Fuzzy laplace transform on two order derivative and solving fuzzy two order differential equations

Int. J.IndustrialMathematisVol. 2,No. 4(2010)279-293

Fuzzy Laplae Transform on Two Order

Derivative and Solving Fuzzy Two Order

Dierential Equations

S.J.RamazanniaTolouti

,M.BarkhordariAhmadi

Department ofMathematis,BandarAbbasBranh, IslamiAzadUniversity,Bandar Abbas,Iran

Reeived3July2010;revised 30November 2010;aepted11Deember2010.

|||||||||||||||||||||||||||||||-Abstrat

In this paper, the laplae transform formula on the fuzzy two order derivative is

inves-tigated by using the strongly generalized dierentiability onept. Then, it is used in a

analytimethodforfuzzytwoorderdierentialequation. Therelatedtheoremsand

prop-erties areproved indetail and themethodis illustratedbysolvingsome examples.

Keywords : Fuzzy-number, Fuzzy-valued funtion, Generalized dierentiability, fuzzy dierential

equation,fuzzylaplaetransform,fuzzyinitialvalueproblem.

||||||||||||||||||||||||||||||||{

1 Introdution

A natural way to model dynamisystems under unortaintyis to use FDEs. Two order

fuzzy dierential equations are one of the simplest FDEs whih may appear in many

appliations. The topi of fuzzy dierential equations (FDEs) has been rapidly growing

in reent years. The onept of the fuzzy derivative was rst introdued by Chang and

Zadeh[22 ℄; itwasfollowedupbyDuboisandPrade[27 ℄,whousedtheextensionpriniple

in their approah. Other methods have been disussed by Puri and Ralesu [44 ℄ and

Goetshel and Voxman [30 ℄. Kandel and Byatt [37 , 38 ℄ applied the onept of FDEs

to the analysis of fuzzy dynamial problems. The FDE and the initial value problem

(Cauhy problem) were rigorously treated by Kaleva [35 , 36 ℄, Seikkala [45 ℄, He and Yi

[40 ℄, Kloeden [39℄ and Menda [42 ℄, and by other researhers (see [10 , 15 , 17 , 16 , 20 ,

26 , 34 ℄). The numerial methods for solving fuzzy dierential equations are introdued

in [1 , 2, 7 , 32℄. A thorough theoretial researh of fuzzy Cauhy problems was given by

Kaleva[35 ℄,Seikkala[45 ℄,OuyangandWu[40 ℄,andKloeden[39 ℄andWu[48 ℄. Kaleva[35 ℄

disussedthepropertiesofdierentiablefuzzyset-valuedfuntionsbymeansoftheonept

of H-dierentiability due to Puri and Ralesu [44 ℄, gave the existene and uniqueness

theorem for a solutionof the fuzzy dierentialequation y 0

=f(t;y); y(t

0 ) =y

0

when f

satises theLipshitz ondition. Further, song and Wu [46 ℄ investigate fuzzy dierential

equations, and generalizethemain resultsofKaleva[35℄. Seikkala[45 ℄, denedthefuzzy

derivativewhihisthegeneralizationofHukuharaderivative,andshowedthatfuzzyinitial

value problem y 0

= f(t;y); y(t

0 ) =y

0

hasa unique solution, for the fuzzy proess of a

real variable whose values are in the fuzzy number spae (E, D), where f satises the

generalized Lipshitz ondition. Strongly generalized dierentiability was introdued in

[12 ℄ and studied in [10℄. The strongly generalized derivative is dened for a larger lass

of fuzzy-valued funtionthan the H-derivative,and fuzzy dierentialequations an have

solutionswhihhave a dereasinglengthof theirsupport. So weuse thisdierentiability

onept in the present paper. The fuzzy Laplae transform method solves FTDEs and

orresponding fuzzy two order and boundary value problems. In this way fuzzy Laplae

transformsreduetheproblemofsolvingaFTDEtoanalgebraiproblem. Thisswithing

from operations of alulus to algebrai operations on transforms is alled operational

alulus, a very important area of applied mathematis, and for the engineer, thefuzzy

Laplae transform method is pratially the most important operational method. The

fuzzyLaplaetransformalso hastheadvantagethatitsolvesproblemsdiretly,fuzzytwo

order value problemswithoutrst determininga generalsolution, and non homogeneous

dierentialequationswithoutrst solvingtheorrespondinghomogeneous equation.

The paperisorganized asfollows:

Setion2 ontains thebasimaterial to be usedin thepaper. Insetion 3 fuzzy Laplae

transform for two order derivative is dened and Proedure for solving FDEs by fuzzy

Laplae transform is proposed. Several examples are given in setion 4, and onlusions

aredrawninsetion 5.

2 Preliminaries

We now reall some denitions needed through the paper. The basi denitionof fuzzy

numbersisgiven in[25 , 31 ℄.

By R ,we denotethe setof all real numbers. A fuzzy numberisa mappingu: R![0;1℄

withthe followingproperties:

(a)u is uppersemi-ontinuous,

(b)u is fuzzyonvex, i.e., u(x+(1 )y)minfu(x);u(y)g forall x;y2R ;2[0;1℄,

()u is normal,i.e.,9x

0

2R forwhih u(x

0 )=1,

(d) supp u = fx 2 R j u(x) > 0g is the support of the u, and its losure l(supp u) is

ompat.

Let E be the set of all fuzzy number on R . The r-level set of a fuzzy number u 2 E,

or1, denotedby[u℄

r

, isdened as

[u℄

r =

fx2Rj u(x)rg if 0r1

l(suppu) if r=0

Itislearthatther-levelsetofafuzzynumberisalosedandboundedinterval[u(r);u (r)℄,

whereu(r)denotestheleft-handendpointof[u℄

r

andu(r)denotestheright-handendpoint

e y(t)=

1 if t=y

o if t6=y

R an be embeddedinE.

Remark 2.1. (See[52 ℄) Let X beCartesianprodut of universesX =X

1

:::X

n , and

A

1 ;:::;A

n

be n fuzzy numbers in X

1

;:::;X

n

, respetively. f is a mapping from X to a

universe Y, y=f(x

1 ;:::;x

n

). Thenthe extension priniple allows us to denea fuzzy set

B in Y by

B =f(y;u(y))jy=f(x

1 ;:::;x

n );(x

1 ;:::;x

n )2Xg

where

u

B (y)=

(

sup

(x1;:::;xn)2f 1

(y) minfu

A

1 (x

1 );:::;u

A

n (x

n

))g; if f 1

(y)6=0;

0 if otherwise:

wheref 1

isthe inverse of f.

For n=1, the extension priniple, of ourse, redues to

B =f(y;u

B

(y))jy =f(x);x2Xg

where

u

B (y)=

(

sup

x2f 1

(y) u

A

(x); if f 1

(y)6=0;

0 if otherwise:

Aording to Zadeh ;

s extensionpriniple,operation ofaddition onE isdened by

(uv)(x)=sup

y2R

minfu(y);v(x y)g; x2R

and salarmultipliationof a fuzzynumberisgiven by

(ku)(x)= (

u(x=k); k >0;

e

0 ; k=0;

where ~

02E:

It iswellknownthat thefollowingproperties aretrue forall levels

[uv℄

r =[u℄

r +[v℄

r

; [ku℄

r =k[u℄

r

From thisharateristi of fuzzy numbers, we see that a fuzzy numberis determined by

theendpointsoftheintervals[u℄

r

. Thisleadstothefollowingharateristirepresentation

of afuzzy numberintermsof thetwo "endpoint" funtionsu (r)and u(r). An equivalent

parametri denitionisalso given in([29, 41 ℄) as:

Denition 2.1. A fuzzy number u in parametri form is a pair (u ;u) of funtions u(r),

u(r); 0r 1, whih satisfy the following requirements:

ontin-2. u(r) is abounded non-inreasingleft ontinuous funtion in (0;1℄, andright

ontin-uousat 0,

3. u(r)u(r); 0r1.

A rispnumberissimplyrepresentedbyu(r)=u(r)=; 0r 1:Wereall that

for a< b < whih a;b; 2 R , the triangular fuzzy numberu =(a;b;) determined by

a;b; is given suh that u(r) =a+(b a)r and u(r)= ( b)r arethe endpoints of

ther-levelsets, forall r2[0;1℄.

For arbitrary u=(u(r);u (r)), v =(v(r);v(r)) and k >0 we deneaddition uv ,

sub-tration u v and saler multipliationbyk as(See[29 , 41 ℄)

(a)Addition:

uv=(u(r)+v(r);u (r)+v(r))

(b)Subtration:

u v=(u(r) v(r);u (r) v(r))

()Multipliation:

uv=(minfu (r)v(r);u (r)v(r);u(r)v(r);u (r)v(r)g;maxfu(r)v(r);u (r)v(r);u (r)v(r);u(r)v(r)g)

(d)Salermultipliation:

ku= (

(ku;ku ); k 0;

(ku;ku ); k <0:

The Hausdordistane betweenfuzzy numbersgiven by D:EE !R

+ S

0,

D(u;v)= sup

r2[0;1℄

maxfju (r) v(r)j;ju(r) v(r)jg;

where u=(u (r);u(r)), v =(v(r);v(r))R is utilized (See [12 ℄). Then, it is easy to see

thatD isa metriinE andhas thefollowingproperties (See[43 ℄)

(i)D(uw;vw)=D(u;v), 8u;v;w2E,

(ii)D(ku;kv)=jkjD(u;v), 8k2R ;u;v2E,

(iii)D(uv;we)D(u;w)+D(v;e), 8u;v;w;e2E,

(iV)(D;E) isa ompletemetrispae.

Theorem 2.1. (See [8 ℄) (i) If we dene e

0 =

0 , then

e

0 2 E is a neutral element with

respet to addition, i.e. u e

0= e

0u=u, for all u2E.

(ii) With respet to e

0, none of u2EnR ,has opposite in E.

(iii) For any a;b 2 R with a;b 0 or a;b 0 and any u 2 E, we have (a+b)u =

aubu; for the general a;b2R ,the above property does not neessarily hold.

(iv) For any 2R andany u;v 2E, we have (uv)=uv;

(v) For any ;2R and any u2E, we have (u)=(:)u;

It is well-known that the H-derivative (dierentiability in the sense of Hukuhara)for

fuzzy mappingswasinitiallyintroduedbyPuri and Ralesu([44 ℄)and it isbased on the

H-dierene of sets,asfollows.

Denition 2.3. Let x;y 2E. If thereexists z2E suh that x=yz, then z is alled

the H-diereneof x andy, and it is denoted by x h

y.

Inthispaper,thesign" h

"alwaysstandsforH-dierene,andalsonotethatx h

y6=

x y.

In thispaperwe onsiderthe followingdenitionwhihwasintroduedby Bedeand Gal

in([12, 13 ℄).

Denition 2.4. Let f : (a;b)!E and x

0

2(a;b). We say that f isstrongly generalized

dierential atx

0

(Bede-Gal dierential). If thereexists an element f 0

(x

0

)2E, suh that

(i) for all h>0 suÆientlysmall,

9f(x

0

+h) f(x

0

); 9f(x

0

) f(x

0 h)

and the limits(in the metriD)

lim

h&0 f(x

0

+h) f(x

0 ) h = lim h&0 f(x 0

) f(x

0 h) h =f 0 (x 0 ) or

(ii) for all h>0 suÆientlysmall,

9f(x

0

) f(x

0

+h); 9f(x

0

h) f(x

0 )

and the limits(in the metriD)

lim h&0 f(x 0 ) f(x 0 +h) h = lim h&0 f(x 0

h) f(x

0 ) h =f 0 (x 0 ) or

(iii) for all h>0 suÆientlysmall,

9f(x

0

+h) f(x

0

); 9f(x

0

h) f(x

0 )

and the limits(in the metriD)

lim

h&0 f(x

0

+h) f(x

0 ) h = lim h&0 f(x 0

h) f(x

0 ) h =f 0 (x 0 ) or

(iv) for all h>0 suÆientlysmall,

9f(x

0

) f(x

0

+h); 9f(x

0

) f(x

0 h)

and the limits(in the metriD)

lim f(x 0 ) f(x 0 +h) = lim f(x 0

) f(x

(h and h atdenominators mean 1

h and

1

h

, respetively)

Theorem2.2. (Seee.g. [21 ℄) Lety:[0;a℄R !R beontinuousandf :[0;a℄E !

E, be the Zadeh s extension of y, i.e., [f(t;x)℄

r

= f(t;[x℄

r

). If y is non-inreasing with

respettothe seondargument,usingthe derivativeinDenition(2.4), ase (ii),the fuzzy

solution of

y 0

=f(t;y); y(t

0 )=y

0

whenever it exists, oinides withthe solution obtained via dierential inlusions.

Remark 2.2. These ase (iii) and (iv) introdued in [12 ℄, in order to ensure a

dieren-tiable swith the ase (i) and ase (ii) in Denition (2.4). Of ourse, as the authors in

[12 ℄and in [21 ℄havestated, the ases (i) and (ii)in Denition (2.4), aremoreimportant

sinease (iii) and (iv) in Denition (2.4) our only on a disrete set of points. As an

example supporting these omments, let us onsider 2EnRbe any fuzzy(non-real)

on-stant and let f :[0;a℄E !E, f(t)=ost; t2[0;a℄. It is natural toexpet that f

isdierentiableeverywherein itsdomain. Let usobserve thatf isdierentiable aording

to Denition (2.4) (ii), on eah sub interval (2k;(2k+1)) and dierentiable aording

to Denition (2.4)(i), on eah interval of the form (2k+1);(2k), k 2Z. But, at the

pointsfkg, k2Z,noneoftheases(i)and(ii)inDenition(2.4)arefullled. Namely,

atthese pointsthe H-dierenesf(k+h) h

f(k) andf(k) h

f(k h) maynot exist

simultaneously. Also,the H-dierenes f(k) h

f(k+h)andf(k h) h

f(k) annot

exist simultaneously, so f is not dierentiable at k in none of the ases (i) and (ii) of

dierentiability in Denition (2.4). Instead, it will be dierentiable as in the ases (iii)

and (iv) in Denition (2.4). Another argument for the importane of the ases (iii) and

(iv) in Denition (2.4), is in the Theorem (2.2). Indeed, above stated theorem dose not

over the ase when f(t;x) has not onstant monotoniity. In these ases (i) and (ii)

of dierentiability in Denition (2.4), so the ases (iii) and (iv) in Denition (2.4) may

beome important as swith points. In the speial ase when f is a fuzzy-valued funtion,

we have the following result.

Theorem 2.3. (See e.g. [21 ℄). Let f : R ! E be a funtion and denote f(t) =

(f(t;r);f(t;r)), for eah r2[0;1℄. Then

(1) If f is(i)-dierentiable, then f(t;r) and f(t;r) are dierentiablefuntions and

f 0

(t)=(f 0

(t;r);f 0

(t;r))

(2) If f is(ii)-dierentiable,then f(t;r) and f(t;r) are dierentiable funtions and

f 0

(t)=(f 0

(t;r);f 0

(t;r))

Denition 2.5. (See [5 , 6℄) Let f : (a;b)E ! E and x

0

2 (a;b). We Dene the

nth-order dierential of f as follow: We say that f is strongly generalized dierentiable

of the nth order at x

0

. If there exists an element f (s)

(x

0

) 2E; 8s= 1;:::;n, suh

that

(i) for all h>0 suÆientlysmall,

9f (s 1)

(x +h) f (s 1)

(x ); 9f (s 1)

(x ) f (s 1)

and the limits(in the metriD) lim h&0 f (s 1) (x 0

+h) f (s 1) (x 0 ) h = lim h&0 f (s 1) (x 0 ) f (s 1) (x 0 h) h =f (s) (x 0 ) or

(ii) for all h>0 suÆientlysmall,

9f (s 1) (x 0 ) f (s 1) (x 0

+h); 9f (s 1) (x 0 h) f (s 1) (x 0 )

and the limits(in the metriD)

lim h&0 f (s 1) (x 0 ) f (s 1) (x 0 +h) h = lim h&0 f (s 1) (x 0

h) f(x

0 ) h =f (s) (x 0 ) or

(iii) for all h>0 suÆientlysmall,

9f (s 1)

(x

0

+h) f (s 1) (x 0 ); 9f (s 1) (x 0 h) f (s 1) (x 0 )

and the limits(in the metriD)

lim h&0 f (s 1) (x 0

+h) f (s 1) (x 0 ) h = lim h&0 f (s 1) (x 0 h) f (s 1) (x 0 ) h =f (s) (x 0 ) or

(iv) for all h>0 suÆientlysmall,

9f (s 1) (x 0 ) f (s 1) (x 0

+h); 9f (s 1) (x 0 ) f (s 1) (x 0 h)

and the limits(in the metriD)

lim h&0 f (s 1) (x 0 ) f (s 1) (x 0 +h) h = lim h&0 f (s 1) (x 0 ) f (s 1) (x 0 h) h =f (s) (x 0 )

(h and h atdenominators mean 1

h and

1

h

, respetively 8s=1:::n)

To more detail about dierent ases of strongly generalized dierentiability see [5 ,6 ℄

Denition2.6. [6℄Letf(x)beontinuousfuzzy-valuefuntion. Supposethat f(x)e px

improper fuzzyRimannintegrableon [0;1),then R

1

0

f(x)e px

dxisalled fuzzylaplae

transforms and isdenoted as:

L[f(x)℄= Z

1

0

f(x)e px

dx (p>0 and integer)

we have

Z

1

f(x)e px

dx=( Z

1

f(x;r)e px

dx; Z

1

f(x;r)e px

also by using the denition of lassial laplae transform:

`[f(x;r)℄ = R

1

0

f(x;r)e px

dx and `[f(x;r)℄

= R

1

0

f(x;r)e px

dx

then, we follow:

L[f(x)℄=(`[f(x;r)℄;`[f(x;r)℄):

Theorem 2.4. [6℄ Let f 0

(x) bean integrable fuzzy-valued funtion,and f(x)isthe

prim-itiveof f 0

(x) on [0;1). Then

L[f 0

(x)℄=pL[f(x)℄ h

f(0) where f is (i) differentiable

or

L[f 0

(x)℄=( f(0)) h

( pL[f(x)℄) where f is (ii) differentiable

Theorem2.5. [6 ℄Letf(x),g(x)beontinuous-fuzzy-valued funtionsand

1 ,

2

are

on-stant. Suppose that f(x)e px

, g(x)e px

are improper fuzzy Rimann-integrable on [0;1),

then

L [(

1

f(x))(

2

g(x))℄=(

1

L[f(x)℄)(

2

L [g(x)℄):

Theorem 2.6. [6℄ Let f beontinuousfuzzy value funtion and L [f(x)℄ =F(p),Then

L[e ax

f(x)℄=F(p a)

wheree ax

is real value funtion and p a>0.

3 Laplae transform formula on two order fuzzy derivative

and its appliations

In this setion, by using denition of laplae transform on rst-order fuzzy derivative,

laplae transform formula on seond-order fuzzy derivative is introdued then laplae

transformmethod forsolvingseond-order fuzzydierentialequation isproposed.

Theorem3.1. . Letf :R!E beafuntion anddenote f(t)=(f(t;r);f(t;r)),for eah

r2[0;1℄. Then

(1) If f;f 0

aredierentiable in the rst form (i) or f;f 0

aredierentiable in the seond

form (ii), then f 0

(t;r) and f 0

(t;r) are dierentiablefuntions and

f 00

(t)=(f 00

(t;r);f 00

(t;r))

(2) If f is (i)-dierentiable and f 0

is dierentiable in the seond form (ii) or f is

(ii)-dierentiable and f 0

is dierentiable in the rst form (i) , then f 0

(t;r) and f 0

(t;r)

are dierentiablefuntions and

f 00

(t)=(f 00

(t;r);f 00

Proof: Sine the proof proedure is similar for all two ases, we onsider ase (1)

withoutlossof generality.

Iff;f 0

isdierentiable intherst form(i), thenfrom theorem(2.3), we have:

f 0

(t)=(f 0

(t;r);f 0

(t;r))

now, onsiderg(t)asfollows:

g(t)=f 0

(t)

ifh>0 and r2[0;1℄, wehave

g(t+h) h

g(t)=(g(t+h;r) g(t;r);g(t+h;r) g(t;r))

=(f 0

(t+h;r) f 0

(t;r);f 0

(t+h;r) f 0

(t;r))

and,multipliedby 1

h

, wehave:

g(t+h) h g(t) h =( f 0

(t+h;r) f 0 (t;r) h ; f 0

(t+h;r) f 0 (t;r) h ) similarly, f 0 (t) h f 0 (t h) h =( f 0

(t;r) f 0

(t h;r)

h

; f

0

(t;r) f 0

(t h;r)

h

)

passingto the limit,wehave:

f 00

(t)=(f 00

(t;r);f 00

(t;r))

and Iff;f 0

isdierentiableinthe rst form(ii), thenfrom theorem (2.3),wehave:

f 0

(t)=(f 0

(t;r);f 0

(t;r))

now, onsiderg(t)asfollows:

g(t)=f 0

(t)

ifh<0 and r2[0;1℄, wehave

g(t+h) h

g(t)=(g(t+h;r) g(t;r);g(t+h;r) g(t;r))

=(f 0

(t+h;r) f 0

(t;r);f 0

(t+h;r) f 0

(t;r))

and,multipliedby 1

h

, wehave:

g(t+h) h g(t) h =( f 0

(t+h;r) f 0 (t;r) h ; f 0

(t+h;r) f 0 (t;r) h ) similarly, f 0 (t) h f 0 (t h) h =( f 0

(t;r) f 0

(t h;r)

h

; f

0

(t;r) f 0

(t h;r)

h

)

passingto the limit,wehave:

f 00

(t)=(f 00

(t;r);f 00

Denition 3.1. For arbitrary u=(u(r);u (r)), h

u and h

( u) are dened as follows:

h

u=( u (r); u(r))

h

( u)=(u(r);u (r))

Theorem 3.2. Let f 00

(x) be an integrable fuzzy-valued funtion, and f(x);f 0

(x) are the

primitive of f 0

(x);f 00

(x) on [0;1). Then

L[f 00

(x)℄ =p 2 L[f(x)℄ h pf(0) h f 0 (0)

where f is (i) differentiable and f 0

is (i) differentiable

or L[f 00 (x)℄ = h ( p) h

( p)L [f(x)℄ h h

( p) h

( 1)f(0) h h

( 1)f 0

(0)

where f is (ii) differentiable and f 0

is (ii) differentiable

or L[f 00 (x)℄ = h ( p 2

)L[f(x)℄ h h

( p)f(0) h h

( 1)f 0

(0)

where f is (i) differentiable and f 0

is (ii) differentiable

or L[f 00 (x)℄ = h ( p 2

)L[f(x)℄ h h

( p)f(0) h

f 0

(0)

where f is (ii) differentiable and f 0

is (i) differentiable

Proof: By indution, it an be proved easily. we shall now disuss how the laplae

transformmethod solvesfuzzy dierentialequations.

Considerthefollowingfuzzyinitial value problem

y 00

+ay 0

+by=er(t) y(t

0 )= e k 0 ; y 0 (t 0 )= e k 1

withonstant aand b.

By applyingthelaplaetransformmethod on fuzzyinitial value problem,wehave:

L[y 00

℄+aL[y 0

℄+bL[y℄=L[er(t)℄

Then,bysubstituting,laplaetransformformulason rstand seond-order fuzzy

deriva-tive intheorem (2.4) and (3.2) we obtainthefollowingalternativesforsolving:

Case I. Ifweonsider y(t)and y 0

(t) byusing(i)-dierentiable,thenwe have

p 2

L[y(t)℄ h

py(t

0 ) h y 0 (t 0

)apL[y(t)℄ h

ay(t

0

)bL[y(t)℄=L[er(t)℄

Case II. Ifwe onsider y(t) andy 0

(t)byusing(ii)-dierentiable,thenwe have

p 2

L[y(t)℄ h

py(t ) y 0

(t )a( h

ay(t

0

)bL[y(t)℄=L[er(t)℄

CaseIII.Ifweonsidery(t)byusing(i)-dierentiableandy 0

(t)byusing(ii)-dierentiable,

thenwehave

( h

( p 2

))L[y(t)℄( p)y(t

0 ) y

0

(t

0

)apL[y(t)℄ h

ay(t

0

)bL[y(t)℄=L[er(t)℄

CaseIV.Ifweonsidery(t)byusing(ii)-dierentiableandy 0

(t)byusing(i)-dierentiable,

thenwehave

( h

( p 2

))L[y(t)℄( p)y(t

0 )

h

y 0

(t

0 )(

h

p)aL[y(t)℄

( a)y(t

0

)bL[y(t)℄=L[er(t)℄

usingthisrepresentationforfourases, we havethe followingexamples.

4 example

In this setion, we present two examples to illustrate the laplae transform method and

also omparethe resultsofthismethod withother method.

Example 4.1. Consider the one-dimensional heatLetus onsider the seond order fuzzy

dierential equation

8

<

: y

00

3y 0

+2y= e

4

y 0

(0)= e

0

y(0)= e

1

where e

1=(0:8+0:2r;1:5 0:5r) and e

4=(3:2+0:8r;5 r):byusingfuzzylaplaetransform

method, we have:

L[y 00

℄ 3L[y 0

℄2L[y℄ =L[ e

4 ℄

in (i) dierentiable, then by using ase(II), we have

L[y(t;r) ℄=(3:2+0:8r)

1

p(p 1)(p 2)

+(0:8+0:2r)

p 3

(p 1)(p 2)

L[y(t;r)℄=(5 r)

1

p(p 1)(p 2)

+(1:5 0:5r)

p 3

(p 1)(p 2)

Henesolution isas follows:

y(t;r)=(3:2+0:8r)( 1

2 e

t

+ 1

2 e

2t

)+(0:8+0:2r)(2e t

e 2t

)

y(t;r)=(5 r)( 1

2 e

t

+ 1

2 e

2t

)+(1:5 0:5r)(2e t

e 2t

)

Now, if we onsider r=1, then

y(t;1)=y(t;1) =2 2e t

+e 2t

:

By using H-dierentiability and Hukuhara dierentiability onepts the following results

are obtained:

y(t;r)=( 1:6 0:4r)osht 4

sinht+( 0:8+0:2r)osh2t 2

+ 1

3 (e

2t

e 2t

) 1

3 (e

t

e t

)+ 1

3

(5 r)t+ 1

2

(3:2+0:8r) 0:8 0:2r

y(t;r)= 2osht (1:6+0:4r)sinht+osh2t+(0:2r 0:8)sinh2t+ 5

2 1

2 r

The disadvantage of strongly generalized dierentiability of a funtion with respet to

H-dierentiabilityandHukuharadierentiabilityseemstobethatafuzzydierentialequation

has notgot a unique solution. So a fuzzy dierentialequationmayhave several solutions.

the advantage of the existene of these solutions is that we an hoose the solution that

reetsthe behavior of the modelled real-world system, in a betterway.

Example 4.2. onsider the initial value problem equation

8

<

: y

00

+4y = e

4 x

y 0

(0)=0

y(0)= e

1

where e

1= (0:8+0:2r;1:5 0:5r) and e

4 =(3:2+0:8r;5 r): in (ii) dierentiable, then

by using ase(I), we have

L[y(t;r)℄ py(0;r) y 0

(0;r)+4L[y(t;r)℄=L[(3:2+0:8r)t℄

L[y(t;r) ℄ py(0;r) y 0

(0;r)+4L[y(t;r)℄=L[(5 r)t℄

Henesolution isas follows:

y(t;r)=(0:8+0:2r)(x 1

2

sin2t+os2t)

y(t;r)=(5 r)( 1

4 t

1

8

sin2t)+(1:5 0:5r)os2t

Now, if we onsider r=1, then

y(t;1)=y(t;1)=x 1

2

sin2x+os2x

From,examples (4)and (4.2),we seethat thesolution ofa FDE is dependent on the

seletionof thederivative: the(i)-dierentiable orthe(ii)-dierentiable.

5 Conlusion

Developingfuzzy Laplaetransform,weprovidedsolutions to fuzzytwo orderdierential

equationwhihwasinterpretedbyusingthestronglygeneralizeddierentiabilityonept.

Referenes

[1℄ S.Abbasbandy, T.Allahviranloo,Numerial solutionoffuzzy dierentialequationby

Tailor method,Journal of Computational Method in AppliedMathematis 2 (2002)

113-124.

[2℄ S.Abbasbandy,T.Allahviranloo,O.Lopez-Pouso, J.J.Nieto,Numerialmethodsfor

fuzzy dierential inlusions, Computer and Mathematis With Appliations

48/10-11(2004)1633-164 1.

[3℄ T. Allahviranloo, N. Ahmady, E. Ahmady, Numerial solution of fuzzy

dieren-tial equations by preditor-orretor method, Information Sienes, 177/7 (2007)

16331647.

[4℄ T. Allahviranloo, M. Barkhordari, Fuzzy Laplae transforms, Soft

Comput,14(2010)235-243.

[5℄ T. Allahviranloo, N. A. Kiani, N. Motamedi, Solving Fuzzy Dierential Equations,

InformationSienes,179 (2009) 956-966.

[6℄ T.Allahviranloo,N.A.Kiani,M.Barkhordari,TowardtheExistaneanduniqueness

of solutions of seond-order fuzzy Dierential Equations, Information Sienes, 179

(2009) 1207-1215.

[7℄ T. Allahviranloo, Dierene methods for fuzzy partial dierential equations,

Com-putationalmethodsinappliead mathematis, 2(2002),No. 3,PP. 233-242. 169 (2005)

16-33.

[8℄ G. A. Anastassiou, S.G. Gal, On a fuzzy trigonometri approximation theorem of

Weierstrass-type, Journalof Fuzzy Mathematis 9(2004)701-708.

[9℄ B. Bede, S.G. Gal, Generalizations of the dierentiability of fuzzy-number-valued

funtions with appliations to fuzzy dierential equations, Fuzzy Sets and Systems

151(2005)581-599 .

[10℄ B. Bede, I. J. Rudas, A. L. Bensik, First order linear fuzzy dierential equations

undergeneralized dierentiability,informationsienes, 177 (2006)1648-1662.

[11℄ B. Bede, S.G. Gal, Almost periodi fuzzy -number-value funtions, Fuzzy Sets and

Systems147(2004)385-403 .

[12℄ B. Bede, S. G. Gal, Generalizations of the dierentiability of fuzzy-number-valued

funtions with appliations to fuzzy dierential equations, Fuzzy Sets and Systems

151(2005)581-599 .

[13℄ B. Bede, I. Rudas, A. Bensik, First order linear fuzzy dierential equations under

generalizeddierentiablity,informationseieness 177(2006)3627-36 35 .

[14℄ B. Bede, Imre J. Rudas , Attila L., First order linear fuzzy dierential equations

undergeneralized dierentiabilityInformationSienes 177 (2007) 3627-3635 .

[16℄ J.J.Bukly, Simulating ContinuousFuzzy Systems, Springer,2006.

[17℄ J. J. Bukly, T. Feuring, Introdution to fuzzy partial dierential equations. Fuzzy

Setsand Systems105 (1999)241-248.

[18℄ R.B. Bird, W. E.Stewartand E. N.Lightfoot, Transpart Phenomena, 2nd ed,John

Wileyand Sons, NewYork, 2002.

[19℄ JamesJ.Bukly,ThomasFeuring,Introdutiontofuzzypartialdierentialequations,

Fuzzy Setsand Systems105 (1999), PP. 241-248.

[20℄ WuCongxin, S.Shiji,Existenetheorem tothe Cauhyproblemof fuzzydierential

equationsunderompatness-typeonditins.InformationSienes108(1998)123-134.

[21℄ Y.Chalo-Cano,H.Roman-Flores,Onnew solutionsoffuzzy dierentialequations,

Chaos,Solitonsand Fratals 38 (2006) 112-119.

[22℄ S.S.L.Chang,L.Zadeh,Onfuzzymappingandontrol.IEEEtransSystemCybernet,

2(1972) 30-34.

[23℄ S.S.L.ChangandL.A.Zadeh,Onfuzzymappingandontrol,IEEETrans.Systems

ManCybernet.2(1972) 30-34.

[24℄ M. Dehghan, A nite dierene method for a non-loal boundary value

prob-lem for two-dimensional heat equation, Applied Mathematis and Computation

112(2000)133-142 .

[25℄ D. Dubios,H. Prade,Fuzzy numbers: an overview,in: J. Bezdek(Ed.), Analysis of

Fuzzy Information,CRCPress, 1987,pp.112-148.

[26℄ Z.Ding, M. Ma,A.Kandel,Existeneof thesolutionsof fuzzydierentialequations

withparameters,InformationSienes 99(1997) 205-217.

[27℄ D. Dubios, H.Prade, Towards fuzzy dierential alulus, Fuzzy Sets and Systems 8

(1982) 1-7,105-116, 225-233.

[28℄ M. Friedman,M. Ming, A.Kandel,Fuzzy linear systems,FSS, 96 (1998) 201-209.

[29℄ M.Friedman,M.Ming,A.Kandel,Numerialsolutionoffuzzydierentialandintegral

equations,Fuzzy Setsand System 106 (1999) 35-48.

[30℄ R. Goetshel, W. Voxman, Elementeryalulus, Fuzzy Sets and Systems 18 (1986)

31-43.

[31℄ S.G. Gal,Approximationtheoryinfuzzysetting,in: G.A.Anastassiou (Ed.),

Hand-book of Analyti-Computational Methods in Applied Mathematis, Chapman Hall

CRCPress, 2000, pp.617-666.

[32℄ M. Ghanbari, Numerial solution of fuzzy initial value problems under generalized

dierentiabilitybyHPM,Int.J. IndustrialMathematis,Vol. 1,No.1 (2009) 19-39.

[34℄ L.J. Jowers, J. J. Bukly, K. D. Reilly, Simulating ontinuous fuzzy systems,

Infor-mationSienes 177 (2007)436-448.

[35℄ O.Kaleva,Fuzzy dierentialequations, fuzzy Setsand Systems24 (1987) 301-317.

[36℄ O. Kaleva, The Cauhy problem for fuzzy dierential equations, Fuzzy Sets and

Systems35 (1990) 389-396.

[37℄ A. Kandel, Fuzzy dynamial systems and the nature of their solutions, in: P. P.

Wang,S.K.Chang(Eds),FuzzySetsTheoryandAppliationtoPoliyAnalysisand

InformationSystems,PlenumPress, NewYork, 1980,PP,. 93-122.

[38℄ A.Kandel,W.J.Byatt,Fuzzydierentialequations,in:ProeedingoftheInterational

Confereneon CybernetisandSoiety, Tokyo,Novermber 1978, PP. 1213-12160.

[39℄ P. Kloeden, Remarks on Peano-like theorems for fuzzy dierential equations, Fuzzy

Setsand Systems44(1991) 161-164.

[40℄ OuyangHe,WuYi,onfuzzydierentialequations,FuzzySetsandSystems32(1989)

321-325.

[41℄ M. Ma, M. Friedman, A. Kandel,Numerial solutionof fuzzy dierentialequations,

Fuzzy Setsand Systems105 (1999) 133-138.

[42℄ W.Menda,LinearfuzzydierentialequationsystemsonR 1

,JounalofFuzzySystems

Mathematis2 (1)(1988)51-56, inhinese.

[43℄ M. L. Puri, D. Ralesu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986)

409-422.

[44℄ M.L.Puri,D.Ralesu,Dierentialforfuzzyfuntion,J.Math.Anal.Appl.91(1983)

552-558.

[45℄ S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987)

319-330.

[46℄ S.Song, C. Wu, Existene and uniqueness of solutions to Cauhy problem of fuzzy

dierentialequations,FuzzySet andSystem 110 (2000) 55-67.

[47℄ H. C. Wu, The improper fuzzy Riemann integral and its numerial integration, In

form.si.111(1999), 109-137.

[48℄ H.C. Wu, Thefuzzy Riemannintegraland its numerialintegration,Fuzzy Setand

Systems110(2000) 1-25.

[49℄ J. Xu, Z. Liao, Z. Hu, A lass of linear dierential dynamial systems with fuzzy

initialondition,FuzzySets and Systems, 158 (2007) 2339-2358.

[50℄ L.Zadeh,Towardageneralizedtheoryofunertainty(GTU) anoutline,Information

Sienes,172 (2005) 140.

[51℄ L.Zadeh, Isthereaneedforfuzzy logi,InformationSienes,178 (2008)2751-2779.

[52℄ H.J.Zimmerman,Fuzzyprogrammingandlinearprogrammingwithseveralobjetive