Legendre Wavelets for Systems of Fredholm Integral Equations of the Second Kind

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1 World Applid Scincs Journal 9 (9): 8-, ISSN IDOSI Publications, Lgndr Wavlts for Systs of Frdhol Intgral Equations of th Scond Kind a,b tb (t)= a, a,b a R, a. J. Biazar and H. Ebrahii Dpartnt of Mathatics, Faculty of Scinc, Univrsity of Guilan, P. O. Bo , P.C , Rasht, Iran Abstract: In this papr, systs of Frdhol intgral quations of th scond kind hav bn studid. A nurical thod for solving ths systs is prsntd. h thod is basd upon Lgndr wavlt approiations. So apls ar prsntd to illustrat th ability of th thod. Ky words:systs of Frdhol intgral quations of th scond kind Mothr wavlt Lgndr wavlts Oprational atri INRODUCION Lgndr wavlts n, () t = ( k, n ˆ,, t ) hav four argunts k ˆn = n, n =,,...,, k is any positiv intgr, is th ordr of Lgndr polynoials and t is th noralizd ti. hy ar dfind on th intrval [, ] as follows: Orthogonal functions and polynoials hav bn usd by any authors for solving various probls. h ain ida of using orthogonal basis is that a probl rducs to solving a syst of linar or nonlinar algbraic quations by truncatd sris of orthogonal basis functions for solution of probl and using th oprational atrics. Hr w us Lgndr wavlts basis on intrval [, ]. So of its applications ar nonlinar Voltrra- Frdhol intgral quation [], Frdhol intgral quations of th first kind [3], Abl s intgral quations [4], nonlinar intgral quations [5], diffrntial quations of Lan-Edn typ [6], variational probls [7] and so othr probls. Systs of Frdhol intgral quations of th scond kind hav bn solvd by othr thods as wll, Adoian dcoposition thod [9], aylor-sris pansion thod [], Sinc- collocation thod [], Hootopy prturbation thod []. Lgndr Wavlts and hir Proprtis Wavlts and Lgndr Wavlts: Wavlts constitut a faily of functions constructd fro dilation and translation of a singl function calld th othr wavlt, []. Whn th dilation paratr, and th translation paratr, b vary continuously w hav th following faily of continuous wavlts as. n k nˆ nˆ P ( k + + t n ˆ ), t <, k k ( t ) = ( k,n ˆ,,t ) =, othrwis P ( t ) =, P (t ) = t, + P + (t) = tp (t) P (t), =,, Corrsponding Author: J. Biazar, Dpartnt of Mathatics, Faculty of Scinc, Univrsity of Guilan, P. O. Bo , P.C , Rasht, Iran l: ; Fa: , E ail: jafar.biazar@gail.co 8 () k- =,,..., M- and n =,,..,. P (t). ar th faous Lgndr polynoials of ordr, which ar orthogonal rspct to th wight function w(t) =, on th intrval [-.] and satisfy th following rcursiv forula: h st of Lgndr wavlts ar an orthonoral st, [-8]. () (3)

2 World Appl. Sci. J., 9 (9): 8-, Function Approiation: A function f () L ([,]) ay L F F F b pandd as O L F () P = O O L F, k (4) F f() = c n n(), O O O L n= = F,O and L ar M M atrics givn by c n =(f (), n ()), stands for th innr product. W can () considr truncatd sris in (4), as follows: F =, k M (5) f() c () = C (). n n n= = (3) k- C and () ar M atrics givn by O =, C=c,c,,c M,c,c,,c M,,c k,,c k M =c,c,,c MM,c +,,c k, M and,m () = (), (),, (), (), (), =,,M (),, k (),, k (),M = (), (),, M (), M+ (),, k (). M f (, y) ( ) K ( y). ( M) M3 Also a function f (,y) L ([,] [,]) can b approiatd as (4) k M k- Hr th lnts of atri K =[k ] M will b (5) ( ) obtain by ( ) d = I, ( ( ) ) k (6) (7) L= M3 M3 ( M3) M5 ( M3) M M (8) h intgration of th product of two Lgndr wavlts vctor functions is drivd as: k i j = i( ), f (, y ), j( y ), i, j =,,..., M,. I is an idntity atri. (9) Solution of th Systs of Frdhol Intgral Equations h Oprational Matri for Intgration: h intgration of th Scond Kind via Lgandr Wavlts: Considr th of th vctor (), dfind in (7), can b achivd as. following syst of Frdhol intgral quations of th scond kind () () t dt = P ( ). ui( ) = fi( ) + k( t, ) G( u( t), u( t),..., un( t)) dt, t,, i=,,..., n. k- k- P is th M M oprational atri for intgration, [8]. his atri was obtaind as. (6) 9

3 World Appl. Sci. J., 9 (9): 8-, f i () L ([,]), k (, t) L ([,] [,]) and G ar linar or non-linar functions of u (), u (),..., u () for i, j = n,,...,n and u (), i =,,...,n ar th unknown functions. i W considr th following approiations for syst (6) by using Lgndr wavlts as: u C f i( ) ( ) i, i( ) ( ) Fi, k t (, ) ( K ) ( t), G( u( t), u( t),..., u t C n ( ) ) ( t), i, j =,,..., n C ar colun vctor functions of lnts of th vctor C. i By substituting ths approiations into syst (6), w would hav: ( ) i= ( ) i+ ( ) () () C F K t tc dt = ( F ) i+ ( K ) ( t) ( t) dtc, i=,,..., n. Eapl : Considr th following linar syst of Frdhol intgral quations. 7 + t u ( ) = + + ( u () t + v() t ) dt, v( ) = + + t ( u () t + v() t ) dt,, t ( F ), + ( F ), t 3 u ( ) ( C ), v ( ) ( C ), C C ( ) K ( t), t ( ) K ( t), 3 3 =,,,,,, =,,,,, (9) With th act solutions u () = + and v () = +, [9]. Put For this syst w find hrfor, th following will b obtaind: ( C ) i = ( ) Fi + KC, i=,,..., n. j = (7) h solutions would b achivd as follows 6 6 i i i i i= i= u ( ) = c ( ) = +, v ( ) = c ( ) = +, By ultiplying (), in both sids of syst (7), thn applying ( ) d, w gt th following linar or non-linar syst. Ci KC = Fi, i =,,..., n. (8) Vctor functions C i =,...,n can b obtaind by i solving syst (8). Nurical Eapls: o illustrat th thod so of systs of Frdhol intgral quations of th scond kind hav bn considrd. hs apls ar solvd by k = and M = 6. Which ar act solutions. Eapl : Considr th following linar syst with th - act solutions u () = and v () =, []. + + u tu t dt t ( ) = + () ( ) v() t dt, + v + = + + tu t dt t ( ) () + v() t dt u( ) , () By applying th Lgndr wavlts thod, w hav th following approiat solutions: v( )

4 World Appl. Sci. J., 9 (9): 8-, Fig. : Plots of act and approiat solution for Fig. 3: Plots of act and approiat solution for Eapl Eapl 4 Fig. : Plots of act and approiat solution for Fig. 4: Plots of act and approiat solution for Eapl Eapl 4 Plots of th act and approiat solutions ar prsntd in Figurs and. Eapl 3: considr th following non-linar syst 5 ( ) u ( ) = + u () t + v() t dt, v = + u t + v t ( ) () () dt,, t. with act solution u () = and v () =, [9]. In this apl lt s tak () u ( ) ( C ), v ( ) ( C ), 5 ( ) F, ( ) F, 8 9 K t u t ( ) (), () () t C, 3 hrfor by substituting into syst (), following non-linar syst would b obtaind: ( I K) C C = F, ( IK) C KC = F. ()

5 World Appl. Sci. J., 9 (9): 8-, h act solution will b obtaind by solving 3. Malknjad, K. and S. Sohrabi, 7. Nurical syst (). solution of Frdhol intgral quation of th first kind by using Lgndr wavlts, Applid Eapl 4: In this apl th following two non-linar Mathatics and Coputation, 86: syst of Frdhol intgral quations with act solution 4. Sohrab Ali Yousfi, 6. Nurical solution of u () = and v () = -, ar studid. Abl s intgral quation by using Lgndr wavlts, Applid Mathatics and Coputations, 75: u t u t v t v ( ) = + + ( ) () () + () t dt, 3 5. Mahoudi, Y., 5. Wavlt Galrkin thod for nurical solution of nonlinar intgral 6 3 v t u ( ) = + + () t v() t dt,, t. quation, Applid Mathatics and Coputation, 5 67: 9-9. (3) 6. Yousfi, S., 6. Lgndr wavlts thod for solving diffrntial quations of Lan-Edn typ, By applying th Lgndr wavlts thod, w AMC, 8: driv th following approiat solutions: 7. Razzaghi, M. and S. Yousfi,. Lgndr wavlts dirct thod for variational probls, Mathatics 8 u () , and Coputrs in Siulation, 53: v () Razzaghi, M. and S. Yousfi,. h Lgndr wavlts oprational atri of intgration, Intrnational J. Systs Sci., 3(4): Plots of th act and approiat solutions ar 9. Babolian, E., J. Biazar and A.R. Vahidi, 4. h prsntd in Figurs 3 and 4. dcoposition thod applid to systs of Frdhol intgral quations of th scond kind, CONCLUSION Applid Mathatics and Coputation, 48: Malknjad, K., N. Aghazadh and M. Rabbani, 6. In this papr Lgndr wavlts thod has bn Nurical solution of scond kind Frdhol intgral usd to driv approiat solutions for systs of quations syst by using a aylor-sris pansion Frdhol intgral quations of th scond kind. It can b thod, Applid Mathatics and Coputation, concludd that th thod is vry powrful and usful 76: tchniqu for finding approiat solutions of ths. Rashidinia, J. and M. Zarbnia, 7. Convrgnc of systs. In [9] apls () and (3) wr solvd by approiat solution of syst of Frdhol intgral Adoian dcoposition thod and approiat quations, J. Math. Anal. Appl., 333: 6-7. solutions wr obtaind whil this thod lads act. Javidi, M. and A. Golbabai, 7. A nurical solution. In this papr, w hav usd th packag Mapl solution for solving syst of Frdhol intgral 3 to carry th coputations associatd with ths quations by using Hootopy prturbation apls. thod, Applid Mathatics and Coputation, 89(): REFERENCES. Daubchs, I., 99. n Lcturs on Wavlts, CBMS-NSF.. Yousfi, S. and M. Razzaghi, 5. Lgndr wavlts thod for th nonlinar Voltrra-Frdhol intgral quations, Mathatics and Coputrs in Siulation, 7: -8.