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

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.

This paper is available online at http://stdb.hnue.edu.vn

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

1Faculty of Mathematics, Hai Phong University

2Faculty of Mathematics, Hanoi National University of Education

3Diem 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

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: mt02_02@yahoo.com

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 J a × J b 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

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 ofRnbyK n

C

Let A and B be two nonempty bounded subsets of K n

C Denote by||.|| a norm in R n The

distance between A and B is defined by the Hausdorff metric

d H (A, B) = max

{ sup

a ∈A binf∈B ||a − b|| , sup

b ∈B ainf∈A ||a − b||

}

and (K n

C , d H) is a complete space [11]

Let E n be the space of functions u:Rn → [0, 1] satisfying:

i) there exists a x0 ∈ R n such that u(x0) = 1;

ii) u is fuzzy convex, that is for x, z ∈ R n and 0 < λ ≤ 1,

u(λx + (1 − λ)z) ≥ min[u(x), u(z)];

iii) u is semi-continuous;

iv) [u]0 ={x ∈ R n : u(x) > 0 } is a compact set in R n

If u ∈ E n , u is called a fuzzy set and the α-level of u is defined by

[u] α ={x ∈ R n : u(x) ≥ α} for each 0 < α ≤ 1.

Then from (i) to (iv), it follows that [u] αis inK n

C

The fuzzy sets u ∈ E1 is called fuzzy numbers The triangular fuzzy numbers are

those fuzzy sets in E1 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:

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 × R n → R n is a function, then, according to Zadeh’s extension principle

we can extend g to E n × E n → E nby the function defined by

g(u, u)(z) = sup

z=g(x,z)

min{u(x), u(z)}.

If g is continuous then

[g (u, u)] α = g ([u] α , [u] α)

for all u, u ∈ R n, 0≤ α ≤ 1.

Especially, we will define addition and scalar multiplication of fuzzy sets in E n levelsetwises, that is, for all u, u ∈ R n, 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

x=x1+x2

min{u(x1), u(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

0<α ≤1 H d ([u]

α

, [u] α)

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.

Definition 2.1 Let J a = [0, a], J b = [0, b], J ab = [0, a] × [0, b] A map f : J ab → E n

is called continuous at (t0, s0) ∈ J ab if the multi-valued map f α (t, s) = [f (t, s)] α is continuous at (t0, s0) with respect to the Hausdorff metric d H for all α ∈ [0, 1].

In this paper, we denote C(J ab , E n ) be a space of all continuous functions f : J →

E n with the supremum weighted metric H1defined by

H1(f, g) = sup

(s,t) ∈J

d ∞ (f (t, s) , g (t, s))e −λ(t+s)

Since (E n , d ∞ ) is a complete metric space, it can shown that (C(J ab , E n ), H1) is also a complete metric space (see [11])

Definition 2.2 A map f : J ab × E n → E n is called continuous at point (t0, s0, x0) ∈

J ab × E n provided, for any fixed α ∈ [0, 1] and arbitrary ϵ > 0, there exists δ(ϵ, α) > 0 such that

d H ([f (t, s, x)] α , [f (t0, s0, x0)]α ) < ϵ

whenever max |t − t0| , |s − s0| < δ(ϵ, α) and d H ([x] α , [x0]α ) < δ(ϵ, α) for all (t, s, x) ∈

J ab × E n

Definition 2.3 A function f : J ab → 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).

Definition 2.4 Let f : J ab → E n The integral of f over J ab , denoted by∫a

0

∫b

0f (t, s) dsdt

is defined by

(∫ a

0

∫ b

0

f (t, s) dsdt

)α

=

∫ a

0

∫ b

0

f α (t, s) dsdt

={

∫ a

0

∫ b

0

v (t, s) dsdt |v : J ab → R n

is a measurable selection for f α }

for all α ∈ (0, 1] (see [3]) A function f : J ab → E n is integrable on J ab if

∫a

0

∫b

0f (t, s) dsdt is in E n

Let f, g : J ab → E n be integrable and λ ∈ R The intergral has the elementary

properties as follows

i) ∫a

0

∫b

0[f (t, s) + g(t, s)]dsdt =∫a

0

∫b

0f (t, s)dsdt +∫a

0

∫b

0g(t, s)dsdt,

ii) ∫a

0

∫b

0λf (t, s)dsdt = λ∫a

0

∫b

0f (t, s)dsdt,

ii) d ∞(∫a

0

∫b

0f (t, s)dsdt,∫a

0

∫b

0(t, s)dsdt) ≤∫a

0

∫b

0d ∞ (f (t, s), g(t, s))dsdt.

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 : J ab → E n The fuzzy partial derivative of f with respect to x at the point (x0, y0)∈ J is a fuzzy set ∂f (x0, y0)

∂x ∈ E n which is defined by

∂f (x0, y0)

∂x = limh →0

f (x0+ h, y0)− f (x0, y0)

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)∈ J ab are defined similarly (see [6]).

The aim of this section is to consider the existence and uniqueness of the fuzzy solutions for the general hyperbolic partial differential equation

∂2u(x, y)

∂x∂y +

∂(p1(x, y)u(x, y))

∂(p2(x, y)u(x, y))

∂y + c(x, y)u(x, y) = f (x, y, u(x, y))

(3.1)

for (x, y) ∈ J ab The local initial conditions are

u(0, 0) = u0, u(x, 0) = η1(x), u(0, y) = η2(y), (x, y) ∈ J ab , (3.2)

where p i ∈ C(J ab , R); i = 1, 2, c ∈ C(J ab , R), η1 ∈ C(J a , E n ), η2 ∈ C(J b , E n) are

given functions; u0 ∈ E n and f : J ab × C(J ab , 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) ∈ J ab

Definition 3.1 A function u ∈ C(J ab , E n ) is called a solution of the problem (3.1), (3.2)

if it satisfies

u(x, y) =q1(x, y) −

∫ y

0

p1(x, s)u(x, s)ds −

∫ x

0

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

−

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

q1(x, y) = η1(x) + η2(y) − u0+

∫ y

0

p1(0, s)η2(s)ds +

∫ x

0

p2(t, 0)η1(t)dt

for all (x, y) ∈ J ab

By using the new weighted metric H1 in the space C(J ab , 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(J ab , E n )

Proof Let p1 = sup(t,s) ∈J a ×J

b |p1(t, s) |, p2 = sup(t,s) ∈J a ×J

b |p2(t, s) |, c =

sup(t,s) ∈J a ×J b |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(J ab , E n)→ C(J ab , E n) defined as follows:

N (u)(x, y) =q1(x, y) −

∫ y

0

p1(x, s)u(x, s)ds −

∫ x

0

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

−

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

We will show that N is a contraction operator Indeed, if u, u ∈ C(J ab , E n ) and α ∈ (0, 1]

then

N (u(x, y)) =q1(x, y) −

∫ y

0

p1(x, t)u(x, t)dt −

∫ x

0

p2(s, y)u(s, y)dx

−

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

and

N (u(x, y)) =q1(x, y) −

∫ y

0

p1(x, t)u(x, y)dt −

∫ x

0

p2(s, y)u(s, y)ds

−

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

From the properties of supremum metric, we have:

d ∞ (N (u(x, y)),N (u(x, y))) ≤ d ∞(

∫ y

0

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

∫ y

0

p1(x, s)u(x, s)ds) + d ∞(

∫ x

0

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

∫ x

0

p2(t, y)u(t, y)dt) + d ∞(

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

+ d ∞(

∫ x

0

∫ y

0

f (t, s, u(t, s))dtds,

∫ x

0

∫ y

0

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

d ∞(

∫ y

0

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

∫ y

0

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

(t,s) ∈J a ×J b

|p1(t, s) |d ∞(

∫ y

0

u(x, s)ds,

∫ y

0

u(x, s)ds)

≤ p1

∫ y

0

d ∞ (u(x, s), u(x, s))ds.

Hence for each (x, y) ∈ J a × J b, one gets

e −λ(x+y) d ∞(

∫ y

0

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

∫ y

0

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

≤ p1e −λ(x+y)

∫ y

0

d ∞ (u(x, s), u(x, s))e −λ(x+s) e λ(x+s) ds

≤ p1H1(u, u)e −λ(x+y)

∫ y

0

e λ(x+s) ds

≤ p1

λ H1(u, u).

Similarly, we obtain:

e −λ(x+y) d ∞(

∫ x

0

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

∫ x

0

p2(t, y)u(t, y)dt) ≤ p2

λ H1(u, u).

Nevertheless

d ∞(

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

(t,s) ∈J a ×J b

|c(t, s)|

∫ x

0

∫ y

0

d ∞ (u(s, t), u(s, t))dsdt

≤ c

∫ x

0

∫ y

0

d ∞ (u(s, t), u(s, t))dsdt.

Hence

e −λ(x+y) d ∞(

∫ x

0

∫ y

0

c(t, s)u(t, s)dtds,

∫ x

0

∫ y

0

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

≤ ce −λ(x+y)∫ x

0

∫ y

0

d ∞ (u(s, t), u(s, t))e −λ(t+s) e λ(t+s) dsdt

≤ cH2(u, u)e −λ(x+y)

∫ x

0

∫ y

0

e λ(t+s) dsdt

≤ c

λ2H1(u, u).

Moreover, one gets

d ∞(

∫ x

0

∫ y

0

f (t, s, u(t, s))dtds,

∫ x

0

∫ y

0

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

≤ K

∫ x

0

∫ y

0

d ∞ (u(t, s), u(t, s))dsdt.

It implies that

e −λ(x+y) d ∞(

∫ x

0

∫ y

0

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

∫ x

0

∫ y

0

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

≤ Ke −λ(x+y)∫ x

0

∫ y

0

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

≤ KH1(u, u)e −λ(x+y)

∫ x

0

∫ y

0

e λ(t+s) dsdt

≤ K

λ2H1(u, u).

That shows

H1(N (u(x, y)), N (u(x, y))) ≤ [ p1+ p2

c + K

λ2 ]H1(u, u).

Since we can choose λ > 0 satisfying

p1+ p2

c + K

λ2 < 1,

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

Example 4.1 The hyperbolic equation has the form

∂2u(x, y)

∂x∂y =−C1, (x, y) ∈ [0, 1] × [0, 1] (4.1)

with the local conditions

u(x, 0) = u(0, y) = u(0, 0) = C2, (4.2)

where C1, C2are triangular fuzzy numbers in [0, M ], M > 0 with the following level sets

[C1]α = [C11α , C12α ], [C2]α = [C21α , C22α]

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

[∂

2u(x, y)

∂x∂y ]

α

= [∂

2u α

1(x, y)

∂x∂y ,

∂2u α

1(x, y)

∂x∂y ].

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

[−C1]α = [min{−C α

11, −C α

12} , max {−C α

11, −C α

12}]

= [−C α

12, −C α

11]

for α ∈ [0, 1] and (x, y) ∈ [0, 1]×[0, 1] Thus the equation (4.1) is equivalent to the system

∂2u α1(x, y)

∂x∂y =−C α

12, ∂

2u α2(x, y)

∂x∂y =−C α

The local conditions (4.2) is equivalent to the following system

u α1(x, 0) = u α1(0, y) = u α1(0, 0) = C21α , (4.4)

u α2(x, 0) = u α2(0, y) = u α2(0, 0) = C22α (4.5)

The solutions of system (4.3) with conditions (4.4), (4.5) are

u α1(x, y) = −C α

12xy + C21α , u α2(x, y) = −C α

11xy + C22α Hence, the solution of problem (4.1), (4.2) has level sets

[u] α = [−C α

12xy + C21α , −C α

11xy + C22α]

for α ∈ [0, 1] and (x, y) ∈ [0, 1] × [0, 1] We can write u(x, y) = −C1xy + C2.

Example 4.2 Consider the fuzzy hyperbolic equation

∂2u(x, y)

∂x∂y +

∂u(x, y)

∂u(x, y)

∂y + u(x, y) = 4Ce

x+y , (x, y) ∈ [0, 2] × [0, 2], (4.6)

with the local conditions

u(x, 0) = Ce x , u(0, y) = Ce y , u(0, 0) = C, (4.7)

where C is a fuzzy triangular number in [0, M ], M > 0 C has level sets [C] α = [C α

1, C α

2]

for α ∈ [0, 1].

We have f (x, y, u(x, y)) = 4Ce x+y then f satisfies condition (H) with K = 1 Hence, the

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(D x , D y )U (x, y) = ∂

2u(x, y)

∂x∂y +

∂u(x, y)

∂u(x, y)

∂y + u(x, y) then φ(D x , D y )U (x, y) also has level sets

[φ(D x , D y )U (x, y)] α =[∂

2u α1(x, y)

∂x∂y +

∂u α1(x, y)

∂u α1(x, y)

∂y + u

α

1(x, y),

∂2u α

2(x, y)

∂x∂y +

∂u α

2(x, y)

∂u α

2(x, y)

∂y + u

α

2(x, y)]

for all α ∈ [0, 1] and (x, y) ∈ [0, 2] × [0, 2] By the extension principle, we have

[4Ce x+y]α = [min{4C α

1e x+y , 4C2α e x+y }, max{4C α

1e x+y , 4C2α e x+y }]

= [4C1α e x+y , 4C2α e x+y]

Similarly

[Ce x]α = [C1α e x , C2α e x ], [Ce y]α = [C1α e y , C2α e y]

for all α ∈ [0, 1] and (x, y) ∈ [0, 2] × [0, 2] Hence, the equation (4.6) is equivalent to the system

∂2u α

1(x, y)

∂x∂y +

∂u α

1(x, y)

∂u α

1(x, y)

∂y + u

α

1(x, y) = 4C1α e x+y , (4.8)

∂2u α

2(x, y)

∂x∂y +

∂u α

2(x, y)

∂u α

2(x, y)

∂y + u

α

2(x, y) = 4C2α e x+y (4.9)

The local conditions (4.7) are equivalent to the following

u α1(x, 0) = 4C1α e x , u α1(0, y) = 4C1α e x , u α1(0, 0) = 4C1α , (4.10)

u α2(x, 0) = 4C2α e x , u α1(0, y) = 4C2α e x , u α2(0, 0) = 4C2α (4.11)

Solving the problems (4.8), (4.10) and (4.9), (4.11) we have the solutions:

u α1(x, y) = 4C1α e x+y , u α2(x, y) = 4C2α e x+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 e x+y , 4C2α e x+y]

and we can write u = 4Ce x+y