the existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations

Fuzzy Optim Decis MakingDOI 10.1007/s10700-014-9186-0

The existence and uniqueness of fuzzy solutionsfor 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 for α-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

H. V. Long (B)Department of Basic Sciences, University of Transport and Communications, Hanoi, Vietname-mail: [email protected]

N. T. K. SonDepartment of Mathematics, Hanoi University of Education, Hanoi, Vietname-mail: [email protected]

N. T. M. HaDepartment of Mathematics, Hai Phong University, Hai Phong, Vietname-mail: [email protected]

L. H. SonCentre for High Performance of Computing, VNU University of Science,Vietnam National University, Hanoi, Vietname-mail: [email protected]

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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 by Lakshmikantham and Moha-patra (2003) and the references cited therein. The concept of a fuzzy derivative wasfirst introduced by Chang and Zadeh (1972), later Dubois 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 infor-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 papersof Chalco-Cano and Roman-Flores (2008), Lupulescu and Abbas (2012), Rodríguez-López (2013) and Nieto et al. (2011), etc. However, there is still lack of qualitative andquantitative researches for fuzzy PDEs. Fuzzy PDEs were first introduced by Buckleyand Feuring (1999). And up to now, the available theoretical results for this kind ofequations are included in some researches of Allahviranloo et al. (2011), Arara et al.(2005), Bertone et al. (2013) and Chen et al. (2009). Some other efforts were suc-ceeded in modeling some real world processes by fuzzy PDEs. For instance, Jafelice etal. (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 of Nikravesh 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 paper Bertone et al. (2013) the authors considered the existence and unique-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 withan improvement in technique to ensure that the fuzzy solutions exist without anycondition on data and the boundary of the domain. New weighted metrics are used

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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 sideof 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 in Nikravesh 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. Section 2 presents some neces-sary preliminaries of fuzzy analysis, that will be used throughout this paper. In Sect. 3,we concern with the existence and uniqueness of fuzzy solutions for wave equationsin 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 Theorem 3.1 without anycondition in the domain by using a new weighted metric H1 in 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)

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The results about the solvability of this problem are given in Theorem 3.2 withmetric H and Theorem 3.3 by using new weighted metric H2. The general hyperbolicPDEs in the form

∂2u(x, y)

∂x∂y+ ∂ (p1(x, y)u(x, y))

∂x+ ∂(p2(x, y)u(x, y))

∂y+ c(x, y)u(x, y)

= f (x, y, u(x, y)) , (5)

for (x, y) ∈ [0, a]× [0, b], are considered in the Sect. 4. Some results of the existenceand uniqueness of fuzzy solutions for these equations with local initial conditionsand integral boundary conditions are also exhibited in Theorems 4.1 and 4.2. Someillustrated examples for our results are given in Sect. 5 with some numerical simulationsfor α-cuts of the solutions. Finally, some conclusions and future works are discussedin 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 to Lakshmikanthamand Mohapatra (2003).

Let En be space of functions u : Rn → [0, 1], which are normal, fuzzy convex,

semi-continuous and bounded-supported functions. We denote CC(Rn) by the set ofall nonempty compact, convex subsets of R

n . The α-cuts of u are [u]α = {x ∈ Rn :

u(x) ≥ α} for 0 < α ≤ 1. Obviously, [u]α is in CC(Rn). And CC(Rn) is completemetric space with Hausdorff metric defined by

Hd(A, B) = max

{supb∈B

infa∈A

{||b − a||}, supa∈A

infb∈B

{||b − a||}}, A, B ∈ CC(Rn)

where || · || is usual Euclidean norm in Rn . Furthermore,

(i) Hd(t A, t B) = |t | · Hd(A, B)

(ii) Hd(A + A′, B + B ′) ≤ Hd(A, B) + Hd(A′, B ′);(iii) Hd(A + C, B + C) = Hd(A, B)

where A, B, C, A′, B ′ ∈ CC(Rn) and t ∈ R.Let (En, d∞) be a complete metric space with supremum metric d∞ defined by

d∞(u, u) = sup0<α≤1

Hd([u]α , [u]α

), u, u ∈ En .

If g : Rn ×R

n → Rn is a function, then, according to Zadeh’s extension principle we

can extend g to En × En → En by the function defined by

g(u, u)(z) = supz=g(x,z)

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

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

0 ≤ α ≤ 1.Let J = [x1, y1]×[x2, y2] is a rectangular of R

2. A map f : J → En is called con-tinuous at (t0, s0) ∈ J ⊂ R if multi-valued map fα (t, s) = [ f (t, s)]α is continuousat (t, s) = (t0, s0) with respect to Hausdorff metric Hd for all α ∈ [0, 1]. C(J, En) isdenoted a space of all continuous functions f : J → En with the supremum metricH defined by

H( f, g) = sup(s,t)∈J

d∞ ( f (s, t) , g (s, t)) .

It can be shown that (C(J, En), H) is also a complete metric space.Mapping f : J × En → En is called continuous at (t0, s0, u0) ∈ J × En provided,

for any fixed α ∈ [0, 1] and arbitrary ε > 0, there exists δ(ε, α) > 0 such that

Hd([ f (t, s, u)]α , [ f (t0, s0, u0)]

α)

< ε

whenever max{|t − t0| , |s − s0|} < δ(ε, α) and Hd ([u]α , [u0]α) < δ(ε, α), (t, s, u)

∈ J × En .The integral of f : J = [x1, y1]×[x2, y2] → En , denoted by

∫ y1x1

∫ y2x2

f (t, s) dsdt ,is defined by

⎡⎣

y1∫

x1

y2∫

x2

f (t, s) dsdt

⎤⎦

α

=y1∫

x1

y2∫

x2

fα (t, s) dsdt

=⎧⎨⎩

y1∫

x1

y2∫

x2

v (t, s) dsdt |v : J → Rn

is a measurable selection for fα

⎫⎬⎭, for α ∈ (0, 1].

A function f : J → En is integrable if∫ y1

x1

∫ y2x2

f (t, s) dsdt is in En .For any f : J → En , fuzzy partial derivative of f with respect to x at point

(x0, y0) ∈ J is a fuzzy set∂ f (x0, y0)

∂x∈ En which is defined by

∂ f (x0, y0)

∂x= lim

h→0

f (x0 + h, y0) − f (x0, y0)

h.

Here the limitation is taken in the metric space (En, d∞) and u − v is the Hukuharadifference of u and v in En . The fuzzy partial derivative of f with respect to y andhigher order of partial derivatives of f are defined similarly.

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3 The fuzzy solutions of the hyperbolic PDEs

Denote Ja = [0, a], Jb = [0, b], with a, b > 0. In this part of the paper, we considerthe hyperbolic PDE (1), where f : Ja × Jb × En → En is a given function, whichsatisfies following hypothesis.

Hypothesis (H) There exists K > 0 such that

Hd([ f (s, t, u)]α, [ f (s, t, u)]α) ≤ K Hd

([u]α, [u]α)

for all (s, t) ∈ Ja × Jb and all u, u ∈ En .

In Arara et al. (2005) studied the existence of fuzzy solutions of the equation (1)with local initial conditions (2), where η1 ∈ C(Ja, En), η2 ∈ C(Jb, En) are givenfunctions and u0 ∈ En .

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(Ja × Jb, En) satisfying ∂2u(x,y)∂x∂y =

f (x, y, u(x, y)) on Ja × Jb and u(0, 0) = u0, u(x, 0) = η1(x), u(0, y) = η2(y) forall (x, y) ∈ Ja × Jb.

Proposition 3.1 (Arara et al. 2005) Assume that the hypothesis (H) holds. If K ab < 1,then the problem (1)–(2) has a unique fuzzy solution in the space C(Ja × Jb, En).

Remark 3.1 This result based on the condition K ab < 1. That condition is strict forthe domain Ja × Jb to satisfy if the Lipschitz constant K is big enough. On the otherhand, it depends on the large scale of the domain. To relax this restriction, we use aweighted metric H1 in the space C(Ja × Jb, En)

H1( f, g) = sup(s,t)∈J

{d∞ ( f (s, t) , g (s, t)) e−λ(s+t)

},

where λ is a suitable positive number. It is not difficult to check that (C(Ja × Jb,

En), H1) is also a complete metric space.

Definition 3.2 A function u ∈ C(Ja × Jb, En) 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) ∈ Ja × Jb.

Theorem 3.1 Assume that the condition (H) holds. Then the problem (1)–(2) has aunique solution in C(Ja × Jb, En).

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Proof From Definition 3.2, we realize that fuzzy solution of problem (1)–(2) (if itexists) is a fixed point of the operator N : C(Ja × Jb, En) → C(Ja × Jb, En) definedas follows

N (u(x, y)) = q1(x, y) +x∫

0

y∫

0

f (s, t, u(s, t)) dsdt. (6)

We will show that N is a contraction operator. Indeed, for u, u ∈ C(Ja × Jb, En) andα ∈ (0, 1] then from the properties of supremum metric and (6), we have

d∞ (N (u(x, y)), N (u(x, y)))

≤ d∞

⎛⎝

x∫

0

y∫

0

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

x∫

0

y∫

0

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

⎞⎠

≤ K

x∫

0

y∫

0

d∞ (u(s, t), u(s, t)) dsdt.

It implies that

e−λ(x+y)d∞

⎛⎝

x∫

0

y∫

0


x∫

0

y∫

0

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

⎞⎠

≤ K e−λ(x+y)

x∫

0

y∫

0

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

≤ K H1(u, u)e−λ(x+y)

x∫

0

y∫

0

eλ(s+t)dsdt ≤ K

λ2 H1(u, u)

for all (x, y) ∈ Ja × Jb. That shows

e−λ(x+y)d∞ (N (u(x, y)), N (u(x, y))) ≤ K

λ2 H1 (u, u) , (x, y) ∈ Ja × Jb.

Therefore

H1 (N (u), N (u)) ≤ K

λ2 H1(u, u), for all u, u ∈ C(Ja × Jb, En).

By choosing λ = √2K we have K

λ2 = 12 . Hence, N is a contraction operator and by

Banach fixed point theorem, N has a unique fixed point, that is the solution of theproblem (1)–(2). The proof is completed. �

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H. V. Long et al.

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

Definition 3.3 A function u ∈ C(Ja×Jb, En) 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


where q(x, y) = g1(x) + g2(y) − g1(0) + k1(0)∫ b

0 g2(t)dt , for (x, y) ∈ Ja × Jb.

Theorem 3.2 Let k1 = sups∈Ja|k1(s)|, k2 = supt∈Jb

|k2(t)|. Assume that the hypoth-esis (H) is satisfied. If (1 + k1b)(1 + k2a) < 2 − K ab, then the problem (1), (3), (4)has a unique fuzzy solution in C(Ja × Jb, En).

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

u(x, y) = q(x, y) −b∫

0


0

k2(y)u(x, y)dx

− k1(0)

b∫

0

a∫

0


0

y∫

0


where q(x, y) = g1(x) + g2(y) − g1(0) + k1(0)∫ b

0 g2(t)dt, for all (x, y) ∈ Ja × Jb.We will prove that fuzzy solution of problem (1), (3), (4) is a fixed point of operator

N : C(Ja × Jb, En) → C

(Ja × Jb, En)

defined as follows

N (u(x, y)) = q(x, y) −b∫

0


0

k2(y)u(x, y)dx

− k1(0)

b∫

0

a∫

0


0

y∫

0

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

For u, u ∈ C(Ja × Jb, En) arbitrary and α ∈ (0, 1], one gets

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Hd([N (u(x, y))]α , [N (u(x, y))]α

)

≤ Hd

⎛⎝⎡⎣

b∫

0

k1(x)u(x, y)dy

⎤⎦

α

,

⎡⎣

b∫

0

k1(x)u(x, y)dy

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

a∫

0

k2(y)u(x, y)dx

⎤⎦

α

,

⎡⎣

a∫

0

k2(y)u(x, y)dx

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣k1(0)

b∫

0

a∫

0

k2(y)u(x, y)dxdy

⎤⎦

α

,

⎡⎣k1(0)

b∫

0

a∫

0

k2(y)u(x, y)dxdy

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

x∫

0

y∫

0


⎤⎦

α

,

⎡⎣

x∫

0

y∫

0


⎤⎦

α⎞⎠

≤ k1

b∫

0

Hd([u(x, y)]α , [u(x, y)]α

)dy + k2

a∫

0

Hd([u(x, y)]α , [u(x, y)]α

)dx

+ |k1(0)|k2

b∫

0

a∫

0

Hd([u(x, y)]α , [u(x, y)]α

)dxdy

+x∫

0

y∫

0

Hd([ f (s, t, u(s, t))]α , [ f (s, t, u(s, t))]α

)dsdt

≤ (k1b + k2a + k1k2ab)d∞(u(x, y), u(x, y))

+x∫

0

y∫

0

K .Hd([u(s, t)]α , [u(s, t)]α

)dsdt

≤ (k1b + k2a + k1k2ab)H(u, u) + K

a∫

0

b∫

0

d∞(u(x, y), u(x, y))dxdy

≤ (k1b + k2a + k1k2ab + K ab)H(u, u), for all(x, y) ∈ Ja × Jb.

Hence

H (N (u), N (u)) = sup(x,y)∈Ja×Jb

d∞ (N (u (x, y)), N (u (x, y)))

= sup(x,y)∈Ja×Jb

(sup

0<α≤1Hd

([N (u(x, y))]α , [N (u(x, y))]α

))

≤ (k1b + k2a + k1k2ab + K ab)H (u, u) .

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Because k1b + k2a + k1k2ab + K ab < 1. By Banach fixed point theorem, N has aunique fixed point, which is the fuzzy solution of the problem (1), (3), (4). The theoremis proved completely. �Remark 3.2 The result in Theorem 3.2 still depends on the large scale of the domain.The question now is that how to reduce this condition to get better results? We havethe answer in special case when a = b and integral boundary conditions have thefollowing forms

u(x, 0) + k1(x)

x∫

0

u(x, t)dt = g1(x), x ∈ Ja, (7)

u(0, y) + k2(y)

y∫

0

u(s, y)ds = g2(y), y ∈ Ja, (8)

where k1, k2 ∈ C(Ja, R), g1, g2 ∈ C(Ja, En) are given functions. Technique of theproof bases mainly on a new weighted metric H2 defined as follows:

H2( f, g) = sup(s,t)∈Ja×Ja

{d∞( f (s, t) , g (s, t))e−λ max{s,t}

}, λ > 0.

Lemma 3.1 Problem (1), (7) and (8) is equivalent to following integral equation

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

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt

− K2(y)

y∫

0

⎡⎣

s∫

0

y∫

0


⎤⎦ ds

+x∫

0

y∫

0


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)

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Multiplying both sides of this equation by k1(x), then taking integral with respect tothe second variable we have

x∫

0

k1(x)u(x, t)dt = k1(x)x[u(x, 0) − u(0, 0)] + k1(x)

x∫

0

u(0, t)dt

+ k1(x)

x∫

0

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt . (10)

From Eq. (8) we implies

x∫

0

u(0, t)dt =x∫

0

g2(t)dt −x∫

0

t∫

0

k2(t)u(s, t)dsdt (11)

Combinate (10) and (11) we have

x∫

0

k1(x)u(x, t)dt = Q1(x) + xk1(x)u(x, 0) − k1(x)

x∫

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

g2(t)dt .

From boundary condition (7) we have

g1(x) = u(x, 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

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt,

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H. V. Long et al.

So that we have

u(x, 0) = Q(x) + k1(x)

xk1(x) + 1

x∫

0

t∫

0

k2(t)u(s, t)dsdt

− k1(x)

xk1(x) + 1

x∫

0

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt, (13)

where

Q(x) = 1

xk1(x) + 1[g1(x) − Q1(x)] .

By doing the same arguments we have

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

yk2(y) + 1

y∫

0

s∫

0

k1(s)u(s, t)dtds

− k2(y)

yk2(y) + 1

y∫

0

⎡⎣

s∫

0

y∫

0


⎤⎦ ds, (14)

where

P(y) = 1

yk2(y) + 1

⎡⎣g2(y) + yk2(y)u(0, 0) − k2(y)

y∫

0

g1(s)ds

⎤⎦ .

Substituting (13) and (14) into (9) we have

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

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt

− K2(y)

y∫

0

⎡⎣

s∫

0

y∫

0


⎤⎦ ds

+x∫

0

y∫

0


123


where

G(x, y) = Q(x) + P(y) − u(0, 0)

K1(x) = k1(x)

xk1(x) + 1, K2(y) = k2(y)

yk2(y) + 1. (15)

�Definition 3.4 u ∈ C(Ja × Ja, En) is a fuzzy solution of (1), (7) and (8) if u satisfiesthe integral equation in Lemma 3.1.

By using the new weighted metric H2 we can reduce the conditions in Theorem 3.2to 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(Ja × Ja, En).

Proof Consideration N : C(Ja × Ja, En) → C(Ja × Ja, En), 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

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt

− K2(y)

y∫

0

⎡⎣

s∫

0

y∫

0


⎤⎦ ds

+x∫

0

y∫

0


For each (x, y) ∈ Ja×Ja and u, u ∈ C(Ja×Ja, En) 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)

123

H. V. Long et al.

+ d∞

⎛⎜⎝K1(x)

x∫

0

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt, K1(x)

×x∫

0

⎡⎢⎣

x∫

0

t∫

0


⎤⎥⎦ dt

⎞⎟⎠

+ d∞

⎛⎝K2(y)

y∫

0

⎡⎣

s∫

0

y∫

0


⎤⎦ ds, K2(y)

×y∫

0

⎡⎣

s∫

0

y∫

0


⎤⎦ ds

⎞⎠

+ d∞

⎛⎝

x∫

0

y∫

0


x∫

0

y∫

0


⎞⎠ . (16)

For simplicity we denote

c1 = sup(x,y)∈Ja×Ja

|K1(x)| |k2(y)|, c2 = sup(x,y)∈Ja×Ja

|K2(y)| |k1(x)|c3 = K sup

x∈Ja

|K1(x)| , c4 = K supy∈Ja

|K2(y)| .

Then we have

e−λ max{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

⎞⎟⎠

≤ e−λ max{x,y}c1

x∫

0

t∫

0

d∞(u(s, t), u(s, t))dsdt

= c1e−λ max{x,y}x∫

0

t∫

0

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

= c1 H2(u, u)e−λ max{x,y}x∫

0

t∫

0

eλ max{s,t}dsdt


0

t∫

0

eλt dsdt = c1 H2(u, u)e−λ max{x,y}x∫

0

teλt dt

≤ c1x

λH2(u, u)e−λ max{x,y}eλx ≤ ac1

λH2(u, u). (17)

123


Similarly we have

e−λ max{x,y}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

⎞⎠

≤ ac2

λH2(u, u). (18)

We now will estimate the third term in the right side of (16)

e−λ max{x,y}d∞

⎛⎜⎝K1(x)

x∫

0

x∫

0

t∫

0

f (s, t, u(s, t)) dsdtdt,

K1(x)

x∫

0

x∫

0

t∫

0

f (s, t, u(s, t)) dsdtdt

⎞⎟⎠

≤ c3e−λ max{x,y}x∫

0

x∫

0

t∫

0

d∞(u(s, t), u(s, t))dsdtdt

= c3e−λ max{x,y}x∫

0

x∫

0

t∫

0

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


0

x∫

0

t∫

0

eλ max{s,t}dsdtdt .

One gets

x∫

0

x∫

0

t∫

0

eλ max{s,t}dsdtdt

=x∫

0

⎡⎢⎣

t∫

0

⎛⎝

t∫

0

eλ max{s,t}ds +x∫

t

eλ max{s,t}ds

⎞⎠ dt

⎤⎥⎦ dt

=x∫

0

⎡⎢⎣

t∫

0

⎛⎝

t∫

0

eλt ds +x∫

t

eλsds

⎞⎠ dt

⎤⎥⎦ dt

=x∫

0

⎡⎢⎣

t∫

0

(teλt + 1

λeλx − 1

λeλt

)dt

⎤⎥⎦ dt

123

H. V. Long et al.

=x∫

0

[1

λteλt − 2

λ2 eλt + 1

λeλx t + 2

λ2

]dt

=(

x

λ2 + x2

2λ

)eλx + 2x

λ2 + 3

λ3

(1 − eλx)

≤(

x

λ2 + x2

2λ

)eλx + 2x

λ2 .

Thus we have

e−λ max{x,y}x∫

0

x∫

0

t∫

0

eλ max{s,t}dsdtdt

≤ e−λ max{x,y}[(

x

λ2 + x2

2λ

)eλx + 2x

λ2

]

=⎧⎨⎩

e−λx[(

xλ2 + x2

2λ

)eλx + 2x

λ2

]if x ≥ y

e−λy[(

xλ2 + x2

2λ

)eλx + 2x

λ2

]if y > x

≤ 3a

λ2 + a2

2λ.

That leads to

c3 H2 (u, u) e−λ max{x,y}x∫

0

x∫

0

t∫

0

eλ max{s,t}dsdtdt ≤ c3

(3a

λ2 + a2

2λ

)H2(u, u). (19)

Similarly, we have following estimation for the fourth term in the right side of (16)

e−λ max{x,y}d∞(

K2(y)

y∫

0

s∫

0

y∫

0

f (s, t, u(s, t)) dsdtds,

K2(y)

y∫

0

s∫

0

y∫

0

f (s, t, u(s, t)) dsdtds

)

≤ c4

(3a

λ2 + a2

2λ

)H2(u, u). (20)

Finally, we have

e−λ max{x,y}d∞

⎛⎝

x∫

0

y∫

0


x∫

0

y∫

0


⎞⎠

123


≤ K e−λ max{x,y}x∫

0

y∫

0

d∞(u(s, t), u(s, t))dsdt

= K e−λ max{x,y}x∫

0

y∫

0

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

= K H2(u, u)e−λ max{x,y}x∫

0

y∫

0

eλ max{s,t}dsdt.

+) If x < y then

x∫

0

y∫

0

eλ max{s,t}dsdt =x∫

0

⎡⎣

s∫

0

eλ max{s,t}dt +y∫

s

eλ max{s,t}dt

⎤⎦ ds

=x∫

0

⎡⎣

s∫

0

eλsdt +y∫

s

eλt dt

⎤⎦ ds =

x∫

0

[seλs + 1

λeλy − 1

λeλs

]ds

= 1

λxeλx + 1

λeλy x − 2

λ2 (eλx − 1) ≤ 1

λxeλx + 1

λeλy x .

It follows

e−λ max{x,y}x∫

0

y∫

0

eλ max{s,t}dsdt = e−λy

x∫

0

y∫

0

eλ max{s,t}dsdt

≤ e−λy(

1

λxeλx + 1

λeλy x

)≤ 2x

λ≤ 2a

λ.

+) If y ≤ x , we also get

e−λ max{x,y}x∫

0

y∫

0

eλ max{s,t}dsdt = e−λx

x∫

0

y∫

0

eλ max{s,t}dsdt

≤ e−λx(

1

λyeλy + 1

λeλx y

)≤ 2y

λ≤ 2a

λ.

It implies

K H2(u, u)e−λ max{x,y}x∫

0

y∫

0

eλ max{s,t}dsdt ≤ 2K a

λH2(u, u)

123

H. V. Long et al.

or

e−λ max{x,y}d∞

⎛⎝

x∫

0

y∫

0


x∫

0

y∫

0


⎞⎠

≤ 2K a

λH2(u, u). (21)

By multiplying e−λ max{x,y} to both sides of (16), and from (17) to (21), we receive

H2(N (u), N (u)) ≤[

a(c1 + c2 + 2K )

λ+

(3a

λ2 + a2

2λ

)(c3 + c4)

]H2(u, u).

We can choose λ > 0 satisfying

a(c1 + c2 + K )

λ+

(3a

λ2 + a2

2λ

)(c3 + c4) < 1.

This follows that N has unique fixed point. The theorem is proved completely. �Remark 3.3 From Theorems 3.1 and 3.3, by using different weighted metrics in thespace C(Ja×Jb, En), we can receive the various results in the existence and uniquenessof fuzzy solutions of the problem without conditions on the boundary of the domain.

4 Fuzzy solutions of general hyperbolic PDEs

We consider problem in general cases (5), where (x, y) ∈ Ja× Jb, f : Ja× Jb×En →En and c, pi ∈ C(Ja × Jb, R); i = 1, 2. And the local initial conditions are in (2),in which η1 ∈ C(Ja, En), η2 ∈ C(Jb, En) are given functions and u0 ∈ En .

Definition 4.1 A function u ∈ C(Ja × Jb, En) is called a fuzzy solution of theequations (5) with local condition (2) if it satisfies the equation

u(x, y) = q1(x, y) −y∫

0

p1(x, t)u(x, t)dt −x∫

0

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

−x∫

0

y∫

0

c(s, t)u(s, t)dsdt +x∫

0

y∫

0


where q1(x, y) = η1(x)+η2(y)−u0 + ∫ y0 p1(0, t)η2(t)dt + ∫ x

0 p2(s, 0)η1(s)ds, for(x, y) ∈ Ja × Jb.

Theorem 4.1 Let p1 = sup(s,t)∈Ja×Jb|p1(s, t)|, p2 = sup(s,t)∈Ja×Jb

|p2(s, t)|, c =sup(s,t)∈Ja×Jb

|c(s, t)|. If Hypothesis (H) is satisfied and (bp1+ap2)+ab(c+K ) < 1,then problem (5) and (2) has a unique solution in C(Ja × Jb, En).

123


Proof By doing the same previous arguments, we consider the operator N1 : C(Ja ×Jb, En) → C(Ja × Jb, En) defined as follows

N1(u(x, y)) = q1(x, y) −y∫

0

p1(x, t)u(x, t)dt −x∫

0

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

−x∫

0

y∫

0

c(s, t)u(s, t)dsdt +x∫

0

y∫

0


Let u, u ∈ C(Ja × Jb, En) and α ∈ (0, 1], (x, y) ∈ Ja × Jb. Then

Hd([N1(u(x, y))]α , [N1(u(x, y))]α

)

≤ Hd

⎛⎝⎡⎣

y∫

0

p1(x, t)u(x, t)dt

⎤⎦

α

,

⎡⎣

y∫

0

p1(x, t)u(x, t)dt

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

x∫

0

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

⎤⎦

α

,

⎡⎣

x∫

0

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

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

x∫

0

y∫

0

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

⎤⎦

α

,

⎡⎣

x∫

0

y∫

0

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

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

x∫

0

y∫

0


⎤⎦

α

,

⎡⎣

x∫

0

y∫

0


⎤⎦

α⎞⎠

≤b∫

0

|p1(x, t)|Hd([u(x, t)]α, [u(x, t)]α) dt

+a∫

0

|p2(s, y)|Hd([u(s, y)]α, [u(s, y)]α

)ds

+a∫

0

b∫

0

|c(s, t)|Hd([u(s, t)]α, [u(s, t)]α

)dsdt

+ K

a∫

0

b∫

0

Hd([u(s, t)]α, [u(s, t)]α

)dsdt

≤ (bp1 + ap2 + ab(c + K )) d∞ (u(x, y), u(x, y)) .

It implies

H (N1(u), N1(u)) = sup(x,y)∈Ja×Jb

d∞(N1(u (x, y)), N1 (u ((x, y)))

123

H. V. Long et al.

= sup(s,t)∈Ja×Jb

(sup

0<α≤1Hd

([N1(u(x, y))]α , [N1(u)(x, y)]α

))

≤ (bp1 + ap2 + ab(c + K )) H (u, u) .

That shows the contractive property of the operator N1. �We continue for the problem of the equation (5) with the integral boundary condi-

tions (3) and (4)

Definition 4.2 Function u ∈ C(Ja × Jb, En) is called a fuzzy solution of problem(5), (3) and (4) if it satisfies following fuzzy integral equation

u(x, y) = P (x, y, u(x, y)) + Q (x, y, u(x, y)) + R (x, y, u(x, y)) ,

where

Q(x, y, u(x, y)) = −y∫

0

p1(s, t)u(s, t)dt −x∫

0

p2(s, t)u(s, t)ds

−x∫

0

y∫

0

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

+x∫

0

y∫

0

f (s, t, u(s, t))dsdt, (22)

P(x, y, u(x, y)) = −b∫

0


0

k1(y)u(x, y)dx

−a∫

0

b∫

0

k2(0)k1(x)u(x, y)dxdy

−x∫

0

b∫

0

p2(s, 0)k1(x)u(x, y)dsdy

−y∫

0

a∫

0

p1(0, t)k2(y)u(x, y)dxdt (23)

and

R(x, y, u(x, y)) = g1(x) + g2(y) − g2(0) +a∫

0

k2(0)g1(x)dx

+y∫

0

p1(0, t)g2(t)dt +x∫

0

p2(s, 0)g1(s)ds, (24)

for all (x, y) ∈ Ja × Jb.

123


Theorem 4.2 Let k1 = supt∈Ja|k1(t)|, k2 = sups∈Jb

|k2(s)|. If hypothesis (H) issatisfied and a(1 + k1) + b(1 + k2) + ab(c + K + p2k1 + p1k2 + k1k2) < 1, thenproblem (5), (3) and (4) has a unique solution in Ja × Jb.

Proof Integrating both sides in [0, x] × [0, y], one has

u(x, y) = P (x, y, u(x, y)) + Q(x, y, u(x, y)) + R (x, y, u(x, y)) ,

where P, Q and G are defined by (22), (23) and (24), respectively.We will prove that the operator N2 : C(Ja × Jb, En) → C(Ja × Jb, En) defined

by

N2(u(x, y)) = P(x, y, u(x, y)) + Q(x, y, u(x, y)) + R(x, y, u(x, y))

has a unique fixed point. Let u, u ∈ C(Ja × Jb, En), α ∈ (0, 1] and (x, y) ∈ Ja × Jb,then

Hd([N2(u)(x, y)]α , [N2(u)(x, y)]α

) ≤ Hd([P(u(x, y))]α , [P (u (x, y))]α)

+ Hd([Q(u(x, y))]α, [Q (u(x, y))]α

) + Hd([R(u(x, y))]α , [R (u(x, y))]α

)

Following Theorem 3.1 we have

Hd([Q(u(x, y))]α , [Q(u(x, y))]α

) ≤ (bp1 + ap2 + abc + K ab) H(u, u).

Moreover

Hd([P(u(x, y))]α , [P(u(x, y))]α)

≤ Hd

⎛⎝⎡⎣

b∫

0

k1(x)u(x, y)dy

⎤⎦

α

,

⎡⎣

b∫

0

k1(x)u(x, y)dy

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

a∫

0

k2(x)u(x, y)dx

⎤⎦

α

,

⎡⎣

a∫

0

k2(x)u(x, y)dx

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

x∫

0

b∫

0

p2(s, 0)k1(s)u(s, t)dsdt

⎤⎦

α

,

⎡⎣

x∫

0

b∫

0

p2(s, 0)k1(s)u(s, t)dsdt

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

a∫

0

y∫

0

p1(0, t)k2(t)u(x, t)dxdt

⎤⎦

α

,

⎡⎣

a∫

0

y∫

0

p1(0, t)k2(t)u(x, t)dxdt

⎤⎦

α⎞⎠

+ Hd

⎛⎝⎡⎣

a∫

0

b∫

0

k2(0)k1(x)u(x, y)dxdy

⎤⎦

α

,

⎡⎣

a∫

0

b∫

0

k2(0)k1(x)u(0, y)dxdy

⎤⎦

α⎞⎠

≤ [k1b + k2a + ab(p2k1 + p1k2 + k1k2)] · H(u, u).

123

H. V. Long et al.

Or

Hd([N2(u(x, y))]α, [N2 (u(x, y))]α)

≤ [a(1 + k1) + b(1 + k2) + ab (c + K + p2k1 + p1k2 + k1k2)] · H(u, u).

It follows

H(N2(u), N2(u)) = sup(x,y)∈Ja×Jb

d∞ (N2 (u (x, y)) , N2 (u(x, y)))

= sup(s,t)∈Ja×Jb

(sup

0<α≤1Hd

([N2(u(x, y))]α , [N2(u)(x, y)]α

))

≤ [a(1 + k1) + b(1 + k2) + ab(c + K + p2k1 + p1k2 + k1k2)] H (u, u) .

This implies that N2 is a contraction operator. And problem (5), (3), (4) have uniquesolution. The theorem is completely proved. �

5 Examples

In this section, we present some examples showing the existence of fuzzy solutionsfor hyperbolic PDEs. Firstly, we fuzzify the deterministic solution (see Allahviranlooet al. 2011; Buckley and Feuring 1999) to built fuzzy solution. After that, by usingthe continuity of Zadeh’s extension principle and numerical simulation we show somegraphical representations of the fuzzy solution.

Example 5.1 Consider the following fuzzy hyperbolic equation

uxy = cey = f (x, y, c) (25)

where c is a constant in J = [0, M], with M > 0, (x, y) ∈ [0, 1] × [0, 1]. And theboundary conditions are

u(x, 0) + 1

4

1∫

0

u(x, y)dy = 1

4[cx(e + 3) + (e + 3)] , (26)

u(0, y) + 1

4

1∫

0

u(x, y)dx =(

1

8c + 5

4

)ey; (27)

The function f (x, y, u) is independent on u, so Hd([ f (s, t, u)]α, [ f (s, t, u)]α) = 0.So the hypothesis (H) is satisfied with any positive number, for example K = 1.That follows the conditions in the Theorems 3.1 and 3.3 hold with k1 = k2 = 1

4 anda = b = 1. Therefore there exists a unique fuzzy solution of this problem.

The solution of the deterministic equations corresponding to (24)–(25) is

u(x, y) = g(x, y, c) = cxey + ey, (x, y) ∈ [0, 1] × [0, 1] .

123


By fuzzifying number c, we have fuzzy number C with α-cut [C]α = [c1(α), c2(α)].The fuzzy solution G(x, y, C) can be defined as fuzzification of this crisp solution,built from Zadeh’s extension principle, as we explain further. By setting

[G]α = [g1(x, y, α), g2(x, y, α)]

where

g1(x, y, α) = min {g(x, y, c)|c ∈ [C]α} = c1(α)xey + ey,

g2(x, y, α) = max {g(x, y, c)|c ∈ [C]α} = c2(α)xey + ey,

and similarly

[F]α = [ f1(x, y, α), f2(x, y, α)] = [c1(α)ey, c2(α)ey] ,

since the partials of f and g with respect to c1 are all positive. Now we checkto see if G is differentiable. Consider differential operator ϕ(Dx , Dy)u(x, y) =∂2u(x,y)

∂x∂y . We compute [ϕ(Dx , Dy)g1(x, y, α), ϕ(Dx , Dy)g2(x, y, α)], which equals[c1(α)ey, c2(α)ey], which are the α-cuts of fuzzy number Cey . Hence G(x, y) isdifferentiable and

ϕ(Dx , Dy)G(x, y, C) = F(x, y, C).

Since all partials of G and F with respect to c are positive and boundary conditionsare all satisfied

g1(x, 0, α) + 1

4

1∫

0

g1(x, y, α)dy = 1

4[c1(α)x(e + 3) + (e + 3)] ,

g2(x, 0, α) + 1

4

1∫

0

g2(x, y, α)dy = 1

4[c2(α)x(e + 3) + (e + 3)] ,

g1(0, y, α) + 1

4

1∫

0

g1(x, y, α)dx =(

1

8c1(α) + 5

4

)ey,

g2(0, y, α) + 1

4

1∫

0

g2(x, y, α)dx =(

1

8c2(α) + 5

4

)ey .

We conclude that G(x, y, C) is the solution of (25)–(27) in the sense of Buckleyand Feuring (see in Allahviranloo et al. 2011; Buckley and Feuring 1999). By usingTriangular fuzzy numbers we can simulate some α-cuts of fuzzy solutions, the resultsis shown in Fig. 1.

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Fig. 1 Numerical simulation for fuzzy solution of (25)–(27) with Triangular fuzzy number [C]α = [0.1α+0.9; 1.1 − 0.1α]

Example 5.2 The hyperbolic PDE is

uxy + ux + uy + u = f (x, y, c) = xy + cy + y + x + c + 1, (28)

where (x, y) ∈ Ja × Jb = [0, 18 ] × [0, 1

8 ], c is a constant in J = [0, M], M > 0.And the local condition are

u(0, 0) = u(x, 0) = 0; u(0, y) = cy. (29)

We have a = b = 18 , p1(x, y) = p2(x, y) = c(x, y) = 1. We recognize that the right

side of the equation is f (x, y, c) = xy + cy + y + x + c + 1, so the hypothesis (H)is satisfied with arbitrary positive number. So all conditions of the Theorem 4.1 hold.Therefore there exists a unique fuzzy solution of this problem.

Indeed, for any fixed c ∈ [0, M], the deterministic solution of (28)–(29) is

u(x, y) = g(x, y, c) = cy + xy.

We fuzzify c, f and g by using the extension principle. Fuzzy function F is com-puted from f and G is computed from g. After that we will show that G is the fuzzysolution of (24)–(25).

Indeed, let

[G]α = [g1(x, y, α), g2(x, y, α)] = [c1(α)y + xy, c2(α)y + xy]

and

[F]α = [ f1(x, y, α), f2(x, y, α)]

= [xy + c1(α)y + y + x + c1(α) + 1, xy + c2(α)y + y + x + c2(α) + 1] .

We consider differential operator

ϕ(Dx , Dy)u(x, y) = uxy + ux + uy + u.

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Fig. 2 Numerical simulation for fuzzy solution of (28)–(29) with Gaussian fuzzy number [C]α =[1 − 0.1

√ln 1

α , 1 + 0.1√

ln 1α ]

We compute

[ϕ(Dx , Dy)g1(x, y, α), ϕ(Dx , Dy)g2(x, y, α)]

the result is

[xy + c1(α)y + y + x + c1(α) + 1, xy + c2(α)y + y + x + c2(α) + 1]

which are the α-cuts of fuzzy number xy + Cy + y + x + C + 1. Hence G(x, y) isdifferentiable and ϕ(Dx , Dy)G(x, y, C) = F(x, y, C). Since all partials of G and Fwith respect to c are positive and boundary conditions are all satisfied

g1(0, 0, α) = g2(0, 0, α) = 0, g1(x, 0, α) = g2(x, 0, α) = 0,

g1(0, y, α) = c1(α)y, g2(0, y, α) = c2(α)y.

Therefore, G(x, y, C) is the fuzzy solution of (28)–(29).In this example, we fuzzify crisp number c by Gaussian fuzzy numbers C with

membership function C(t) = exp(−100(t − c)2). The α−cuts of C are

[c1(α), c2(α)] =[

c − 0.1

√ln

1

α, c + 0.1

√ln

1

α

].

The continuity of extension principle states that fuzzy solution of (28)–(29) is

[G(x, y, C)]α =[(

c − 0.1

√ln

1

α

)y + xy,

(c + 0.1

√ln

1

α

)y + xy

].

The simulation of some α-cuts of this fuzzy solution is shown in Fig. 2.

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Example 5.3 In this example, we consider hyperbolic PDE

uxy(x, y) − 2xux (x, y) + 3y2uy(x, y) − xyu(x, y)

= c(−x2 y2 − 2xy + 3y2x + 1), (30)

where (x, y) ∈ Ja × Jb = [0, 16 ] × [0, 1

6 ], c a constant in J = [0, M], M > 0. Theboundary conditions are

u(x, 0) +1/6∫

0

xu(x, y)dy = 1

72cx2, (31)

u(0, y) +1/6∫

0

y2u(x, y)dx = 1

72cy3. (32)

This example states a = b = 16 , p1(x, y) = −2x, p2(x, y) = 3y2, c(x, y) =

−xy, k1(x) = x, k2(y) = y2 and f (x, y, u) = c(−x2 y2 − 2xy + 3y2x + 1). Itimplies Hd([ f (s, t, u)]α, [ f (s, t, u)]α) = 0 and conditions in Theorem 4.2 hold forany positive number K , like K = 1

6 . Therefore there exists a unique fuzzy solution ofthis problem.

By fuzzifying c we have fuzzy number C and fuzzy solution of (30)–(32) is

[G(x, y, C)]α = [c1(α)xy, c2(α)xy].

Now we consider Parabolic fuzzy number with membership function

C(t) ={

1 − 4(b−a)2

(t − a+b

2

)2if t ∈ [a, b]

0 other

Fig. 3 Numerical simulation for fuzzy solution of (30)–(32) with parabolic fuzzy number [C]α =[2 − 0.2

√1 − α; 2 − 0.2

√1 − α]

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Fuzzy solution of (30)–(32) is

[G(x, y, C)]α =[(

a + b

2− b − a

2

√1 − α

)xy,

(a + b

2+ b − a

2

√1 − α

)xy

].

Fig. 3 shows some numerical simulations for level-sets of fuzzy solution of (30)–(32) with parabolic fizzy number.

6 Conclusions and future works

The main goals of this article have been to investigate the fuzzy solutions of some classof hyperbolic PDEs with local initial conditions and integral boundary conditions. Wehave achieved these goals by using Banach fixed point theorem and illustrated theseresults by some numerical examples. Our results provide the theoretical foundation formany fuzzy PDEs optimal algorithms and support some decision making processes.The next step in our future research is to investigate the existence and uniqueness ofsolution of some classes of fuzzy functional PDEs and extend the existence intervalfrom a restricted domain to the whole domain by using some extension methods.

Acknowledgments The authors would like to thank Editor-in-Chiefs, Prof. Shu-Cherng Fang; AssociateEditor; anonymous reviewers; Prof. Bui Cong Cuong (VAST), Dr. Tran Dinh Ke (HNUE) for their commentsand their valuable suggestions that improved the quality and clarity of the paper. This work is supported byNAFOSTED, Vietnam under contract No.102.01-2012.14.

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