Numerical Solution of Linear Fredholm Fuzzy Integral Equations by Modified Homotopy Perturbation Method

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1 Austrln Journl of Bsc nd Appled Scences, 4(): , ISS umercl Soluton of Lner Fredholm Fuzzy Integrl Equtons y Modfed Homotopy Perturton Method S.M. Khorsn Ksr, M. Khezerloo, 3 M.H. Dogn Aghcheghloo Deprtment of mthemtce, Shhr-E-Rey Brnch,Islmc Azd Unerversty,Tehrn, Irn Islmc Azd Unerversty, Young Resercher Clu, Ardl Brnch, Ardl, Irn 3 Pure mth Deprtment of the shhd Bhonr Unversty of Kermn.Kermn, Irn Astrct: By usng prmetrc form of fuzzy numers we convert Fredholm Fuzzy Integrl Equton to lner system of ntegrl equtons of the second knd n crsp cse.we use Modfed homotopy perturton method nd fnd the pproxmte soluton of ths system nd hence otn n pproxmton for fuzzy soluton of the lner fuzzy Fredholm ntegrl equton. Key words: Fuzzy ntegrl equtons, Homotopy perturton method ITRODUCTIO The topcs of fuzzy ntegrl equtons (FIE) whch growng nterest for sometme, n prtculr n relton to fuzzy control, hve een rpdly developed n recent yers. Pror to dscussng fuzzy dfferentl nd ntegrl equtons nd ther ssocted numercl lgorthms, t s necessry to present n pproprte ref ntroducton to prelmnry topcs such s fuzzy numers nd fuzzy clculus.the concept of fuzzy sets, ws orgnlly ntroduced y Zdeh (975), led to the defnton of fuzzy numers nd ts mplementton n fuzzy control nd pproxmte resonng prolems. The sc rthmetc structure for fuzzy numers ws lter developed y Duos nd Prde (978), ll of them oserved fuzzy numers s collecton of levels. Addtonl relted mterl cn e found n. Goetschel nd Voxmn (986) suggested new pproch. They represented fuzzy numer s prmeterzed trple (see Secton ) nd then emedded the set of fuzzy numers nto topologcl vector spce. Ths enled them to desgn the scs of fuzzy clculus. Prelmnres: In ths secton the most sc nottons used n fuzzy clculus re ntroduced. Defnton.: A fuzzy numer s fuzzy set u : R I =[,] whch stsfes. u s upper sem contnuous. cd,. ux ( )= outsde some ntervl 3. there re rel numer, : cd for whch: c,. ux ( ) s monotonc ncresng on d,. ux ( ) s monotonc decresng on.3 ux ( )=, x An lterntve defnton or prmetrc form of fuzzy numer whch yelds the sme E s gven y Klev. A fuzzy numer u s pr ( ur ( ), ur ( )) of functons ur (), ur ();rwhch stsfy the followng Correspondng Author: S.M. Khorsn Ksr, Deprtment of mthemtce, Shhr-E-Rey Brnch,Islmc Azd Unerversty,Tehrn, Irn 646

2 Aust. J. Bsc & Appl. Sc., 4(): , requrements:. ur () s monotonclly ncresng left contnuous functon.. ur () s monotonclly decresng left contnuous functon. 3. ur () ur (),r. A populr fuzzy numer s trpezodl fuzzy numer wth tolernce ntervl wdth we use the notton: u =(,,, ) Its prmetrc form: ur ( ) = ( r), ur ( ) = ( r),, left wdth nd rght If = then trpezodl trnsform to trngulr fuzzy numer nd we denote ll of the trngulr fuzzy numer wth FT ( R). Let v =( vr ( ), vr ( )), u =( ur ( ), ur ( )). Some results of pplyng fuzzy rthmetcs on fuzzy numers v, u re s follows: (Zdeh, L.A., 975) ; x >: x=( xv( r), xv( r)) x <: x=( xv( r), xv( r)) vu =( vr ( ) ur ( ), vr ( ) ur ( )) vu =( vr ( ) ur ( ), vr ( ) ur ( )) Defnton.(Gl, S.G., ) For rtrry fuzzy numers ( ur ( ), ur ( )), ((),()) vr vr the quntty Duv (, )= mx sup vr ( ) ur ( ), sup vr ( ) ur ( ) r r u v s the dstnce etween nd. If the fuzzy functon f(t) s contnuous n the metrc D, ts defnte ntegrl exsts. Furthermore f (, trdt ) f(, trdt ) f (, trdt ) f(, trdt ) () 3 Fuzzy Integrl Equton: The ntegrl equtons whch re dscussed n ths secton re the Fredholm equtons of the second knd(ffie-) s follow: ut ()= f() t Kstusds (,) () () 647

3 Aust. J. Bsc & Appl. Sc., 4(): , where >, Kst (, ) s n rtrry kernel functon over the squre s, t nd f(t) s functon of t: t. If f(t) s crsp functon then the solutons of Eq. () re crsp s well. However, f f(t) s fuzzy functon these equtons my only possess fuzzy solutons. Suffcent condtons for the exstence of unque soluton to the fuzzy Fredholm ntegrl equton of the second knd,.e. to Eq. () where f(t) s fuzzy functon, re gven n(guerr, M.L., L. Stefnn, 5). ow we ntroduce prmetrc form of FFIE- wth respect to Defnton.. Let ( f ( tr, ), f( tr, )) nd ( utr (, ), utr (, )), r nd t[, ] re prmetrc form of f(t) nd u(t), respectvely then, prmetrc form of FFIE- s s follows: utr (, )= f(, tr) v( stusr,, (, ), usr (, )) ds utr (, )= f(, tr) v( stusr,, (, ), usr (, )) ds (3) where Kstusr (,)(, ) Kst (,) v ( s, t, u( s, r), u( s, r))= Kstusr (,)(, ) Kst (,)< nd Kstusr (,)(, ) Kst (,) v( s, t, u( s, r), u( s, r))= Kstusr (,)(, ) Kst (,)< t for ech r nd. In next secton, we expln Homotopy perturton method(hpm) s numercl lgorthm for pproxmtng of lner ntegrl equtons n crsp cse then, we fnd pproxmte solutons for utr (, ) nd utr (, ) for ech r nd t. 4 Modfy HPM: To llustrte the HPM, J-Hun He (999) consdered the followng nonlner dfferentl equton: Au ( )= f( r), r (4) wth oundry condtons u B =( u, )=, r n Where A s generl dfferentl opertor, B s oundry opertor, f(r) s known nlytc functon, s the oundry of the domn. Suppose the opertor A cn e dvded nto two prts: M nd. Therefore, (3) cn e rewrtten s follows: (5) Mu ( ) u ( )= fr ( ) (6) He n (999) constructed homotopy (, r p): [,] R whch stsfes 648

4 Aust. J. Bsc & Appl. Sc., 4(): , H(, p)= M( ) M( y ) pm( y ) p[ ( ) f( r)]= r p (,] y where nd s n meddng prmeter, nd s n ntl pproxmton of (3). Hence, t s esy to see H(,) = M( ) M( y ) =, H(, p)= A( ) f( r)= nd chngng the vrton of p from to s the sme s chngng H(, p) from M( ) M( y ) to A( ) f( r). In topology, ths s clled deformton, M( ) M( y) nd A( ) f( r) re clled homotopc. (7) Owng to the fct tht p cn e consdered s smll prmeter, y pplyng the perturton technque, we cn ssume tht the soluton of (6) nd (7) cn e expressed s seres n p, s follows: = p p... (8) when p, Eqs. (6) nd (7) correspond to Eqs. (5) nd (8) ecomes the pproxmte soluton of Eq. (5),.e. ux ( )= lm =... p The seres (9) s convergent for most of the cses, nd lso the rte of convergence depends on how we choose A(v). ow we propose scheme to ccelerte the rte of convergence of HPM ppled to lner Fredholm (9) ntegrl equtons wth kernels of the form perturton s follows:[5] Hupm (,, )=( pfu ) ( ) plu ( ) p( p) mgx ( ) K( x, t)= g( x) h( t). We defne new convex homotopy () where Fu ( )= ux ( ) f( x) nd Lu ( )= ux ( ) f( x) g( xhtutdt ) ( ) ( ) = hence we cn wrte ( ) () () ( ) ( )= u f pg x h t u t dt mpg x mp g x () now y usng (3) n (7), nd equtng the terms wth dentcl power of p, we otn 649

5 Aust. J. Bsc & Appl. Sc., 4(): , p u f x u f x : ( )= = ( ) p : u mg( x) g( x) h( t) u ( t) dt =, u = ( c m) g( x), c= h( t) f( t) dt p : u mg( x) g( x) h( t) u ( t) dt = u = [ m( cm) ] g( x), = h( t) g( t) dt 3 p : u3 = h() t u() t dt nd n generl = u ( ) ( ), =,3,... h t u t dt n n n now we fnd m such tht u =, snce f = then u = u =... =, nd the exct soluton wll e u 3 4 otned s ux ( )= u( x) u( x), hence for ll vlues of x we should hve m( cm) = or c [ ht ( ) f( tdt ) ][ ht ( ) gtdt ( ) ] m = = htgtdt () () or kttdt (,) m= ( ) ( ) h t f t dt kttdt (,) provded tht kttdt (,). ow consder the generl cse kxt (, )= g( xht ) ( ) here we choose the convex homotopy s follows: Hupm (,, )=( pfu ) ( ) plu ( ) p( p) mg ( x) = y dong smlr mnpultons, we otn 64

6 Aust. J. Bsc & Appl. Sc., 4(): , u = f u = mg ( x) g ( x) h( t) u ( t) dt = hf ( tdt ) m g( x)*3mm u = mg ( x) g ( x) h( t) u ( t) dt = m hudt g( x) u = g ( x) h( t) u ( t) dt, =,3,..., n n we try to fnd the prmeters m, =,,...,, such tht u = u =... = 3 hence from u we should hve m h() t u() t dt = now y susttutng u n (9), we otn m h() t h () t f() t dtm g () t dt = let j j j j= c = h ( t ) f ( t ) dt, = h ( t ) g ( t ) dt j j j j j then m ( c m ) =, =,,..., j j j j= () under certn condton,the vlues of m, =,,..., n (). Let the mtrx B nd the vectors m nd c e defned s follows: B=[ ], m=[ m ], c=[ c ] j j j therefore from () we cn wrte ( B I) m= Bc, cn e otned from the system of lner equtons nd f ( B I) s nonsngulr then 64

7 Aust. J. Bsc & Appl. Sc., 4(): , m=( B I) Bc In the cse of non-degenerte kernels,y usng Tylor expnson for functons of two vrles, we cn wrte kxt (, ) (f possle) s follows: kxt (, )= g( xht ) ( ) = nd y pplyng Hupm (,, ) method we cn pproxmte the soluton of the gven ntegrl equton. 5 umercl Exmples: Exmple 5.: Consder the fuzzy Fredholm ntegrl equton wth f t r t 3 r r 5 5 r r f t r t 3 r r 5 5 r r 3 (, )= ( ( ) (4 )) 3 (, ) = ( ( ) (4 )) nd kernel Kst (, ) = st st, The exct soluton n ths cse s gven y 3 3 utr t t r r r r (, )=( )( ( ) (4 )) 3 3 utr t t r r r r (, )=( )( ( ) (4 )) Some frst terms of MHPM seres re: u t r t r r r r 5 3 (, )= (3( ) (4 )) 3 u t r t r r r r 3 (, )= (3( ) (4 )) u( t, r)= nd u t r 3 t r r r r 3 (, )= ( ) (4 )) 64

8 Aust. J. Bsc & Appl. Sc., 4(): , 3 3 u t r t r r r r (, )= ( ( ) (4 )) u( t, r)= As we oserve, fter two terms the exct soluton s otned. Concluson: In ths work we llustrted numercl lgorthm for solvng fuzzy Fredholm ntegrl equtons of the second knd, usng Modfy HPM method. We feel tht ths work whch presents pplcle computtonl methods, my help to nrrow the exstng gp etween the theoretcl reserch on FIEs. ACKOWLEDGMETS The uthors would lke to thnk the Prof Asndy nd Prof Allhvrnloo for ther creful redng of the pper nd helpful suggestons nd nonymous referees. REFERECES Asndy, S., 7. Applcton of He s homotopy pertruton method to functonl ntegrl equtons. Chos,soltons & Frctls, 3(5): He, J.H., 999. Homotopy pertruton technque. Comput Method Appl Mech Eng., 73(3-4): Duos, D., H. Prde, 98. Towrds fuzzy dfferentl clculus: Prt 3, dfferentton, Fuzzy Sets nd Systems, 8: Gl, S.G.,. Approxmton theory n fuzzy settng, n: G.A. Anstssou (Ed.), Hndook of Anlytc- Computtonl Methods n Appled Mthemtcs, Chpmn Hll CRC Press, Gol, A., B. Kermt, 8. Modfed homotopy perturton method for solvng Fredholm ntegrl equtons, Chos, Solton nd Frctls, 37: Guerr, M.L., L. Stefnn, 5. "Approxmte fuzzy rthmetc operton usng monotonc nterpolton", Fuzzy Sets nd Systems, 5: Fredmn, M., M. Mng, A. Kndel, 998. Fuzzy lner systems, Fuzzy Sets nd Systems, 96: -9. Zdeh, L.A., 975. "The concept of lngustc vrle nd ts pplcton to pproxmte resonng", Informton Scences, 8: Bdrd, R., 984. Fxed pont theorem for fuzzy numers Fuzzy sets systems, 3: 9-3. Duos, D., H. Prde, 978. Operton on fuzzy numers, J. System Sc., 9: Goetschel, R., W. Voxmn, 986. Elemtry Clculus, Fuzzy sets system, 8: Wu, C., M. M, 99. On the ntegrls, seres nd ntegrl equtons of fuzzy set-vlued functons, J. Hrn Inst. Technol., :