JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58. No. 7, pp. 27-38 This paper is available online at http;//stdb.hnue.edu.vn
F U Z Z Y SOLUTIONS F O R GENERAL H Y P E R B O L I C PARTIAL D I F F E R E N T I A L EQUATIONS W I T H L O C A L INITIAL CONDITIONS N g u y e n T h i ]Vly H a \ N g u y e n T h i K i m Son^ a n d H a T h i T h a n h T a m ^
^Faculty of Mathematics, Hai Phong University
^Faculty of Mathematics, Hanoi National University of Education
^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 [ I I ] .
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: [email protected]
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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 [I]. 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 x Jf,. 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 M" by /CJ.
Let A and B be two nonempty bounded subsets of/CJ. Denote by ||.|| a norm mM". The distance between A and B is defined by the Hausdorff metric
dH{A,B) = maxl supinf \\a- 6||,supinf \\a- b\\ >
and (^c,c?//) is a complete space [II].
Let £•" be the space of functions u: R" -^ [0,1] satisfying:
i) there exists a a:o 6 R'* such that u{xo) = 1;
ii) u is fuzzy convex, that is for x,z EW and 0 < A < 1, u{Xx-^ (1 - A)^) > min[u{x),u{z)]\
iii) u is semi-contmuous;
iv) {u]° = {x EW^ : u{x) > 0} is a compact set in R'^.
If u e £'", u is called a fuzzy set and the a-level of u is defined by [u]° - {a: 6 R" : u{x) > a} for each 0 < a < 1.
Then from (i) to {iv), it follows that [if]"^ is in /C^.
The fuzzy sets u E E^ is called fiizzy numbers. The triangular fuzzy numbers are those fuzzy sets in E^ for which the sendograph is a triangle. A triangular fiizzy number u is defined by three numbers ai < 02 < 03 such that [u]° = [oi, 03], u^ = 02. We write u > 0 if ai > 0, u > 0 if fli > 0, tt < 0 if 03 < 0 and u < 0 if as < 0. The a-Ievel set of a fuzzy number is presented by an ordered pair of function [ui{a),u2ia)],0 < a < 1 which satisfies the following requirements:
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ruzzy solutions jor general tiyperbolic partial Oifferential equations with local initial conditions i) ui[oi) is a bounded left continuous non-decreasing function of a,
ii) ^2(0) is a bounded left continuous non-increasing function of a, iii) U i ( l ) < U 2 { l ) .
If p : M" X M" -?• R " is a function, then, according to Zadeh's extension principle we can extend g \.o E'^ x E'^ -)• E^ by the function defined by
g{u,u){z) = sup mm{u{x),u{'z)].
If 5 is continuous then
[g{u,u)r ^ giiu]" ,[u]'') forallw.we-R'*, 0 < a < 1.
Especially, we will define addition and scalar multiplication of fuzzy sets in E"
levelsetwises, that is, for Mu,u EW,0 < a < 1, k ER,k ^ 0 [u + u]° = [u]" -!- [u]°
and
[kyf = k[u]°
where
{u-\-u){x) = sup min{u(xi),u(x2)}
and
ku{x) = u{x/k).
Supremum metric is the most commonly used metric on E" defined by the Hausdorff metric distance between the level sets of the fuzzy sets
dooiu,u) = sup Hd{[uY',[u]°) 0 < a < l
for all u,u E £ " . It is obvious that ( £ " , d ^ ) is a complete metric space [11]. From the properties of Hausdorff metric, we have:
i) dcx,{cu,cv) = \c\.doo{u,v),
ii) doo{u + u',v-hv') < d<^{u,v)-\-doo{u',v'), iii) doc{u + w,v-\-w) = doo{u,v)
for all u, V, u', v\ w E E^ and c 6 E.
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Definition 2.1. Let J a = [0,a],Jb = [0,6], 7^6 = [0,a] x [0,6]. A map f : Jab ^ S"
is called continuous at {IQ, SQ) E Jab if ihe multi-valued map fa{t,s) = lf{t,s)]"'is continuous at {tQ,so) with respect to the Hausdorff metric d a far all a E [0,1].
In this paper, we denote C{Jab, £'") be a space of all continuous functions f : J ^ E^ with the supremum weighted metric Hi defined by
i ^ i { / , 5 } - sup d^{f{t,s).g{t,s))e-^^'^''^
Since {E^, d^) is a complete metric space, it can shown that {C{Jab, £"•), Hi) is also a complete metric space (see [11]).
Deflnition 2.2. A map f : Jab 'X- E'^ ^ E'^ is called continuous at point {to, SQ, XQ] E Jab X E^ provided, for any fixed a e [0,1] and arbitrary e > 0, there exists 6{e, a) > 0 such that
d«([/(f,s,3;)r,[/(f„,5o,x„)n<f
whenever max \t - toi, | s - so| < Si(,a)and dn Hx]" , [xo]") < 6ie,a)forallit,a,x) 6
Jab X E".
Definition 2.3. A function f : J^b —^ E" is called integrably bounded if there exists an integrable function h E I,'(J, R") such that \\y\\ < his,t) for ally 6 fois.t).
Definition 2.4. Let f : J^b^ E". The integral off over Jab, denoted by /„"/„'/ iUs) dsdt is defined by
j I fit, s) dsdt) = I I f^it,s)dsdt JQ JO I 7o Jo
={rfvi
Jo Jo
J it, s) dsdt\v : Jab -> Wis a measurable selection for fa}
for all c 6 (0,1] (see ]3]). A function f Jab -> £ " is integrable on Jab if
lofo f f^' ^) ^^'^^ i^'" E".
Let /, g : Jab -* E" be integrable and A 6 R. The intergral has the elementary properties as follows
i) /o7o [/(*. s) + git, s)]dsdt = /„7„V(t, s)dsdt + f^jllgit, s)dsdt, ii) /„7o V(*. ^'^dsdt = A/„7„V(i, s)dsdt,
ii) d„i£J^fit,s)dsdt,J„'f^it,s)dsdt) < j;j^dMit,s),git,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'^. If there exists z E E"^ such that x = y + z then we call z the Hukuhara-difference of X and y, denoted x — y (see [11 J).
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^. The fuzzy partial derivative of f with respect to x at the point (XQ, T/Q) € J is a fuzzy set ^ E E^ which is defined by
ox
df{^-o,yo) ^ j . ^ f[^-o + h,yo) - f{xo,yo) dx ll-^0 h
Here the limit is taken in the metric space [E^, d^) and u — v is the Hukuhara-difference ofu and v in £ " . The fuzzy partial derivative off with respect to y and the higher order of fuzzy partial derivative of f at the point {.7:0.1/o) E Jab f^^s defined .'similarly (.see [6]).
3. ]V[ain result
The aim of this section is to consider the existence and uniqueness of the fuzzy solutions for the general hyperbolic partial differential equation
9 ^ g ^ ^ dip,ix,y)u{x,y)) ^ dip2ix,y)uix,y)) ^ ^^^_^^^^^^ y) = fix,y,«(x,y))
x y X y ^^^^
for ix, y) e Jab- The local initial conditions are
ti(0, 0) = UQ, uix, 0) = tiiix), «(0, y) = ri2iy). ix, y) 6 Jab, (3.2) where p, 6 C ( J . j , R ) , i = 1,2, c £ C ( J „ t , R ) , Tji £ C ( J < „ £ " ) , jjj £ CiJb,E") are given functions; Uo £ £ " and / : J ^ x C ( J , t , E") -> E" which satisfies the following hypothesis:
Hypothesis H
There exists K > 0 such that
docifix,y,uix,y)),fix,y,uix,y))) < Kd„iuix,y),uix,y)) holds for all u,ue E", ix, y) £ Jab-
Dermition 3.1. A function u £ C ( J ^ , E") is called a solution of the problem (3.1), (3.2) if it sadsfies
uix,y)=qiix,y)- / piix,s)uix,s)ds - / p2(*,y)ti(t,y)df Jo Jo - f I c(«,s)i/(i,s)ds(i«+ / / fit,s,uit,s))dsdt,
Jo Jo JO Jo
where
qi{x>y) = m{^^) + V2{y)-uo+ / PI{0, s)i]2{s)ds + / p2{t,Q)7)i{t)dt Jo Jo for all [xty] E Jab-
By using the new weighted metric Hi in the space C(J„(,,E"), 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^).
Proof Letpi - sup(,_,)gj^^j^ \pi{t, s)\,p2 = sup^^^^^^j^^j^ \p2{t, s)|, c =
sup(f s)EJa^Jb 1''^^' ^^1' ^^^^ 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"") -)• C[Jab, E"") defined as follows:
N{u){x,y)=qi{x,y)- I pi{x,s)u{x,s)ds - / p2{t,y)u{t,y)dt Jo Jo - c{t,s)u{t,s)dsdt+ / / f{t,s,u{t,s))dsdt.
Jo Jo Jo JQ
We will show that N is a contraction operator. Indeed, if u,u G C{Jab, E^) and a E (0,1]
then
N{u{x,y)) =qi{x,y) - / pi{x,t)u{x,t)dt - / P2{s,y)u{s,y)dx Jo Jo
ox ry rx ry
- c{t,s)u{t,s)dsdt+ / / f{t,s,u{t,s))dsdt Jo Jo Jo Jo
and
N{u{x,y))=qi{x,y)- pi{x,t)u{x,y)dt - p2{s,y)u{s,y)ds Jo Jo - c{t,s)u{t,s)dsdt-\' / / f{t,s,u{t,s))dsdt.
Jo Jo Jo Jo From the properties of supremum metric, we have:
ry ry d^{N{u{x,y)),N{u{x,y)))<d^{ pi{x,s)u{x,s)ds, / pi{x,s)u{x,s)ds)
Jo Jo -d^{ P2{t,y)u{t,y)dt, I P2{t,y)u{t,y)dt)
Jo Jo
i-x ry i-x ry
-dcoi / c{t,s)u{t,s)dsdt, / c{t,s)u{t,s)dsdt) Jo Jo Jo Jo
«{l I fit,s,uit,s))dtds, I I fit,s,uit,s))dsdt) Jo Jo Jo Jo
+ c + c
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Fuzzy solutions for general hyperbolic partial differential equations with local inilial conditions Moreover
ry ry doci Pi{x,s)u{x,s)ds, pi{x,s)u{x,s)ds)
Jo Jo ry ry
< sup |pi(f,.s)Mco( / u{x,s)ds, u{x,s)ds) {t,s)€JaxJb Jo JQ
< P i / d^{u{x,s),u{x,s))ds.
JQ
Hence for each {x, y) E Ja ^ Jb, one gets
fy ry .-^i^+y)d^{ pi{x,s)u{x,s)ds, pi{x,s)u{x,s)ds)
Jo Jo
< p i e - ^ t ^ + ' ' ' rrfoo(w(:r,s),u(.T,s))e-^(^+^'e^f^+^>ds Jo
<PiHi[u,u)e-^^^^''^ ['e^^^-^''>ds Jo
<^Hi{u,u).
Similarly, we obtain:
'=--'f"+^^rfoc(y P2{t,y)u{t,y)dt, j\2[t,y)u{t,y)dt) < ^Hi[u,u).
Nevertheless
rx ry rx ry
doo{ / c{s,t)u{s,t)dsdt, / c{s,t)u{s,t)dsdt) Jo Jo Jo Jo
< sup | c ( i , s ) | / / doo{u{s,t),u{s,t))dsdt (i.5)eJ„xJt Jo ^0
<c / d^{u{s,t)MsJ))dsdt.
Jo Jo Hence
e-^(^+y)doo{ I I c{t,s)u{t,s)dtdsj j c[t,s)u{t,s)dsdt) Jo Jo Jo Jo
^^^-x{x+y) r rd^{u{s,t),u{s,t))e-^^'+''>e^^'+'^dsdt Jo Jo
< cll2{u,u)e'^^''^-'^ r r(i^^'+'Msdi Jo Jo
< -^rhiu,u)-
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Moreover, one gets
dooiT rf{t,s,u{t,s))dtds, f rf{t,s,uit,s))dsdt) Jo Jo JQ JO
<K I I d^{u{t,s).u{t,s))dsdt.
Jo JO It implies that
-^-^f^+^^rfool r r f{t,s,u{t,s))dxdy, [ r f{t,s,u{t,s))dsdf)
Jo Jo JQ JQ
<Ke-^i-+y) r rd^{u{x,y),u{x,y))e-^^'^'''>e^^'+'^dsdt JQ JO
<KHi{u,u)e~^^''+'^ f f e^^'+'^dsdt JO JO
<^H,iu,u).
H,iNiuix,y)),Niuix,y))) < [ ? i ± ^ + ^ ] f f i ( „ , n ) . Since we can choose A > 0 satisfying
Pt + P2 c + K X X^ '
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 d^u{x,y)
dxdy with the local conditions
= - C i , ( x , y ) e [ 0 , l ] x [ 0 , l ] (4.1)
uix,0) = uiO,y) = uiO,0) = C2, (A.I) where Ci, C2 are triangular fuzzy numbers in [0, M], M > 0 with the following level sets
icir = [cs,cf2i,[C2r = [c?„c£]
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Fuzzy solutions for general hyperbolic partial differential equations with local inilial conditions for Q E [0,1] and {x, y) E [0,1] x [0,1].
In this problem, if f{x, y, u{x, y)) = -Ci then condition (H) is .satisfied with K = 2.
Therefore, from Theorem 3 J there exists a solution to this problem.
Next, we find this fuzzy solution. A.ssume that solution u has level sets [u]° = [ui{x,y)'',U2{x,y)°]fbraE [0,l]and {x,y) E [0,1] x [0,1]. We also have
r ^ M ^ j ^ ^ m ^ ,9^u^{^'y) d^n1[x,y) dxdy dxdy ' dxdy Applying the extension principle, the fuzzy number —Ci has level sets
[-CiY = [ m i n { - C J ' i , - C ° } , m a x { - C f i , - C r 2 } ]
= [~C'i2' ~ ^ n ]
for a E [0,1] and {x, y) E [0,1] x [0,1]. Thus the equation (4.1) is equivalent to the .fy.stem dhi°{:i.y) ^ _ ^ ^ d'^u^(x.y) ^ _ ^ „
dxdy ^^' dxdy ^^
The local conditions (4.2) is equivalent to the following system
< ( x , 0 ) = < ( 0 , y ) - < ( 0 , 0 ) = C^„ (4.4)
< ( a ; , 0 ) ^ u^{0,y) = u ^ ( 0 , 0 ) = Cf^- (4.5) The solutions of system (4.3) with conditions (4.4), (4.5) are
< ( 2 : , y) = -C°2^y + C21, u^{x, y) = -Cf^xy + C^^- Hence, the solution of problem (4.1), (4.2) has level sets
M" ^ [-^n^y + ^2v -^u^y + ^"22]
for a E [0,1] and [x, y) E [0,1] x [0,1]. We can write u[x, y) = -Cixy + C2.
Example 4.2. Consider the fuzzy hyperbolic equation
^'"t'^'^) + ? : ± ^ + M ^ + „(x, y) = 4Ce^«, (i, y) S [0, 2] x [0,2], (4.6) dxdy dx dy
with the local conditions
u(a.-, 0) = Ce', u(0, y) = Ce", ti(0,0) = C,
where C is a fiazy triangular number in [0,M\. M > 0. C has level sets [C]" -= [C]
forae [0,1].
We have fix, y, uix, y)) = iCc'*' then f satisfies condition (H) with K = \. Hem
condition of Theorem 3.1 holds. Therefore, there exists a fuzzy soludon of this problem.
Next, we will give a clear solution. Suppose that solution u heu; level sets [u\° = [MI(X, y)", U2ix, y)"] for a £ [0,1] and (.x, y) £ [0, 2] x [0,2]. Define
,^ ^ ,... . d'^uix,y) du(x,y) duix.y) , , AJJ., D,)Uix, y) = - ^ + 4 ^ + - ^ + uix. y) then '^{Dx, Dy)U{x, y) also has level .sets
y^D.,rj,Uix,y)T - f ^ + ^ ^ . ^ . <ix,y), dxdy ''" dx ^ dy +"2t-'^'Wi for all a E [0,1] and (,T, y) E [0, 2] x [0, 2]. By the extension principle, we have
[4Ce^+^]° - [min{4(:7fe^+^4C2°6^+^},max{4Cfe^+^4C2''e^+^}]
= [4Ci''e"-^^4C2^e^+^]
Similarly
[Ce'Y' - [Cf e", C^e% [Ce^f = [ C f e ^ C^e"]
for all a E [0,1] and [x, y) E [0, 2] x [0, 2]. Hence, the equation (4.6) is equivalent to the system
s^|ip) ^ a ^ ^ dj^ ^ ^ ^ ^^^^
dxdy dx dy
d^^i^ + 5 ! i ^ + M ^ + „ = ( , , , ) ^ 4C?e-». (4.9) raoy dx dy
The local conditions (4.7) are equivalent to the following
u^ix, 0) = 4Ci°e', tif (0, y) = 4C^°e^ uj(0, 0) = 4Cf, (4,10) iiJ(.x-,0) = 4CJe*.uJ(0,y) = 4CJe=^, ti5(0, 0) = 4Cj°. (4.11) Solving the problems (4.^), (4.10) fl«rf (4.9), (4.11) we have the solutions:
< ( x , y ) = iC^e'+',uiix,y) = 40^6'+' for all a £ |0,1] and ix, y) £ |0, 2] x [0, 2].
Thus, the solution of problem (4.6), (4.7) is a fuzzy function u, which has level sets lu]'" = [4Cfe'+',4CJe'+"]
and we can write u = ACif^^.
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Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions Example 4.3. We study the following hyperbolic equation:
d'^ulx, y) duix, v) 1 1
- a ^ - - ^ = 2 C , - 2C,(x + y), ix,y) £ [0, ^J x [0, i ] , (4.12)
uix, 0) = C I . T ' + C j , ii(0, y) = Ciy'' + Czsini/ + C3, u(0,0) = C3, (4.13) with ix,y) £ [0, | ] X [0, g] ant/ Cj ;;emg triangular fuzzy numbers having level .sets
\C\f = [Cf,, C%\ > r . = 1, 3, o g [0,1] a^rf (3;, y) e [0, i ] X |0, i ] .
We have fix, y, u) = C,i/ + C2Siuy + C3, It follows that the condition in Theorem 3.1 is satisfied with K = I. Therefore there exists a solution of this problem.
Suppose that solution u has level sets [u]" = [wi(x,T/)'^,U2(x,y)°] for a £ [0,1] and {.T, y) 6 [0, | ] X [0, g]. Using the extension principle, we have
,9'"(-T^,'j) _ ditix,y)-.a ^ .dhj!iix,y) _ du^{x,y) d''u°ix,y) _ a;t?(.r.y) dxdy dx dxdy dx ' dxdy dx and
[2C, - 2Ciix + y)]' = pCf, - 2Ci°j(x + y)], 2C'^2 " 2Cf,(x + y)|
[Cix^ + Cj]" = [C7f,x^ + CS„C-;2X^ + CJ,],
[Ciy^ + Cisiny + C,]" = [C^^y^ + CJjsiny + Cj";, C g y ' + CJ^siny + CJjl Thus equation (4.12) /^ eqiiivaleiu to the following system
d^u^ix,y) du^ix,y) dxdy dx d^u^ix.y) du1ix,y)
= 2 C n ~ 2 ( 7 i 2 ( i + y),
= 2 C f 2 - 2 C f , ( a : + y).
dxdy dx The initial conditions (4.13) are equivalent to the system
uf ( l , 0) = C f i i ^ + CJ,, < ( 0 , y) = Cfiy^ + (^JiSiny + Cfi, tij(0, 0) = Cfj
u^ix, 0) = C^a:' + Cfj, uf (0, y) = Cfjy^ + C^smy + CJ-j, u5'{0, 0) = C^2- The solutions of the system are given by
< = Ctiix + yf + C^isiny + CJi u? = (^^^(x- + y)^ + C^2S^tiy + CJ2
Therefore, the problem (4.12), (4.13) fau a solution ii(a:, y) = Cj (a: + y)^ + Cjsiny + C3 wir/i /eve/ jerj
H ° = [Cni^ + y ) ' + CJiSiny + Cf,, CJ-^Cx + y)^ + C^smy + CJ^]
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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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