Multistage Homotopy Analysis Method for Solving Nonlinear Integral Equations

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1 Avalable at ttp://pvau.edu/aa Appl. Appl. Mat. ISSN: Specal Issue No. (August ) pp. 5 Applcatons and Appled Mateatcs: An Internatonal Journal (AAM) Multstage Hootop Analss Metod for Solvng Nonlnear Integral Equatons H. Jafar and M. A. Froozjaee Departent of Mateatcs Unverst of Mazandaran Babolsar, IRAN jafar@uz.ac.r, 6arab@at.co Receved: Aprl, ; Accepted: Ma 9, Abstract In ts paper, we present an effcent odfcaton of te ootop analss etod (HAM) tat wll facltate te calculatons. We ten conduct a coparatve stud between te new odfcaton and te ootop analss etod. Ts odfcaton of te ootop analss etod s appled to nonlnear ntegral equatons and ed Volterra-Fredol ntegral equatons, wc elds a seres soluton wt accelerated convergence. Nuercal llustratons are nvestgated to sow te features of te tecnque. Te odfed etod accelerates te rapd convergence of te seres soluton and reduces te sze of work. Kewords: Hootop analss etod; Multstage Hootop analss etod; Varatonal teraton etod; Volterra-Fredol ntegral equatons. MSC () No: K8; 5G99. Introducton Te ed Volterra-Fredol ntegral equatons arse n te teor of parabolc boundar value probles, te ateatcal odelng of te spato-teporal developent of an epdec, and varous pscal and bologcal probles. A dscusson of te forulaton of tese odels s gven n Wazwaz () and te references teren.

2 AAM: Intern. J., Specal Issue No. (August ) 5 Te nonlnear ed Volterra-Fredol ntegral equaton s gven n Wazwaz () as u(, ) f (, ) G(,, s, t, u( s, ) dsdt, (, ) were u(,) s an unknown functon, te functons f(,) and G(,,s,t,u) are analtc on D=Ω [,T] and were Ω s a closed subset of (R, n =,,). Te estence and unqueness results for equaton () a be found n Dekann (978), Han et al. (99), and Pacpatta (986). However, few nuercal etods for equaton () are known n te lterature Wazwaz (). For te lnear case, te te collocaton etod was ntroduced n Pacpatta (986) and te projecton etod was presented n Haca (996, ). In Brunner (99) te results of Pacpatta (986) ave been etended to nonlnear Volterra-Haersten ntegral equatons. In Maleknejad et al. (999) and Wazwaz (), a tecnque based on te Adoan decoposton etod was used for te soluton of (). A new etod for solvng () b eans of te Legendre wavelets etod s ntroduced n Yousef et al. (5). Anoter tpe of ed Volterra- Fredol ntegro-dfferental equaton and Fredol ntegro-dfferental equaton tat we are gong to solve approatel b te ultstage ootop analss etod s gven as [,], () ( ) ( n) ( ) f ( ) f ( ) K (, F( ( ) dt K (, G( ( ) dt, K (, G( ( ) dt,, () were f() and te kernels K (, ) and K (, are assued to be n L ( R ) on te nterval t ( ), t and n ( ) s n t dervatve of (). Te ntegro-dfferental equatons (IDEs) arse fro te ateatcal odelng of an scentfc penoena. Nonlnear penoena tat appear n an applcatons n scentfc felds can be odeled b nonlnear ntegro-dfferental equatons. In 99, Lao Rasd et al. (9) eploed te basc deas of te ootop n topolog to propose a general analtc etod for nonlnear probles, nael Hootop Analss Metod (HAM), Lao (), Nadee et al. (), and Rasd et al. (9). Ts etod as been successfull appled to solve an tpes of nonlnear probles Jafar et al. (8), Lao (997), and Rasd et al. (9). In te present paper we use te Multstage Hootop analss etod to obtan solutons of nonlnear ed Volterra-Fredol ntegral equatons and nonlnear ntegral equatons.. Basc Idea of HAM Consder te followng dfferental equaton N [ u( r)], () were N s a nonlnear operator, r denotes ndependent varable, and u(r) s an unknown functon. For splct, we gnore all boundares or ntal condtons, wc can be treated n te slar wa. B eans of generalzng te tradtonal ootop etod, Lao () constructs te so called zero order deforaton equaton

3 6 Jafar et al ( L[ ( r, u( r)] p H( r) N[ ( r, ], () were p [, ] s te ebeddng paraeter, s a nonzero aular paraeter, H (r) s an aular functon, L s an aular lnear operator, u ( r) s an ntal guess of u (r), u ( r, s a unknown functon, respectvel. It s portant, tat one as great freedo to coose aular tngs n HAM. Obvousl, wen p = and p =, t olds ( r;) u( r), ( r;) u( r ). (5) Tus, as p ncreases fro to, te soluton u ( r; vares fro te ntal guesses u ( r) to te soluton u (r). Epandng u ( r, n Talor seres wt respect to p, we ave were ( r ; u ( r) u ( r) p, (6) ( r; u ( r) p. (7)! p If te aular lnear operator, te ntal guess, te aular paraeter, and te aular functon are so properl cosen, te seres () converges at p =, ten we ave u ( r ) u ( r ) u ( r ), (8) wc ust be one of solutons of orgnal nonlnear equaton, as proved b Lao (). As and H(r) =, equaton () becoes ( p ) L[ ( r, u( r)] pn[ ( r, ], (9) wc s used ostl n te ootop perturbaton etod He (6), were as te soluton obtaned drectl, wtout usng Talor seres He (6). Accordng to te Defnton (7), te governng equaton can be deduced fro te zero-order deforaton equaton (6). Defne te vector u u ( r), u( r),..., un ( r) n. Dfferentatng equaton () tes wt respect to te ebeddng paraeter p and ten settng p = and fnall, dvdng te b!, we obtan te t -order deforaton equaton L u ( r) u ( r)] H ( r) ( u ), () [ were

4 AAM: Intern. J., Specal Issue No. (August ) 7 and N[ ( r, ] u () ( )! p ( ) p,, (),, It sould be epaszed tat u (r) for s governed b te lnear equaton () under te lnear boundar condtons tat coe fro orgnal proble, wc can be easl solved b sbolc coputaton software suc as Mateatca. For te convergence of te above etod we refer te reader to Lao's work Lao (). If equaton () adts unque soluton, ten ts etod wll produce te unque soluton. If equaton () does not possess unque soluton, te HAM wll gve a soluton aong an oter (possble) solutons.. HAM for Med Volterra-Fredol Integral Equatons To verf te valdt and te potental of HAM n solvng ed Volterra-Fredol ntegral equatons, we appl t to equaton (). For equaton (), frst we cose and L [ (, ; ] (, ; H (, ). We defne a nonlnear operator as follows: N[ (, ; ] (, ; f (, ) G(,, s, t, ( s, t; ) dsdt. () Te zero order deforaton equaton s: ( L[ (, ; u(, )] p N[ (, ; ]. () Dfferentatng equaton (), tes wt respect to te ebeddng paraeter p and ten settng p = and fnall dvdng te b!, we ave te so-called t -order deforaton equaton for : u, ) u (, ) R ( u ), (5) (

5 8 Jafar et al were R ( u G(,, s, t, ( s, t; ) ) u (, ) dsdt ( ) f (, ) p. p Terefore, we can obtan HAM ters b equaton (5). Now, we ave: u, ) u (, ) u (, ) u (, ).... ( Eaple. Consder te nonlnear ed ntegral equaton Yousef et al. (5): u(, ) e te u( s, dsdt,, (6) u (, ). For start HAM we cooseu ( ), and, ten, of two repeat ootop analss etod we wll obtaned ters HAM: u (, ) u (, ) 8 e e e e u(, ) u (, ) u (, ) u (, )... ( 8 e e e ) Te -curves for ts eaple s presented n Fgure wc were obtaned based on te tree order HAM approatons solutons. B -curves, t s eas to dscover te vald regon of, wc corresponds to te lne segents nearl parallel to te orzontal as.

6 AAM: Intern. J., Specal Issue No. (August ) 9 Fgure. -curve dagra Fgure. Relatve error (coputed eac/eact soluton Fgure. Eact soluton Fgure. Approate solutons b HAM ( =-.) Te tree-densonal plot of te error HAM s sown n Fgure.Usng te frst tree ters of te HAM, Fgure wll be generated. Ts eaple s solved b nne ters He's varatonal teraton etod ncte Nadee et al. () and Legendre wavelets etod Yousef et al. (5).. Te Multstage HAM for Nonlnear Integral Equatons Te soluton gven n HAM s local n nature. To etend ts soluton over te nterval I [, T] we dvde te nterval I nto sub-ntervals I j [ j, j ), j,,,, p,, were... p T. We solve te equaton () n eac subnterval I j. Let ( ) be soluton of equaton () n te subnterval I. For p, () s soluton of equaton () n te subnterval I wt ntal condtons b obtanng te ntal condtons fro te nterval I. ( ) ( ), for,..., p.

7 Jafar et al Te soluton of equaton () for [, T] s gven b were p ( ) I ( ), (7),,,. (8) For equaton (), frst we coose and n ( ; L[ ( ; ], n n () H,,,... ( ) ( ) j ( ), for,.... j We defne a nonlnear operator for ed Volterra-Fredol ntegral equatons as follows: N[ ( ; ] ( ; f ( ) K (, F( ( t; ) dt K (, G( ( t; ) dt Also for Fredol ntegro-dfferental equaton: (9) N[ n ( ; ( ; ] f ( ) K (, F( ( t; ) dt n. () Te zero order deforaton equaton s: ( L[ ( ; u( )] p N[ ( ; ]. Dfferentatng equaton (9) and (), tes wt respect to te ebeddng paraeter p and ten settng p = and fnall dvdng te b! and we ave te so-called t -order deforaton equaton for : L[ ( ) ( )] R ( ). Terefore, we can obtan MHAM ters b equaton () n te nterval I : ( ) ( ) ( ) ( )...

8 AAM: Intern. J., Specal Issue No. (August ) Ten, of equaton (7) we obtaned soluton of equaton (). 5. Illustratve Eaples To gve a clear overvew of te ultstage ootop analss etod, we present te followng eaples. We appl te ultstage ootop analss etod and copare te results wt te ootop analss etod. Eaple. Consder te followng nonlnear Volterra-Fredol ntegral equaton Yousef, and Yousef (5): ( ) ( ( dt ( ( dt, () wt te eact soluton ( ). For utlze of ult stage HAM we dvde nterval [, ] nto subnterval [,.] and [.,] In frst nterval we cose and obtaned: ( ) and usng -curve we fnd -.7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) For te second nterval, we coose and use -curve we select = -.59 and obtan: ( ) [,.] ( ) [.,] ( ). Yousef et al. (5) ave solved ts eaple usng te He's varatonal teraton etod.we draw below graps of HAM and MHAM soluton and copare te eact soluton. Fgures (5) and (6) sow coparson of eact solutons, HAM and MHAM. In ts eaple we etend te nterval [,] n Yousef et al. (5) to [,] and observe tat rapd convergence for te solutons s ver good.

9 Jafar et al Fgure 5. Coparson of eact soluton Fgure 6. Coparson of eact soluton (sold lne) and HAM (das lne) (order ) b = - -stage HAM (das lne) (order ) Eaple. Consder te Fredol ntegro-dfferental proble 6. ''' -t ( ) e e ( dt, () wt ntal condton: ' '' () (), () e. Te ultstage HAM procedure would lead to: Dvde te nterval [,] nto subntervals [,], [,] and [,]. In te frst nterval we coose ( ) ( e) and usng -curve we get =-. and: e e e ( ) ( ) e 8e 8e Terefore, ( ) ( ) ( ) e -.5e -.875e.9 e... wc s te soluton n te frst nterval. For te second nterval, we coose ) ( ) and usng -curve, we ave =-.9. So, ( ( ) ( 6 e 6 e 9e - 55e - 5e e...). Terefore, ( ) ( ) ( ) e.9e.5e.766e. e and n nterval [,], we cose ) ( ) and b -curve we cose. 97, (

10 AAM: Intern. J., Specal Issue No. (August ) ( ) ( 5 9 e 55 e e e e 9e 55e...). Terefore, ( ) ( ) ( ).5 57.e 55.7e e 59.5e 69.86e We obtaned te soluton MHAM: ( ) [,] ( ) [,] ( ) [,] ( ) Fgure 7. Coparson of MHAM soluton, HAM soluton (order6) and te eact soluton () e. Eaple. Consder te Fredol ntegro-dfferental proble Sdfar et al. (n press) '' t ' ( ) e e ( ( ( ) dt, (5) ' wt ntal condton: () (). To utlze MHAM, we dvde te nterval [,] nto subntervals [,.] and [.,] and Fgure 8 sows tat te convergence s faster tan n te seres obtaned b standard HAM Sdfar et al. (n press). In te frst nterval, we coose ( ) and usng -curve coose =-., ( ).6..e.. Terefore,

11 Jafar et al ( ) ( ) ( ).6..e. and n te nterval [.,], coose ) ( ) and b -curve we coose =-.89 and obtan: ( ( ) e.9. Terefore, ( ) ( ) ( )...99e. ) ( ) ( ). ( [,] [,] Fgure 8. Coparson error of MHAM and HAM (order 6) b - Reark. In te above eaples we ave solved nonlnear ntegral equatons b usng MHAM troug few teratons and faster convergence respect to HAM. 6. Concluson Hootop analss etod s a powerful etod wc elds a convergent seres soluton for lnear/nonlnear probles. Ts etod s better tan nuercal etods, as t s free fro roundng off errors, and does not requre large coputer power. In ts paper we ave suggested a odfcaton of ts etod wc s called 'ultstage HAM'. Here, we ave appled ultstage HAM for solvng a nonlnear ed Volterra-Fredol ntegral equatons. Te ultstage HAM elds a seres soluton wc converges faster tan te seres obtaned b HAM and VIM and Legendre wavelet. Illustratve eaples presented clear support for ts cla. REFERENCES Brunner, H. (99). On te nuercal soluton of nonlnear Volterra-Fredol ntegral equatons b collocaton etods, SIAM J. Nuer. Anal., Vol. 7, pp

12 AAM: Intern. J., Specal Issue No. (August ) 5 Dekann, M. O. (978). Tresolds and travelng for te geograpcal spread of nfecton, J. Mat. Bol., Vol. 6, pp. 9-. Fredol ntegral equatons, Mat. Coput. Sulaton, Vol. 7, pp. -8. Haca, L. (996). On approate soluton for ntegral equatons of ed tpe, ZAMM Z. Angew. Mat. Mec., Vol. 76, pp Haca, L. (). Projecton etods for ntegral equatons n epdec, J. Mat. Model. Anal., Vol. 7, pp. 9-. Han, G. and Zang, L. (99). Asptotc error epanson for te trapezodal Nstro etod of lnear Volterra-Fredol ntegral equatons, J. Coput. Appl. Mat., Vol. 5, pp He, J.H. (6). Hootop perturbaton etod for solvng boundar value probles, Ps Lett A, Vol. 5, pp He, J.H. (6). Soe asptotc etods for strongl nonlnear equatons, Int J Mod Ps B, Vol. (), pp Jafar, H. and Sef, S. (8). Hootop Analss Metod for solvng lnear and nonlnear fractonal dffuson-wave equaton, Coun. Nonlnear Sc. Nuer. Sulat. Lao, S. J. (). Beond perturbaton: ntroducton to te ootop analss etod. CRC Press, Boca Raton: Capan & Hall. Lao, S.J. (997). Nuercall solvng non-lnear probles b te ootop analss etod, Coputatonal Mecancs, Vol., pp. 5-5 Maleknejad, K. and Hadzade, M. (999). A new coputatonal etod for Volterra-Fredol ntegral equatons, Coput. Mat. Appl., Vol. 7, pp. -8. Nadee, S., Hussan, A., and Kan, M. (). HAM solutons for boundar laer _ow n te regon of te stagnaton pont towards a stretcng seet, Co. In Nonl. Sc. and Nu. Sul., Vol. 5, Issue, pp Pacpatta, B.G. (986). On ed Volterra-Fredol tpe ntegral equatons, Indan J. Pure Appl. Mat., Vol. 7, pp Rasd, M.M., Doarr, G. and Dnarvand, S. (9). Approate solutons for te Burger and regularzed long wave equatons b eans of te ootop analss etod, Co. n Nonl. Sc. and Nu. S., Vol., Iss., pp Sdfar, A., Molabara, A., Babae, A., and Yazdanan A. (In Press).A seres soluton of te nonlnear ed Volterra-Fredol ntegral equatons. Coputers and Mateatcs Tee, H.R. (977). A odel for te spatal spread of an epdec, J. Mat. Bol., Vol., pp Volterra and Fredol ntegro-dfferental equatons, Coun Nonlnear Sc Nuer Sulat. Wazwaz, A.M. (). A relable treatent for ed Volterra-Fredol ntegral equatons wt Applcatons (n press). Yousef, S. and Razzag, M. (5). Legendre wavelets etod for te nonlnear Volterra-Legendre equatons, Mat. Coput. Sulaton 7, -8. Yousef, S.A., Lotf, A. and Degan, M. (n press). He's varatonal teraton etod for solvng nonlnear ed Volterra-Fredol ntegral equatons, Coputers and Mateatcs wt Applcatons.