DSpace at VNU: The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations

DSpace at VNU: The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations tài liệu, g...

Fuzzy Optim Decis Making

DOI 10.1007/s10700-014-9186-0

The existence and uniqueness of fuzzy solutions

for hyperbolic partial differential equations

Hoang Viet Long · Nguyen Thi Kim Son ·

Nguyen Thi My Ha · Le Hoang Son

© Springer Science+Business Media New York 2014

Abstract Fuzzy hyperbolic partial differential equation, one kind of uncertain

dif-ferential equations, is a very important field of study not only in theory but also inapplication This paper provides a theoretical foundation of numerical solution meth-ods for fuzzy hyperbolic equations by considering sufficient conditions to ensure theexistence and uniqueness of fuzzy solution New weighted metrics are introduced toinvestigate the solvability for boundary valued problems of fuzzy hyperbolic equationsand an extended result for more general classes of hyperbolic equations is initiated.Moreover, the continuity of the Zadeh’s extension principle is used in some illustrativeexamples with some numerical simulations forα-cuts of fuzzy solutions.

Keywords Fuzzy partial differential equation· Fuzzy solution · Integral boundarycondition· Local initial condition · Fixed point theorem

Mathematics Subject Classification 34A07· 34A99 · 35L15

Centre for High Performance of Computing, VNU University of Science,

Vietnam National University, Hanoi, Vietnam

e-mail: sonlh@vnu.edu.vn

1 Introduction

Fuzziness is a basic type of subjective uncertainty initialed by Zadeh via membershipfunction in 1965 The complexity of the world makes events we face uncertain invarious forms Besides randomness, fuzziness is also an important uncertainty, whichplays an essential role in the real world

The theory of fuzzy sets, fuzzy valued functions and necessary calculus of fuzzyfunctions have been investigated in the monograph byLakshmikantham and Moha-patra(2003) and the references cited therein The concept of a fuzzy derivative wasfirst introduced byChang and Zadeh(1972), laterDubois and Prade(1982) definedthe fuzzy derivative by using Zadeh’s extension principle.Seikkala (1987) definedthe concept of fuzzy derivative which is the generalization of Hukuhara derivative.There are many different approaches to define fuzzy derivatives and they become avery quickly developing area of fuzzy analysis Moreover, in view of the develop-ment of calculus for fuzzy functions, the investigation of fuzzy differential equations(DEs) and fuzzy partial differential equations (PDEs) have been initiated (Buckleyand Feuring 1999;Seikkala 1987)

Fuzzy DEs were suggested as a way of modeling uncertain and incomplete mation systems, and studied by many researchers In recent years, there has been asignificant development in fuzzy calculus techniques in fuzzy DEs and fuzzy differ-ential inclusions, some recent contributions can be seen for example in the papers

infor-ofChalco-Cano and Roman-Flores(2008),Lupulescu and Abbas(2012),López(2013) andNieto et al.(2011), etc However, there is still lack of qualitative andquantitative researches for fuzzy PDEs Fuzzy PDEs were first introduced byBuckleyand Feuring(1999) And up to now, the available theoretical results for this kind ofequations are included in some researches ofAllahviranloo et al.(2011),Arara et al

Rodríguez-(2005),Bertone et al.(2013) and Chen et al.(2009) Some other efforts were ceeded in modeling some real world processes by fuzzy PDEs For instance,Jafelice et

suc-al.(2011) proposed a model for the reoccupation of ants in a region of attraction usingevolutive diffusion-advection PDEs with fuzzy parameters In Wang et al (2011),developed a fuzzy state-feedback control design methodology by employing a com-bination of fuzzy hyperbolic PDEs theory and successfully applied to the control of anonisothermal plug-flow reactor via the existing LMI optimization techniques For amore comprehensive study of fuzzy PDEs in soft computing and oil industry we citethe book ofNikravesh et al.(2004) Generally, industrial processes are often complex,uncertain processes in nature Consequently, the analysis and synthesis issues of fuzzyPDEs are of both theoretical and practical importance

In the paperBertone et al.(2013) the authors considered the existence and ness of fuzzy solutions for simple fuzzy heat equations and wave equations withconcrete formulation of the solutions.Arara et al.(2005) considered the local andnonlocal initial problem for some classes of hyperbolic equations However, almostall of previous results based on some complicated conditions on data and domain.Generally, existence theorems need a condition which may restrict the domain to asmall scale This paper deal with some boundary value problems for hyperbolic with

unique-an improvement in technique to ensure that the fuzzy solutions exist without unique-anycondition on data and the boundary of the domain New weighted metrics are used

The existence and uniqueness of fuzzy solutions

and suitable weighted numbers are chosen in order to prove that the existence anduniqueness of fuzzy solutions only depend on the Lipschitz property of the right side

of the equations Moreover, the problems with fuzzy integral boundary conditionswill be introduced and the solvability of these problems will be investigated for moregeneral fuzzy hyperbolic equations As we know that fuzzy boundary value problemswith integral boundary conditions constitute a very interesting and important class ofproblems They include two, three, multi-points boundary value problems and local,nonlocal initial conditions problems as the special cases We can see this fact in manyreferences such as (Agarwal et al 2005;Arara and Benchohra 2006) Therefore theresults of the present paper can be considered as a contribution to the subject

In many cases, it is difficult to find the exact solution of fuzzy DEs and fuzzy PDEs

In these cases, some optimal algorithms to find numerical solution must be introduced(see inNikravesh et al 2004, Chapter: Numerical solutions of fuzzy PDEs and itsapplications in computational mechanics) In another example,Dostál and Kratochvíl

(2010) used the two dimensional fuzzy PDEs to build up of a model for judgmentalforecasting in bank sector The simulation solutions of this equations supports themanagers optimize their decision making to close the branch of the bank for reducingthe costs However, before conducting a numerical method for any fuzzy equations,the question arises naturally whether the problems modeled by fuzzy PDEs are well-posed or not Thus, our study provides a theoretical foundation of numerical solutionmethods for some classes of fuzzy PDEs and ensures the consistency, stability andconvergence of optimal algorithms

The remainder of the paper is organized as follows Section2presents some sary preliminaries of fuzzy analysis, that will be used throughout this paper In Sect.3,

neces-we concern with the existence and uniqueness of fuzzy solutions for wave equations

in the following form

∂2u(x, y)

∂x∂y = f (x, y, u(x, y)), (x, y) ∈ [0, a] × [0, b], (1)

with local conditions

u(0, 0) = u0, u(x, 0) = η1(x), u(0, y) = η2(y), (x, y) ∈ [0, a] × [0, b]. (2)

The existence of fuzzy solutions of this problem is proved in Theorem3.1without any

condition in the domain by using a new weighted metric H1in the solutions space Inthis section, we also consider the Eq (1) with the integral boundary conditions

u (x, 0) +

b

0

k1(x)u(x, y)dy = g1(x), x ∈ [0, a], (3)

u (0, y) +

a

0

k2(y)u(x, y)dx = g2(y), y ∈ [0, b]. (4)

The results about the solvability of this problem are given in Theorem 3.2with

metric H and Theorem3.3by using new weighted metric H2 The general hyperbolic

PDEs in the form

in Sect.6

2 Preliminaries

This section will recall some concepts of fuzzy metric space used throughout thepaper For a more thorough treatise on fuzzy analysis, we refer toLakshmikanthamand Mohapatra(2003)

Let E n be space of functions u : Rn → [0, 1], which are normal, fuzzy convex, semi-continuous and bounded-supported functions We denote CC (R n ) by the set of

all nonempty compact, convex subsets ofRn Theα-cuts of u are [u] α = {x ∈ R n :

u(x) ≥ α} for 0 < α ≤ 1 Obviously, [u] α is in CC (R n ) And CC(R n ) is complete

metric space with Hausdorff metric defined by

H d (A, B) = max

sup

b ∈B ainf∈A {||b − a||}, sup

a ∈A binf∈B {||b − a||}

If g: Rn× Rn→ Rnis 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)}

The existence and uniqueness of fuzzy solutions

When g is continuous we have [g (u, u)] α = g ([u] α , [u] α ) for all u, u ∈ R n ,

0≤ α ≤ 1.

Let J = [x1 , y1]×[x2, y2] is a rectangular of R 2 A map f : J → E nis called tinuous at(t0, s0) ∈ J ⊂ R if multi-valued map f α (t, s) = [ f (t, s)] α is continuous

con-at(t, s) = (t0, s0) with respect to Hausdorff metric H dfor allα ∈ [0, 1] C(J, E n ) is

denoted a space of all continuous functions f : J → E nwith the supremum metric

H defined by

H( f, g) = sup

(s,t)∈J d∞( f (s, t) , g (s, t))

It can be shown that(C(J, E n ), H) is also a complete metric space.

Mapping f : J × E n → E nis called continuous at(t0, s0, u0) ∈ J × E nprovided,for any fixedα ∈ [0, 1] and arbitrary  > 0, there exists δ(, α) > 0 such that

Here the limitation 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

higher order of partial derivatives of f are defined similarly.

3 The fuzzy solutions of the hyperbolic PDEs

Denote J a = [0, a], J b = [0, b], with a, b > 0 In this part of the paper, we consider

the hyperbolic PDE (1), where f : J a × J b × E n → E nis a given function, whichsatisfies following hypothesis

Hypothesis (H) There exists K > 0 such that

In Arara et al.(2005) studied the existence of fuzzy solutions of the equation (1)with local initial conditions (2), whereη1 ∈ C(J a , E n ), η2 ∈ C(J b , E n ) are given

functions and u0 ∈ E n

Definition 3.1 (Arara et al 2005) A function u is called a solution of the

prob-lem (1)–(2) if it is a function in the space C (J a × J b , E n ) satisfying ∂2u (x,y)

the domain J a × J b to satisfy if the Lipschitz constant K is big enough On the other

hand, it depends on the large scale of the domain To relax this restriction, we use a

weighted metric H1 in the space C (J a × J b , E n )

where λ is a suitable positive number It is not difficult to check that (C(J a × J b ,

E n ), H1) is also a complete metric space.

Definition 3.2 A function u ∈ C(J a × J b , E n ) is called a solution of the problem (1),(2) if it satisfies

u (x, y) = q1(x, y) +

x

0

y

0

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

where q1 (x, y) = η1(x) + η2(y) − u0, for (x, y) ∈ J a × J b

Theorem 3.1 Assume that the condition (H) holds Then the problem (1)–(2) has a

unique solution in C(J a × J b , E n ).

The existence and uniqueness of fuzzy solutions

Proof From Definition3.2, we realize that fuzzy solution of problem (1)–(2) (if it

exists) is a fixed point of the operator N : C(J a × J b , E n ) → C(J a × J b , E n ) defined

as follows

N (u(x, y)) = q1(x, y) +

x

0

y

0

We will show that N is a contraction operator Indeed, for u , u ∈ C(J a × J b , E n ) and

α ∈ (0, 1] then from the properties of supremum metric and (6), we have

d∞(N(u(x, y)), N (u(x, y)))

y

0

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

x

0

y

0

y

0

y

0

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

x

0

y

0

y

0

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

≤ K H1 (u, u)e −λ(x+y)

x

0

y

0

e λ(s+t) dsdt≤ K

λ2H1(u, u)

for all(x, y) ∈ J a × J b That shows

e −λ(x+y) d∞(N(u(x, y)), N (u(x, y))) ≤ λ K2H1(u, u) , (x, y) ∈ J a × J b

Therefore

H1(N(u), N (u)) ≤ λ K2H1(u, u), for all u, u ∈ C(J a × J b , E n ).

By choosingλ =√2K we have λ K2 = 1

2 Hence, N is a contraction operator and by

Banach fixed point theorem, N has a unique fixed point, that is the solution of the

problem (1)–(2) The proof is completed

We continue concerning with the existence of fuzzy solutions for hyperbolic PDEs(1) with integral boundary conditions (3) and (4), where k1 (·) ∈ C(J a , R), k2(·) ∈ C(J b , R)g1(·) ∈ C(J a , E n ), g2(·) ∈ C(J b , E n ) are given functions.

Definition 3.3 A function u ∈ C(J a ×J b , E n ) is called a fuzzy solution of the problem

(1), (3) and (4) if u satisfies the following integral equation

u (x, y) = q(x, y) −

b

0

k1(x)u(x, y)dy −

a

0

k2(y)u(x, y)dx

− k1 (0)

b

0

a

0

k2(y)u(x, y)dxdy +

x

0

y

0

hypoth-has a unique fuzzy solution in C(J a × J b , E n ).

Proof Integrating both sides of the equation (1) on[0, x] × [0, y], we have

u (x, y) = q(x, y) −

b

0

k1(x)u(x, y)dy −

a

0

k2(y)u(x, y)dx

− k1 (0)

b

0

a

0

k2(y)u(x, y)dxdy +

x

0

y

0

N (u(x, y)) = q(x, y) −

b

0

k1(x)u(x, y)dy −

a

0

k2(y)u(x, y)dx

− k1 (0)

b

0

a

0

k2(y)u(x, y)dxdy +

x

0

y

0

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

For u , u ∈ C(J a × J b , E n ) arbitrary and α ∈ (0, 1], one gets

The existence and uniqueness of fuzzy solutions

a

0

a

0

y

0

b

0

d∞(u(x, y), u(x, y))dxdy

≤ (k1 b + k2 a + k1 k2ab + K ab)H(u, u), for all(x, y) ∈ J a × J b

Because k1 b + k2 a + k1 k2ab + K ab < 1 By Banach fixed point theorem, N has a

unique fixed point, which is the fuzzy solution of the problem (1), (3), (4) The theorem

u (0, y) + k2(y)

y

0

where k1 , k2 ∈ C(J a , R), g1, g2 ∈ C(J a , E n ) are given functions Technique of the

proof bases mainly on a new weighted metric H2defined as follows:

t

0

k2(t)u(s, t)dsdt

+ K2 (y)

y

0

s

0

k1(s)u(s, t)dtds

− K1 (x)

x

0

t

0

⎡

⎣

s

0

y

0

y

0

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

where G(x, y), K1(x), K2(y) are defined by (15 ).

Proof From Eq (1), we have

u (x, y) = u(x, 0) + u(0, y) − u(0, 0) +

x

0

y

0

f (s, t, u(s, t)) dsdt. (9)

The existence and uniqueness of fuzzy solutions

Multiplying both sides of this equation by k1 (x), then taking integral with respect to

the second variable we have

u (0, t)dt

+ k1 (x)

x

0

t

0

g2(t)dt −

x

0

t

0

t

0

k2(t)u(s, t)dsdt

+ k1 (x)

x

0[

x

0

t

0

f (s, t, u(s, t)) dsdt]dt, (12)where

Q1(x) = −xk1(x)u(0, 0) + k1(x)

x

0

u(x, t)dt

= [xk1 (x) + 1] u(x, 0) + Q1(x) − k1(x)

x

0

t

0

k2(t)u(s, t)dsdt

+ k1 (x)

x

0

t

0

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

⎤

⎥

⎦ dt,

0

⎡

⎢x0

t

0

By doing the same arguments we have

u(0, y) = P(y) + k2(y)

yk2(y) + 1

y

0

s

0

⎡

⎣

s

0

y

0

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

⎤

⎦ ds, (14)where

t

0

k2(t)u(s, t)dsdt

+ K2 (y)

y

0

s

0

k1(s)u(s, t)dtds

− K1 (x)

x

0

t

0

⎡

⎣

s

0

y

0

y

0

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

The existence and uniqueness of fuzzy solutions

By using the new weighted metric H2we can reduce the conditions in Theorem3.2

to milder conditions in the following result

Theorem 3.3 Suppose that the hyperbolic (H) is satisfied Then problem (1), (7)

and (8 ) has a unique fuzzy solution in the space C (J a × J a , E n ).

Proof Consideration N : C(J a × J a , E n ) → C(J a × J a , E n ), defined by

N(u(x, y)) = G(x, y) + K1(x)

x

0

t

0

k2(t)u(s, t)dsdt

+ K2 (y)

y

0

s

0

k1(s)u(s, t)dtds

− K1 (x)

x

0

t

0

⎡

⎣

s

0

y

0

y

0

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

For each(x, y) ∈ J a ×J a and u , u ∈ C(J a ×J a , E n ) we have the following estimations

d∞(N(u(x, y)), N (u(x, y)))

≤ d∞(K1(x)

x

0

t

0

k2(t)u(s, t)dsdt, K1(x)

x

0

t

0

k2(t)u(s, t)dsdt)

+ d∞(K2(y)

y

0

s

0

k1(s)u(s, t)dtds, K2(y)

y

0

s

0

k1(s)u(s, t)dtds)

⎢x0

t

0

t

0

⎡

⎣

s

0

y

0

y

0

y

0

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

x

0

y

0

t

0

k2(t)u(s, t)dsdt, K1(x)

x

0

t

0

t

0

d∞(u(s, t), u(s, t))dsdt

= c1 e −λ max{x,y}

x

0

t

0

d∞(u(s, t), u(s, t))e −λ max{s ,t }e λ max{s ,t }dsdt

= c1 H2(u, u)e −λ max{x,y}

x

0

t

0

e λ max{s ,t }dsdt

= c1 H2(u, u)e −λ max{x,y}

x

0

t

0

e λt dsdt = c1 H2(u, u)e −λ max{x,y}

x

0

te λt dt

≤ c1x

λ H2(u, u)e −λ max{x,y} e λx ≤ ac1