International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pure Appl. Sc. Technol., () (), pp Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl Equtons Newton Kntorovch Method nd Comprson wth HPM nd ADM Mehd Gholm Porshoouh *, Behzd Ghnr nd Bjn Rhm Deprtment of Mthemtcs, Testn Brnch, Islmc Azd Unverst, Testn, Irn * Correspondng uthor, e-ml: (m_gholm_p@hoo.com) (Receved: 5--; Accepted: --) Astrct: The m of ths pper s to present numercl soluton for non-lner Fredholm Integrl equtons Newton Kntorovch Method. Frst we ppl Newton Kntorovch Method, nd we use ths method to solve n exmple. The numercl results hve compred wth the results otned HPM nd ADM. The results revel tht the method s ver effectve nd smple. Kewords: Newton Kntorovch Method; Non-lner Fredholm ntegrl equtons; The Qudrture Method. Introducton Nonlner ntegrl equtons wth constnt ntegrton lmts cn e represented n the form () ( x ) = f ( x ) + ( x, t, ( t )) dt, x [, ]. Where ( x, t, ( t )) s the ernel of the ntegrl equtons nd ( t ) s the unnown functon. Eq.() s clled non-lner Fredholm ntegrl equton of second nd. ll functons n () re ssumed to e contnuous n [, ] [, ]. The frst uthor hve extended the Adomn Decomposton Method (ADM)[], nd the Homotop Perturton Method (HPM)[], for solvng non-lner of Fredholm ntegrl equtons of the second nd. The soluton of

2 Int. J. Pure Appl. Sc. Technol., () (), nonlner ntegrl equtons s complcted prolem of computtonl mthemtcs, whch s relted to dffcultes of oth prncpl nd computtonl chrcter. In ths connecton, methods re developed tht re especll desgned for solvng nonlner equtons, ncludng the Newton Kntorovch method, whch mes t possle to provde nd ccelerte the convergence of terton processes n mn cses.. Soluton of method.. The Newton Kntorovch Method We consder ths method n connecton wth the solve of Eq. (). The terton process s constructed s follows: ( x ) = ( x ) + ϕ ( x ), =,, K. () ϕ ( x ) ε ( x ) x, t, ( t ) ϕ ( t ) dt, = + ε = + ( ) ( x ) f ( x ) x, t, ( t ) dt ( x ), At ech step of lgorthm, lner ntegrl equton for the correcton ϕ x s solved.n ths wor we solve these lner ntegrl equtons pplng The Qudrture Method tht s one of numercl methods to solve lner Fredholm Integrl equtons of frst nd, tht expln followng. () (4).. The Qudrture Method The method of qudrture s method for constructng n pproxmte soluton of n ntegrl equton sed on the replcement of ntegrls fnte sums ccordng to some formul. Such formuls re clled qudrture formuls nd, n generl, hve the form ϕ( x ) dx A ϕ( x ) n = (5) Where x ( = Ln) re the scsss of the prtton ponts of the ntegrton ntervl [, ] or qudrture nodes, A,( = Ln) re numercl coeffcents ndependent of the choce of the functon ϕ ( x ). Smpson s rule (or przmodl formul) s the smplest nd most frequentl used n prctce. We use ths method wth s follows

3 Int. J. Pure Appl. Sc. Technol., () (), ϕ( x ) dx ( ϕ + 4 ϕ + ϕ( )) (6) B ppl Qudrture Method for solvng lner Fredholm ntegrl equton of frst nd, n Eq. () we cn fnd pproxmton of x t ech step. ϕ.. Convergence of the lgorthm nd the exstence. Let the functon ( x, t, ( t )) e jontl contnuous together wth the dervtves x, t, ( t ) x, t, ( t ) wth the respect to the vrles x, t,, where nd x, t nd let the followng condtons hold: - For the ntl pproxmton, the resolvng of the lner ntegrl equton wth the ernel x, t, ( t ) stsfes the condton R( x, t ) dt A, x. - The resdul of Eq.(4) for the pproxmton stsfes the neqult ε ( x ) = f ( x ) + x, t, ( t ) dt ( x ) B <. - In the domn, the followng relton holds sup ( x, t, ( t ) dt D <. 4- The constnt A, B nd C stsf the condton H = ( + A ) BD. In ths cse, under ssumptons - 4, the process () converges to the soluton of Eq. () n the domn ( x ) ( x ) H H ( A ) B, x <. The soluton s unque n the domn ( x ) ( x ) + A B, x <. The rte of convergence s determnte the estmte ( ), <. x x H A B x Thus the ove condtons estlsh the convergence of the lgorthm nd the exstence, the poston nd the unqueness domn of soluton of the nonlner equton (). These condtons mpose certn restrctons on the ntl pproxmton x whose choce s n mportnt ndependent prolem tht hs no unfed pproch. As usul, the ntl pproxmton s determned ether more detled pror nlss of the equton under

4 Int. J. Pure Appl. Sc. Technol., () (), consderton or phscl resonng mpled the essence of the prolem descred ths equton. Under successful choce of the ntl pproxmton, the Newton Kntorovch method provdes hgh rte of convergence of the terton process to otn n pproxmte soluton wth gven ccurc [].. Numercl results In ths secton to llustrte ove theoretcl results, n exmple s presented. In ths exmple, numercl results otn ths method compred ones otned nd ADM's method [] nd HPM `s method []. Exmple. Let us solve the followng non-lner Fredholm ntegrl equton: φ( x) = sn( π x) + cos( π x)sn( π x) ( φ( x) ) dt x [,]. 5 9 For whch the exct soluton s φ( x ) = sn( π x ) + cos( π x ). For the ntl pproxmton we te ( x ) = sn( π x ). Accordng to (4), we fnd the resdul = + 5 cos( π x ) 4 cos( π x ) ε ( x ) sn( π x ) cos( π x )sn( πt ) ( t ) dt ( t ) = sn ( πt ) dt = 5 4 The _dervtve of the ernel (x, t, ), whch s needed n the clcultons,hs the form. Accordng to Eq. (), we form the followng equton for (x, t, ) = sn( π x)cos( π t) ( t ) ϕ ( x ) cos( π x ) ϕ( x ) = + cos( π x ) sn( πt ) ( t ) ϕ( t ) dt, 4 B ppl The Qudrture Method t possle to otn soluton such tht cos( πx ) cos( πx ) π ϕ( x ) = + (sn( π) () ϕ() + 4(sn ϕ) + sn( π) () ϕ(), 4 B smplfcton cos( π x ) cos( π x ) ϕ ( x ) = + 4 ϕ, 4 B compute ϕ n (7) we get ϕ = nd puttng n Eq. (7) we rrve followng (7)

5 Int. J. Pure Appl. Sc. Technol., () (), cos( π x ) ϕ ( x ) =, 4 (8) Now we defne the frst pproxmton to the desred functon: ( x ) = ( x ) + ϕ( x ) = sn( π x ) + cos( π x ). 4 We contnue the terton process nd otn ε( x ) = sn( π x ) + cos( π x )sn( πt ) ( t ) dt ( t ) 5 7 = cos( π x ). 64 And 7 cos( π x ) ϕ( x ) = cos( π x ) + sn( πt ) ( t ) ϕ( t ) dt, 64 5 Smlr ove computton eldng 7 ϕ( x ) = cos( π x ) 64 Then 7 ( x ) = ( x ) + ϕ( x ) = sn( π x ) + cos( π x ) + cos( π x ) = sn( π x ) + cos( π x ). 64 To fnd ( x ) (lst terton) gn we compute ε ( x ), ϕ ( x ) s followng = + 5 ε ( x ) sn( π x ) cos( π x )sn( πt ) ( t ) dt ( t ) = cos( π x ). 684 And cos( π x ) ϕ( x ) = cos( π x ) + sn( πt ) ( t ) ϕ( t ) dt, B solve ths ntegrl equton ϕ( x ) = cos( π x ) 684 Then ( x ) = ( x ) + ϕ( x ) = sn( π x ) + cos( π x ) + cos( π x ) = sn( π x ) + cos( π x ). 684 We stop terton t thrd step. Tle shows result of ppl ths method onl three terton to solve ths Exmple nd compres these results nd ADM' method nd HPM' method. Ths tle shows ccurc nd hgh rte of convergence of the terton process.

6 Int. J. Pure Appl. Sc. Technol., () (), Tle Comprsons etween results otn Newton Kntorovch nd HPM nd ADM for exmple x ADM HPM Newton Kntorovch error soluton error soluton error soluton. 4.9 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Conclusons In ths wor ws ult up n effcent tertve method to solve nonlner ntegrl equton the convergence of method hs een proved nd the convergence estlshed.the lgorthm llustrte exmple the perform hs een compred wth other method nd showed tht t ehves equl or etter thn the other two methods. The dvntge of ths method s ts smple procedure nd es computton nd ts rpd convergence. It seems to e ver es to emplo wth relle to solve non-lner Fredholm ntegrl equtons of second nd. The computtons ssocted wth the exmples were performed usng Mple. References [] J. Bzr nd A. Rnjr, A comprson etween Newton s method nd A.D.M.for solvng especl Fredholm ntegrl Equtons, Interntonl Mthemtcl Forum, (7), 5. [] J. Bzr nd H. Ghzvn, Numercl soluton for specl non-lner Fredholm ntegrl equton HPM, Appled Mthemtcs nd Computton, 95, (8), [] A. D. Polnn nd A. V. Mnzhrov, Hndoo of Integrl Equtons, Chpmn & Hll/CRC Press, Boc Rton London, 998.