∂x

and
α
α

α
α
[2C1 − 2C1 (x + y)]α = [2C11
− 2C12
(x + y)], 2C12
− 2C11
(x + y)]
α 2
α
α 2
α
[C1 x2 + C3 ]α = [C11
x + C31
, C12
x + C32
],
α 2
α
α
α 2
α
α
[C1 y 2 + C2 siny + C3 ]α = [C11
y + C21
siny + C31
, C12
y + C22
siny + C32
].

Thus equation (4.12) is equivalent to the following system
∂ 2 uα1 (x, y) ∂uα2 (x, y)
−
= 2C11 − 2C12 (x + y),
∂x∂y
∂x
∂ 2 uα2 (x, y) ∂uα1 (x, y)
α
α
−
= 2C12
− 2C11
(x + y).
∂x∂y
∂x
The initial conditions (4.13) are equivalent to the system
α 2
α
α 2
α
α
α
uα1 (x, 0) = C11
x + C31
, uα1 (0, y) = C11
y + C21
siny + C31
, uα1 (0, 0) = C31

and

α 2
α
α 2
α
α
α
uα2 (x, 0) = C12
x + C32
, uα1 (0, y) = C12
y + C22
siny + C32
, uα1 (0, 0) = C32
.

The solutions of the system are given by
α
α
α
uα1 = C11
(x + y)2 + C21
siny + C31

and
α
α
α
.
siny + C32
(x + y)2 + C22
uα2 = C12

Therefore, the problem (4.12), (4.13) has a solution u(x, y) = C1 (x + y)2 + C2 siny + C3
with level sets
α
α
α
α
α
α
].
siny + C32
(x + y)2 + C22
, C12
siny + C31
(x + y)2 + C21
[u]α = [C11

37


Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam

5. Conclusion
We have investigated the existence and uniqueness of the fuzzy solution for
the general hyperbolic partial differential equation with local conditions.This result is
illustrated by some examples. The next step in the direction proposed here is to study
the fuzzy solution for the general hyperbolic partial differential equation with nonlocal
conditions and integral boundary conditions.
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