A NUMERICAL SOLUTION OF THE URYSOHN-TYPE FREDHOLM INTEGRAL EQUATIONS

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1 A NUMERICAL SOLUTION OF THE URYSOHN-TYPE FREDHOLM INTEGRAL EQUATIONS A JAFARIAN 1,, SA MEASOOMY 1,b, ALIREZA K GOLMANKHANEH 1,c, D BALEANU 2,3,4 1 Deprmen of Mhemics, Urmi Brnch, Islmic Azd Universiy, POBox 969, Oromiyeh, Irn E-mil : jfrin5594@yhoocom E-mil b : mesoomy@yhoocom E-mil c : lirezkhlili22@yhoocoin 2 Deprmen of Chemicl nd Merils Engineering, Fculy of Engineering, King Abdulziz Universiy, PO Box 824, Jeddh, 21589, Sudi Arbi 3 Deprmen of Mhemics nd Compuer Science Çnky Universiy, 653 Ankr, Turkey 4 Insiue of Spce Sciences, POBox MG-23, RO-77125, Mgurele-Buchres, Romni E-mil: dumiru@cnkyedur Received Mrch 24, 214 In he presen pper, combinion of he Bernsein polynomils nd rificil neurl neworks (ANNs) is presened for solving he non-liner Urysohn equion These polynomils re uilized o reduce he soluion of he given problem o he soluion of sysem of non-liner lgebric equions The remining se of nonliner equions is solved numericlly by using he ANNs pproch o yield runced Bernsein series coefficiens of he soluion funcion Severl illusrive exmples wih numericl simulions re provided o suppor he heoreicl clims Key words: Urysohn-ype Fredholm inegrl equion; Bernsein polynomils mehod; Arificil neurl neworks pproch; Approxime soluion PACS: 23Rz,26Cb,26Nm,75Mh 1 INTRODUCTION The opics of inegrl equions hve been n incresing ineres in he ps yers, becuse hese kinds of equions pper in vrious fields of pplied science nd engineering So, geing soluions wih high level of ccurcy for he inegrl equions is very imporn sk Considering h mny rel-world mhemicl problems, especilly in he re of pplied mhemics re oo compliced o be solved in exc erms, he using of numericl mehods hs been swifly developed recenly There re mny numericl mehods for pproximing soluions of he liner nd non-liner Fredholm inegrl equions in one nd wo-dimensionl spces In he lierure cied below, mong he numerous works h hve been suggesed, some fmous pproches re lised s well Eskndrughlu e l in [1], presened RJP Rom 59(Nos Journ Phys, 7-8), Vol , Nos 7-8, (214) P , (c) Buchres, 214

2 626 A Jfrin e l 2 numericl mehod bsed on using spline piece-wise funcions nd Picrd s mehod for solving he Urysohn-ype inegrl equions The Adomin decomposiion mehod (ADM) for obining pproxime series soluion of Urysohn inegrl equions ws presened, see Ref [2] The numericl pproximion soluion of he Urysohn inegrl equion wih wo mehods hs been worked ou [3] The numericl pproximion of his kind of equion hs been sudied by mens of he sinc pproximion wih he double exponenil rnsformions in Ref [4] This numericl mehod combined he sinc Nysrom mehod wih he Newon ierive process h involves solving nonliner sysem of equions Sberi-Ndjfi nd Heidri in Ref [4], offered combinion of he Newon-Knorovich mehod nd qudrure mehods for solving nonliner inegrl equions The mehod solved he nonliner inegrl equions of he Urysohn form in sysemic procedure Also, in Ref [5, 6], wo ierive schemes bsed on he homoopy nlysis mehod hve been used o he numericl soluion of differenil equions [7 9] On he oher hnd, he rificil neurl neworks (ANNs) pproch is one of he more pplicble mehods h hs been used for pproximing soluions of differen kinds of inegrl equions For furher informion on ANNs in his respec, see Refs [1, 11] The ANNs is pplied o solve boh ordinry nd pril differenil equions nd iniil nd boundry vlue problems [12] In mny problems in science nd engineering, we hve some unknown funcions which re oo compliced o be deermined The Bernsein polynomils mehod is one of he erlies nlyic-numeric lgorihms for pproximing he unknowns in differen kinds of mhemicl problems This is n exremely useful wy of expressing compliced funcion in erms of simple polynomils The only requiremen is h he given funcion should be smooh In oher words, poin of ineres i mus be possible o differenie he funcion s ofen s we plese In our erlier works, he liner inegrl equions sysem ws hndled by using hese kinds of polynomils [1] In his pper, we re going o pply combinion of he Bernsein series mehod nd neurl neworks pproch o pproxime soluions of non-liner Fredholm inegrl equion in he Urysohn form, which hs been considered in he sndrd form: b k(x,,y())d = F (x,y(x)), α x β, (1) where k(x,,y()) is he kernel of he inegrl equion nd y(x) is he unknown funcion h will be deermined I should be noed h, he form (1) includes he Fredholm equions of he second nd firs kind Moreover, ll funcions in (1) re ssumed o be coninuous nd usully he cses of α = nd β = b re considered For more deils on he inegrl equions, he reder is referred o he book [13] Supposedly he presen problem hs n unique soluion, comfor compuionl form of he bove problem is obined when he Bernsein polynomils re used s

3 3 A numericl soluion of he Urysohn-ype Fredholm inegrl equions 627 bsis funcions Now, puing s = s i (for i =,,n), rnsforms he resuling sysem o sysem of non-liner lgebric equions for he unknown funcions If more nd more erms re used from he Bernsein series, hen he polynomil represenions beer nd beer pproxime he unknowns I is cler h he n-h order series soluion converges o he exc soluion, if he unknown funcions re polynomils of degree up o n Then, we hve imed o employ hree lyer feed-bck neurl nework (FNN) wih bck-propgion supervised lerning lgorihm h is bsed on he grdien descen mehod o indirec numericl pproximion of he resuling sysem, by sring wih n iniil guess I is cler h he series soluion converges o he exc soluion if such soluion exiss The meril presened below cn be divided ino five prs Secion 2 inends o describe how o find pproxime soluion of n Urysohn ype Fredholm inegrl equion by using he presen pproch In Sec 3, he error nlysis of pproxime soluion corresponding o he presen inegrl equion is given To demonsre he efficiency nd relibiliy of he mehod, some numericl es exmples wih comprisons re given in Sec 4 Finlly, Sec 5 concludes he pper 2 DESCRIPTION OF THE METHOD In mny cses, here exiss n ierive pproximion mehod h cn be pplied o solve some ypes of inegrl equions h occur mos frequenly in pplicions Below, under he ssumpion h he inegrl equion problem (1) hs soluion in he clss indiced bove, we ouline he generl scheme o give h soluion 21 DISCRETIZATION OF THE PROBLEM The Bernsein polynomils represenion is exremely imporn, from mny poins of view This pproch cn be used o pproxime more compliced funcions in erms of simpler polynomils The min im of his secion is consrucing n pproxime soluion of Eq (1) bsed on he replcemen of unknown funcion by he Bernsein series of rbirry posiive ineger degree n, which in generl hs he form: y n (x) = ξ p β n (x), (2) where ξ p is numericl coefficien dependen on he choice of he funcion y(x), nd β n (x) is he Bernsein bsis polynomil, which is defined s [14, 15]: ( np ) (x α) p (β x) n p β n (x) = (β α) n, p =,,n (3)

4 628 A Jfrin e l 4 Theorem 1 [16] For ny funcion y in C[α,β], he sequence {y n (x), n = 1,2,3,} converges uniformly o y, where y n is defined by Eq (2) Proof See [16] Suppose we crry his rgumen, nd ry o deermine polynomil pproximions of degree n for he soluion of he given problem To do his, firs y n (x) is subsiued for y(x) in he Eq (1) Now, we ge: b k(x,, β n ()ξ p )d = F (x, β n (x)ξ p ) (4) Afer some lgebric mnipulions, we obin: b k(x,, ζ p ()ξ p )d = F (x, where i= j= ζ p (x)ξ p ), (5) ζ p () = ( ) p n p ( n p ) ( ) n p p i j ( 1) n j i α p i β j n p j+i (6) Le us choose consn inegrion sep h, nd consider he discree se of poins x q = α+h(q 1), where q =,,n Choosing x q (for q =,,n) o be he bscisss of he priion nodes in x, we rech he following sysem of non-liner lgebric equions: b k(x q,, ζ p ()ξ p )d F (x q, ζ p (x q )ξ p ), q =,,n (7) I is cler h he soluion of he bove sysem generes soluion of he given inegrl equion problem The soluion of he resuling sysem (7) cn lso be obined by n pproch bsed on he rificil neurl neworks, which is presened below 22 ANNS APPROACH The rificil neurl neworks re usully orgnized ino lyers of informion processing unis A view on his nework hs emerged h our undersnding of he srucure nd funcion of he biologicl neurl neworks is key o he success in his field An ANNs re more described s prllel nd disribued processing These kinds of neworks re usully presened s sysems of inerconneced neurons h cn compue vlues These neworks re cpble of lerning, orgnizing nd represening he informion Since ls decdes, rificil neurl neworks hve been used in mny fields Generlly, hey re implemened in lmos every compliced scienificl nd echnologicl field Noe h vs lierure is devoed o

5 5 A numericl soluion of he Urysohn-ype Fredholm inegrl equions 629 Hidden unis Inpu uni z () zp() p x x p Oupu uni + y n () zn() x n n Fig 1 Schemic digrm of he proposed neurl rchiecure neurl neworks, nd he reder cn find hem in Refs [17, 18] Here, in order o ge n ierive scheme for esiming he given non-liner equions sysem (7), brief frmework of he proposed neurl neworks rchiecure is offered Consider he hree-lyer feed-forwrd neurl rchiecure shown in Figure 1 The inpu-oupu relion of ech uni of he proposed neurl ne cn be wrien s follows: Inpu uni: o 1 = (8) Hidden unis: o 2 p = ne 2 p, (9) Oupu uni: where ne 2 p = o1 ζ p () (1) y n () = ne 3, (11) ne 3 = (o 2 p ξ p ) (12) In his pr of our sudy, n rchiecure of ANNs hs been srucured for priculr pplicion in which he nework mus be rined before i becomes useful To sr his procedure, he iniil prmeer (weigh) ξ p mus be chosen rndomly Then, opionlly, rining or lerning begins Noe here h, he lerning rule specifies how o djus he nework prmeers for given rining pern In oher words, he nework needs o be rined wih lerning rule which is described below

6 63 A Jfrin e l Cos funcion Consider he hree-lyer feed-forwrd rchiecure shown in Figure 1 This rchiecure is represenion of he Bernsein series corresponding o he unknown funcion The civion funcion of he hidden unis is ssumed o be he ideniy funcion The inpu signls x q (for q =,,n) re represened o he nework nd hen y n (x q ), which is represening he nework oupu upon he presenion of ξ p (for p =,,n), is clculed Now, suible cos funcion h serves s crierion funcion is designed so s o minimize n error mesure beween he nework s oupu nd he corresponding rge oupu We use commonly error funcion, nmely he men-squred error o rnsform he problem o minimizing on he inpu-oupu spce s follows: e q = 1 b 2 ( k(x q,, ζ p ()ξ p )d F (x q, ζ p (x q )ξ p )) 2, (13) wih he ol error e = e q (14) q= 222 Proposed lerning rule Lerning in rificil neurl neworks is viewed s serch for prmeers (weighs nd bises) h essenilly drives he oupu error o zero The gol is o employ n pproprie lerning lgorihm such h for ech inpu signl x q, he error e q mches zero Considering he bove ide, we sr by deriving n unsupervised grdien descen-bsed lerning procedure, which is nurl generlizion of he del lerning rule, for djusing he prmeer ξ p Now, djusmen rule cn be wrien s follows: ξ p (r + 1) = ξ p (r) + ξ p (r), p =,,n, (15) ξ p (r) = η e q + α ξ p (r 1), (16) ξ p where r is he number of djusmens, η nd α re he smll consn lerning re nd he momenum erm consn, which normlly re chosen beween nd 1, respecively In ddiion, i,j (+1) nd i,j () represen he upded nd curren weigh vlues, respecively The nework prmeers re upded in he mnner h reduces he error nd his work yields h he nework oupu converges for ech given inpu o he desired oupu To do his, he new vlue for ech connecion weigh is found by king he curren weigh nd dding n moun h is proporionl o he slope of rining Now, he pril derivive eq ξ p is o be evlued he curren weigh

7 7 A numericl soluion of he Urysohn-ype Fredholm inegrl equions 631 vlues Using he chin rule for differeniion, one my express he presen pril derivive s: e q = 1 ξ p 2 b ( k(x q,, ζ p ()ξ p )d F (x q, ζ p (x q )ξ p )) 2 = (17) ξ p ( b ( b k(x q,, ζ p ()k (x q,, ζ p ()ξ p )d F (x q, ζ p (x q )ξ p )) ζ p ()ξ p )d ζ p (x q )F (x q, ζ p (x q )ξ p )) Now, upon subsiuing bove relions ino Eq (16) nd using Eq (15), he desired lerning rule will succeed Considering he fc h regulr nes re universl pproximors, herefore he suggesed rchiecure is obviously preferble o pproxime soluion of he resuling sysem (7) o ny desired degree of ccurcy 3 CONVERGENCE ANALYSIS In his secion we inend o prove h he presened numericl mehod converges o he exc soluion of sysem (1) Theorem 2 Le y n (x) be Bernsein series of degree n on x [α,β] h is consn coefficiens hve been produced by solving he non-liner lgebric sysem (7) Then he given polynomil converges o he exc soluion of inegrl equion (1) for n + Proof (see for exmple [1]) 4 NUMERICAL EXAMPLES In his secion, he mehod presened in his pper is used o find numericl soluion of wo illusrive exmples The soluion of he equions obined here, lso will be compred by Newon-Knorovich-qudrure mehod All clculions in he following bles re performed using Mlb v78 Below, we use he specificions s follows: (1) Lerning re: η = 1, (2) Momenum consn: α = 1, (3) The number of ierions: r = 3, 5

8 632 A Jfrin e l 8 Exmple 41 [4] Le us solve he Urysohn inegrl equion: 1 cos(πx)sin(π)y 3 ()d = 5(y(x) sin(πx)), 1 x 1, (18) by he explined Bernsein series mehod The exc soluion corresponding o his equion is y(x) = sin(πx)+ 1 3 (2 391)cos(πx) For n = 4, he originl inegrl equion is reduced o fundmenl non-liner sysem of he form (7), s: cos(πx q )sin(π)( ζ p ()ξ p ) 3 ()d 5( ζ p (x q )ξ p sin(πx q )), q =,,4 (19) To sr he neurl nework process, firs he connecion weighs ξ p, (for p =,,4) re qunified wih smll rndom vlues Now, in order o rin he nework, he presened bck-propgion unsupervised lerning lgorihm begins Afer r ierions, numericl resuls hve been obined nd given in Tble 1 This exmple is going o show he difference beween proposed lgorihm nd Newon-Knorovichqudrure (NKQ) mehod wih dividing he inegrion inervl ino 1 equl subinervls Also, he exc nd pproxime soluions re compred for r = 3 in Fig 2 Tble 1 Numericl resuls for Exmple 41 P roposed mehod s i = 1i Exc soluion NKQ mehod r = 3 r = 5 i = i = i = i = i = i = i = i = i = i = i =

9 9 A numericl soluion of he Urysohn-ype Fredholm inegrl equions Exc soluion Approxime soluion x Fig 2 Comprison of he exc nd pproxime soluions for Exmple Exc soluion Approxime soluion x Fig 3 Comprison of he exc nd pproxime soluions for Exmple 42

10 634 A Jfrin e l 1 Tble 2 Numericl resuls for Exmple 42 P roposed mehod s i = 1i Exc soluion NKQ mehod r = 3 r = 5 i = i = i = i = i = i = i = i = i = i = i = Exmple 42 [5] Consider he Urysohn ype non-liner Fredholm inegrl equion: 1 (x )y 2 ()d = y() ln(4(x + 1)(1 xln2 + x) 2 ) + 2x + 5 4, wih he exc soluion y(x) = ln(x + 1) nd use he presen mehod for finding is pproxime soluion While he iniil nework prmeers re chosen bsed on ssumpions which re given in he previous exmple, he ierive process yields he resuls which hve been ghered in Tble 2 Figure 3 shows he exc soluion nd he pproxime soluion for r = 3 I follows from he resuls of hese exmples h y n (x) converges s r,n o he exc soluion y(x) of he inegrl equion A successful choice of he zeroh pproximion ξ p () cn resul in rpid convergence of he procedure 5 CONCLUSION In his pper, we presened n useful numericl mehod h origined minly from he Bernsein polynomils for solving Urysohn ype inegrl equion As we explined bove, his mehod convers he presen problem o sysem of non-liner lgebric equions which my no be solvble esily Therefore, globl ierive mehod ws discussed by mens of ANNs pproch which is suied for pproximing undeermined Bernsein series coefficiens on bounded inervl Hving deermined he unknown Bernsein coefficiens of he soluion funcion, he series soluion is produced for numericl purposes immediely I is imporn o be noed h, he more erms mus be evlued o he higher ccurcy level Addiionlly,

11 11 A numericl soluion of he Urysohn-ype Fredholm inegrl equions 635 he proposed echnique hs been compred wih he Newon-Knorovich-qudrure mehod The obined numericl resuls from nlyzed exmples illusred h in pplicions involving compuions wih polynomils, he Bernsein form offers n efficien lgorihm versus he NKQ rule, for mny bsic funcions REFERENCES 1 M Eskndrughlu, HE Derili Gherjlr, H Mohmmdiki, A Arzhng, Ausr J Bsic Appl Sci 7(1), (213) 2 R Singh, G Nelkni, J Kumr, The Sci World J 214, Aricle ID (214) 3 K Mleknejd, K Nedisl, J In Equ Appl 25(3), (213); K Mleknejd, K Nedisl, B Mordi, Proceedings of he World Congress on Engineering, 213, Vol I 4 J Sberi-Ndjfi, M Heidri, Compu Mh Applicions 6, (21) 5 A Jfrin, Z Esmilzdeh, L Khoshbkhi, Appl Mh Sci 7(28), (213) 6 F Awwdeh, A Adwi, In Mh Forum 17, (29) 7 AK Golmnkhneh, NA Porghoveh, D Blenu, Rom Rep Phys 65, (213); D Rosmy, M Alipour, H Jfri, D Blenu, Rom Rep Phys 65, (213); AK Golmnkhneh, Ali K Golmnkhneh, D Blenu, Rom Rep Phys 63, (211); AK Golmnkhneh, Ali K Golmnkhneh, D Blenu, Sig Proce 91(3), (211) 8 A Kdem, D Blenu, Rom J Phys 56, (211); XJ Yng, D Blenu, WP Zhong, Proc Romnin Acd A 14, (213) 9 AMO Anwr, F Jrd, D Blenu, F Ayz, Rom J Phys 58, (213); A Jfrin e l, Rom J Phys 58, (213); JJ Rosles Grci, MG Clderon, JM Oriz, D Blenu, Proc Romnin Acd A 14, (213) 1 A Jfrin, S Mesoomy Ni, J Hypersruc 2(1), (213); A Jfrin, S Mesoomy Ni, Appl Mh Model 37(7), (213); A Jfrin, S Mesoomy Ni, In J Mh Mod Numer Opn 4(3), (213); A Jfrin, S Mesoomy Ni, AK Golmnkhneh, D Blenu, Adv Diff Equ DOI:11186/ S Effi, R Buzhbdi, Neurl Compu Appl, DOI:117/s y 12 IE Lgris, A Liks, DI Foidis, IEEE Trnsc Neur New 9(5), (1998) 13 AD Polynin, AV Mnzhirov, Hndbook of Inegrl Equions (2nd edn, Chpmn nd Hll/CRC Press, 28) 14 BN Mndl, S Bhchry, Appl Mh Compu 19, (27) 15 A Sei, Appl Mh Compu 23, 2 27 (214) 16 MJD Powel, Approximion heory nd mehods (Cmbridge Universiy Press, 1981) 17 D Grupe, Principles of rificil neurl neworks (2nd edn, World Scienific Publishing, 27) 18 M Hnss, Applied Fuzzy Arihmeic: An Inroducion wih Engineering Applicions (Springer- Verlg, Berlin, 25)