Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations

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Int. J.IndustrialMathematisVol. 3,No. 2(2011)67-77

Fuzzy Bivariate Chebyshev Method for Solving

Fuzzy Volterra-Fredholm Integral Equations

M.BarkhordariAhmadi a

, M.Khezerloo b

(a)Department ofMathematis,BandarAbbasBranh, IslamiAzadUniversity,

BandarAbbas,Iran.

(b)Young Researher Club, ArdabilBranh,IslamiAzadUniversity, Ardabil,Iran.

Reeived28August 2010;revised24 January2011;aepted 29January2011.

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In this paper, the fuzzy bivariate Chebyshev method is proposed for solving the fuzzy

Volterra-Fredholmintegralequations(FVFIE).FVFTEisonvertedto adualfuzzy linear

system that an be solved by the proposed method in [10 ℄ . And nally, the method is

explainedwith illustrativeexamples.

Keywords: Fuzzynumbers;FuzzyVolterra-Fredholmintegralequation;FuzzybivariateChebyshev

method; Dualfuzzylinearsystem;Nonnegativematrix.

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1 Introdution

The fuzzy dierential and integral equations are important parts of the fuzzy analysis

theoryand they have theimportantvalueof theory andappliation inontroltheory.

The fuzzy mapping funtion was introdued by Chengand Zadeh[4 ℄. Later, Dubois

and Prade [6 ℄ presented an elementary fuzzy alulus based on the extension priniple

[22 ℄. Puri and Ralesu [21 ℄ suggested two denitions for fuzzy funtion. The onept of

integration of fuzzy funtionswas rstintroduedbyDubois and prade[6 ℄.

Park et al. [18 ℄ onsidered theexistene of solutionof fuzzy integralequation in Banah

spae. ParkandJeong[19 ,20 ℄studiedtheexisteneofsolutionoffuzzyintegralequations

of theform

x(t)=f(t)+ Z

t

0

f(t;s;x(s))ds; t0

wheref and xare fuzzy-valuedfuntions (f;x:(a;b)!E whereE istheset ofallfuzzy

numbers) and k is a risp funtion on real numbers. Alternative approahes were later

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suggestedbyGoetshelandVoxman[11 ℄,Kaleva[13 ℄,Matloka[16 ℄,Nanda[17 ℄andothers,

whileGoetshelandVoxman[11 ℄andlaterMatloka[16 ℄preferredaRiemannintegraltype

approah, Kaleva [13 ℄ hose to dene the integral of fuzzy funtion, usingthe Lebesgue

type onept for integration. A Numerial method is introdued by Allahviranloo and

Otadi[1℄ forsolvingfuzzy integrals.

The topisoffuzzy integralequationswhih attratedgrowinginterestforsometime,

inpartiular inrelationto fuzzy ontrol, have beenrapidlydevelopedinreent years.

Thispaperis organized asfollows:

In Setion2,thebasionept offuzzy numberoperationis brought. In Setion3,fuzzy

solutionof fuzzy dualsystem isrepresented. In Setion4,the mainsetion ofthe paper,

fuzzy Volterra-Fredholm integralequations aresolved byusingfuzzy bivariate Chebyshv

method. TheproposedideaisillustratedbytwoexamplesinSetion5. Finallyonlusion

isdrawninSetion6.

2 Preliminaries

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

numbersisgiven in[6, 12 ℄.

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

withthe followingproperties:

(a) u isuppersemi-ontinuous,

(b) uisfuzzyonvex, i.e.,u(x+(1 )y)minfu(x);u(y)gforallx;y2R ;2[0;1℄,

() u isnormal,i.e.,9x

0

2R forwhihu(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 Ebe theset of all fuzzy numberon R . The r-level setof a fuzzy numberu2E;o

r1,denoted by[u℄

r

, is denedas

[u℄

r =

(

fx2Rju(x)rg if 0r 1

l(supp u) if r=0

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

whereu(r)denotestheleft-handendpointof[u℄

r

andu(r)denotestheright-handendpoint

of [u℄

r

. Sineeah y 2R an beregarded asafuzzy numberey denedby

e y(t)=

1 if t=y

o if t6=y

R an be embeddedinE.

Remark 2.1. [23 ℄, Let X be Cartesian produt of universes X = 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

(3)

where

u

B (y)=

(

sup

(x

1 ;:::;x

n )2f

1

(y) minfu

A

1 (x

1 );:::;u

An (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([9 ,14 ℄) 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:

1. u(r) isa bounded non-dereasing leftontinuousfuntion in (0;1℄, andright

ontin-uousat 0,

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℄.

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sub-(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:

3 Fuzzy Solution of Dual Fuzzy Linear System

Denition 3.1. [10 ℄,The fuzzy linear system

BX =AX+Y (3.1)

is alled a dual fuzzy linear system, where A = (a

ij

);B = (b

ij

);1 i;j n are risp

oeÆient matrix and Y a fuzzy number vetor.

Theorem 3.1. [10℄, Let A = (a

ij

);B = (b

ij

);1 i;j n be nonnegative matries the

dualfuzzy linearsystem(3.1) has aunique fuzzysolutionif andonlyif theinverse matrix

of B A existsand has only nonnegative entries.

system(3.1) annotbeequivalentlyreplaedbythefuzzylinearequationsystem(B

A)X =Y. The dualfuzzy linear system(3.1) istransformed to

y

1 +t

11 x

1

++t

1n x

n +t

1;n+1 ( x

1

)++t

1;2n ( x

n )

=s

11 x

1

++s

1n x

n +s

1;n+1 ( x

1

)++s

1;2n ( x

n )

.

y

n +t

n1 x

1

++t

nn x

n +t

n;n+1 ( x

1

)++t

n;2n ( x

n )

=s

n1 x

1

++s

nn x

n +s

n;n+1 ( x

1

)++s

n;2n ( x

n )

y

1 +t

n+1;1 x

1

++t

n+1;n x

n +t

n+1;n+1 ( x

1

)++t

n+1;2n ( x

n )

=s

n+1;1 x

1

++s

n+1;n x

n +s

n+1;n+1 ( x

1

)++s

n+1;2n ( x

n )

.

y

n +t

2n;1 x

1

++t

2n;n x

n +t

2n;n+1 ( x

1

)++t

2n;2n ( x

n )

=s

2n;1

x ++s

2n;n x +s

2n;n+1 ( x

1

)++s

2n;2n ( x

n )

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wheres

ij and t

ij

are determinedasfollows:

b

ij

0; s

ij =b

ij

; s

i+n;j+n =b

ij

b

ij

<0; s

i;j+n = b

ij ; s

i+n;j = b

ij

a

ij

0; t

ij =a

ij

; t

i+n;j+n =a

ij

a

ij

<0; t

i;j+n = a

ij ; t

i+n;j = a

ij

(3.3)

whileall theremainings

ij and t

ij

aretaken zero.

The followingtheoremguarantees theexistene ofa fuzzy solutionforageneral ase.

Theorem 3.2. [10 ℄, Thedualfuzzy linear equation system(3.1) has a uniquefuzzy

solu-tion if and onlyif the inverse matrix of S T exists andnonnegative.

4 Solution of Fuzzy Volterra-Fredholm Integral Equation

In this setion, by using fuzzy bivariate Chebyshv method, we obtain solution of fuzzy

Volterra-Fredholm integralequation.

Denition 4.1. Considerthe following linear fuzzy Volterra-Fredholm integral equation

e

u(x;y)= e

f(x;y)+ Z

y

1 Z

1

k(x;y;s;t)eu (s;t)dsdt (4.4)

where k(x;y;s;t) is a known funtion and eu(x;y), e

f(x;y) are unknown and known fuzzy

valued funtions, respetively.

We tryto solve Eq. (4.4) byusingdoubleChebyshev series:

e

u(x;y)= N

X

i=0 0

N

X

j=0 0

ea

ij T

ij

(x;y); 1x;y1 (4.5)

whereN isa positiveintegerand

e

u(x;y) = 1

4 e a

00 T

00

(x;y)+ 1

2 e a

01 T

01

(x;y)+ 1

2 e a

02 T

02

(x;y)+:::+ 1

2 e a

0N T

0N (x;y)

+ 1

2 ea

10 T

10

(x;y)+ea

11 T

11

(x;y)+ea

12 T

12

(x;y)+:::+ea

1N T

1N (x;y)

+ 1

2 ea

20 T

20

(x;y)+ea

21 T

21

(x;y)+ea

22 T

22

(x;y)+:::+ea

2N T

2N (x;y)

.

+ 1

2 ea

N0 T

N0

(x;y)+ea

N1 T

N1

(x;y)+ea

N2 T

N2

(x;y)+:::+ea

NN T

NN (x;y)

P

0

denotes a sum whoserst term ishalved, and T

mn

(x;y)=T

m (x)T

n

(y),where T

m (x)

denote the Chebyshv polynomial of the rst kind degree m, and ea

ij

are the Chebyshv

fuzzy oeÆientsto be determined. Thematrix formof (4.5) an bewritten asfollows:

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where

T(x;y)=

T

00 T

01

::: T

0N T

10 T

11

::: T

N0 T

N1

::: T

NN and e A= 1 4 e a 00 1 2 e a 01 ::: 1 2 e a 0N 1 2 e a 10 e a 11

::: ea

1N ::: 1 2 e a N0 e a N1

::: ea

NN

T

where T and e

A are1(N +1) 2

and (N +1) 2

1 matries,respetively. u(x;e y) an be

omputedbyusinga doubleChebyshvseriesinthefollowingsteps:

We substitute theseleted pointsof Chebyshvpolynomialinto Eq.(4.4):

e u(x i ;y j )= e f(x i ;y j )+ e I(x i ;y j

); (i;j=0;:::;N) (4.7)

wherex

i

=os( i

N

) and y

j

=os( j

N

),i;j=0;:::;N that

e I(x i ;y j )= Z y j 1 Z 1 1 k(x i ;y j

;s;t)eu (s;t)dsdt; (i;j=0;:::;N): (4.8)

Similarly, k(x

i ;y

j

;s;t) foreah (i;j =0;:::;N) an byexpanded to the turnated double

Chebyshvseriesin theform

k(x

i ;y

j

;s;t)= N X i=0 0 N X j=0 0 k (i;j) l ;p T l p (s;t)

and theChebyshvoeÆientsare determinedasfollows:

k (i;j)

l ;p =

4

(N +1) 2 N X r=0 N X q=0 k(x i ;y j ;s r ;t q )T l p (s r ;t q

); (p;l=0;:::;N)

and

s

r

=os((2r+1)=2(N +1)); r=0;:::;N

t

q

=os((2q+1)=2(N +1)); q =0;:::;N

So,thematrix representationofk(x

i ;y

j

;s;t) an begiven by

[k(x

i ;y

j

;s;t)℄=K (i;j)

:T T

(s;t) ; (4.9)

where

T(s;t)=

T

00 T

01

::: T

0N T

10 T

11

::: T

N0 T

N1

::: T

NN and K (i;j) = h 1 4 e k 00 1 2 e k 01 ::: 1 2 e k 0N 1 2 e k 10 e k 11 ::: e k 1N ::: 1 2 e k N0 e k N1 ::: e k NN i

Substitutingtheexpressions(4.6) and (4.9) into (4.8), we have

[I(x;y )℄=K (i;j)

:Q(y ): e

(7)

where

L:=T T

(s;t):T(s;t)=[L

m;n ℄

(N+1) 2

and

Q(y

j )=[

Z

y

j

1 Z

1

Ldsdt℄=[q

m;n ℄

(N+1) 2

(N+1)

2; m;n=1;:::;(N +1) 2

; j=0;:::;N

Hene,we obtainthematrix [eu (x

i ;y

j )℄ by:

[eu (x

i ;y

j

)℄=T(x

i ;y

j ):

e

A ; (i;j=0;:::;N) (4.11)

Then,substitutingthe expressions(4.10) and (4.11) into(4.7), we have:

T(x

i ;y

j ):

e

A=[ e

f(x

i ;y

j )℄+K

(i;j)

:Q(y

j ):

e

A ; (i;j=0;:::;N) (4.12)

Now, we supposethat foreah i=0;1;:::;N and j=0;1;:::;N

[W

i(N+1)+j+1 ℄:=K

(i;j)

:Q(y

j ); [f

i(N+1)+j+1 ℄:=[

e

f(x

i ;y

j

)℄ (4.13)

So,from(4.13),Eq. (4.12)istransformedtodualfuzzylinearsystemandwetrytoobtain

thesolutionof thedualfuzzy linear system:

AX =BX +Y

whereA=T(x

i ;y

j

),B =[W

1 W

2

::: W

(N+1)

2℄are(N+1) 2

(N+1) 2

rispmatries

and X = e

A, Y =[f

1 f

2

::: f

(N+1)

2℄ are (N+1) 2

1 fuzzymatries. By solvingthe

above-mentioneddualfuzzy linear system,the unknown oeÆients a

ij

an be omputed

andtherebywendthesolutionoffuzzyVolterra-Fredholmintegralequationintrunated

bivariateChebyshevseries.

5 Numerial example

Example 5.1. Consider the linear fuzzy Volterra-Fredholm integral equationas

e

u(x;y)=(+1;3 )(x+y)+ Z

y

0 Z

1

(x+s)eu(s;t)dsdt

Theapproximated solutionu(x;e y) by fuzzy bivariate Chebyshvmethod asfollows:

rstly, we ompute olloation points for N =2 as:

x

0 =y

0

=1; x

1 =y

1

=0; x

2 =y

2 = 1

and

s

0 =t

0 =

p

3

2 ; s

1 =t

1

=0; s

2 =t

2 =

p

3

2

We are goingto obtainthe unknown matrix

e

A=

1

e a

00 1

e a

01 1

e a

02 1

e a

10 e a

11 e a

12 1

e a

20 e a

21 e a

22

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We have

T(x;y) = T 00 T 01 T 02 T 10 T 11 T 12 T 20 T 21 T 22 =

1 2y 4y 2

1 4x 4xy 8xy 2 2x 4x 2 1 8yx 2 2y (4x 2 1)(4y 2 1) and K (0;0) =K (0;1) =K (0;2) = 1 4

0 1 1 0 4 1 0 4 K (1;0) =K (1;1) =K (1;2) =

0 0 0 1 0 4 0 0 0 K (2;0) =K (2;1) =K (2;2) = 1 4

0 1 1 0 4 1 0 4

and also, Q(y 0 )= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4 0 4 3

0 0 0

4 3 0 4 9 0 4 3

0 0 0 0 0

4 9 0 4 3 0 28 45

0 0 0

4

9 0

28

45

0 0 0

4

3 0

4

9

0 0 0

0 0 0 0

4

9

0 0 0 0

0 0 0

4

9 0

28

45

0 0 0

4

3 0

4

9

0 0 0

28 45 0 28 45 0 4 9

0 0 0 0 0

28 45 0 4 9 0 28 45

0 0 0

28 45 0 196 225 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Q(y 1 )= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 1 2 3

0 0 0

2 3 1 3 2 9 1 2 3

0 0 0 0

1 3 2 9 0 2 3 0 14 15

0 0 0

2

9 0

14

45

0 0 0

2 3 1 3 2 9

0 0 0

1

3 2

9

0 0 0 0

0 0 0

2

9 0

14

15

0 0 0

2 3 1 3 2 9

0 0 0

14 15 7 15 14 45 1 3 2 9

0 0 0 0

7 15 14 45 0 2 0 14

0 0 0

(9)

and

Q(y

2 )=

2

6

4

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 3

7

5

So,

A= 2

6

4

1 2 3 4 4 6 3 6 9

1 0 1 4 0 2 3 0 3

1 2 3 4 4 6 3 6 9

1 2 3 0 0 0 1 2 3

1 0 1 0 0 0 1 0 1

1 2 3 0 0 0 1 2 3

1 2 3 4 4 6 3 6 9

1 0 1 4 0 2 3 0 3

1 2 3 4 4 6 3 6 9

3

7

5

and

B= 2

6

4

0:111 0 1:756 0:444 0 2:044 1:756 0 2:573

0:056 0:083 0:256 0:222 0:333 3:511 0:256 0:383 1:176

0 0 0 0 0 0 0 0 0

0 0 0 0:444 0 2:044 0 0 0

0 0 0 0:222 0:333 3:511 0 0 0

0 0 0 0 0 0 0 0 0

0:111 0 1:756 0:444 0 2:044 1:756 0 2:573

0:056 0:083 0:256 0:222 0:333 3:511 0:256 0:383 1:176

0 0 0 0 0 0 0 0 0

3

7

5

and

T

(10)

Then,

e

A T

= 2

6

4

(7:617 7:617; 7:617+7:617)

(31:215 31:215; 31:215 +31:215)

(43 43; 43+43)

(22:16 22:16; 22:16+22:16)

(14:55 14:55; 14:55+14:55)

(4:813 4:813; 4:813+4:813)

(42:53 42:53; 42:53+42:53)

(8:225 8:225; 8:225+8:225)

(38:58 38:85; 38:58+38:85)

3

7

5

6 Conlusion

Inthiswork,wetriedtoobtainthesolutionoffuzzyVolterra-Fredholmintegralequations

byusingfuzzy bivariate Chebyshvmethod. FVFIE was onverted to a dual fuzzy linear

systemthatan beapproximatedbythemethodthatwasproposedin[10 ℄. TheeÆieny

of methodwas illustratedbyone numerial example.

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