Numerical Methods for Finding Multiple Solutions of a Superlinear Problem

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1 ISSN , Eglad, UK Joral of Iformatio ad Comptig Sciece Vol 2, No 1, 27, pp 27- Nmerical Methods for Fidig Mltiple Soltios of a Sperliear Problem G A Afrozi +, S Mahdavi, Z Naghizadeh Departmet of Mathematics, Faclty of Basic Scieces, Mazadara Uiversity, Babolsar, Ira (Received Je, 26, Accepted Agst 2, 26) Abstract Usig two merical methods, we will obtai merical positive soltios of the eqatio = λf () with Dirichlet bodary coditio i a boded domai, where λ > ad f () is a sperliear fctio of We stdy the behavior of the braches of merical positive soltios for varyig λ Keywords: Sperliear eqatio; Mltiple positive soltios; Motai pass Lemma; sb ad spersoltios AMS Sbject Classificatio: 5J6, 5B 1 Itrodctio We are iterested i the positive soltios of the problem x ( ) = λ f( xx, ( )), x where is a boded ad smooth domai i x ( ) =, x N R with bodary, λ > is a real parameter, ad 2 f ( ) = a b + c ( a, b, c > ) I this paper we stdy merical soltios of the eqatio (11) that arises i wide fields of physics, ad so it has bee stdied by several athors Amog others it describes problems of thermal self-igitio, diffsio pheomea idced by oliear sorces or a ball of isothermal gas i gravitatioal I this paper we cocetrate o the merical positive soltios of temperatre distribtio i a object heated by the applicatio of a iform electric crret sggested i [4] I fact we show that the first eigevale of the problem ( x) = λa( x), x (12) ( x) =, x This is a bifrcatio poit of the brach of merical soltios ad so there is aother brach of soltios that for ay positive λ admits a merical soltio where 1,2 Let H be the Sobolev space H ( ) with ier prodct (see [1]), v = v We defie F( ) = J : H R by f ( t) dt Ad we have (11) 1 2 J ( ) = ( F( )) (1) 2 + address: afrozi@mzacir Pblished by World Academic Press, World Academic Uio

2 28 G A Afrozi, et al: Nmerical methods for fidig Mltiple soltios of a sperliear problem J ( ) v = J ( ), v = { f ( ) v} (14) Defie γ ( ) = J ( ) = J ( ), = { 2 f ( )} (15) We ca do itegratio by parts o (14) to get J ( )( v, w) = { w f ( ) vw} (16) J ( ) v = { + f ( )} v Therefore a classical soltio to the PDE (1) is a critical poit of J By defiitio, critical poits of J are wea soltio to (1) By reglarity theory for elliptic bodary vale problems, is a classical soltio to or problem if ad oly if is a wea soltio to (1) I other words, critical poits of the actio fctioal J precisely the classical soltios of the PDE We cosider i this paper S = { H {} : J (, ) = }, where we ote that otrivial soltios to (1) are i S, ad S is a closed sbset of H S is ow as the Nehari maifold (see [5]) ad it is clear that all critical poits of J() mst lie i S It ca be show that all of the coditio of Motai Pass Lemma (see [2]) are satisfied for fctioal J, so there exists a critical poit ˆ of J that we call it motai pass type soltio ad withot loss of geerality we may assme ˆ is positive I the ext sectio we preset sefl merical methods ad itrodce the framewor of the procedre to fidmerical soltios 2 Nmerical Reslts I this sectio we preset or merical reslts are based o motai pass lemma ad sb ad spersoltios Motai pass algorithm: At first we preset a merical algorithm for fidig motai pass type soltio Cosider the problem x ( ) = λ f( xx, ( )), x where x ( ) =, x f () is sperliear Give a ozero elemet H ad a piece-wise smooth regio N R, we will se the otatio to represet a array of real mbers agreeig with o a grid We will tae the grid to be reglar At each step of the iterative process, we are reqired to project ozero elemets of H oto the sbmaifold S Projectio of J () o to the ray { λ : λ > } is give by <, > γ ( ) ( J ( )) = = <, > p 2 Let be a ozero elemet of H, represeted by over the grid Let optimally determied small positive costat Defie = ad We will se otatio mltiple of lyig o S p ( ) = lim 1 ( ) = + γ + s, 1 1 2, s = 5 1 or aother perhaps provided that limit exists, to represet iqe positive JIC for cotribtio: editor@jicorg

3 Joral of Iformatio ad Comptig Sciece, Vol 2 (27) No1, pp We se followig algorithm to fid the soltio: 1 Iitialize with appropriate iitial gess 2 Project o to S (ray projectio i ascet directio elemet oto S, that we explai it above) The stadard Begi loop with = 2 L gradiet is ot the gradiet we are cosiderig, < J ( ), v > = J ( ) v = = = ( ( )( ) {( + ( ) =< ( ) ( f ( ) v) f ( ), v > f ( ) v) f ( ) )} 1 Solve liear system grad = f ) for grad allows oe to explicitly costrct the array J ( ) grad, represetig J () 2 Tae gradiet descet Reproject o to S ( = s J ( ) 2 4 Icremet ad repeat step (1), (2), () til covergece criteria are met: J ( ), + f ( ) The obtaied reslts shows there is a array of soltios that has the orm above the horizotal asymptote 8 whe we defie = = sp ( x) x [,1] For brevity we express jst some of those merical reslts: Approximatio of for λ = eps x \ y = We exected or code for λ = eps, that eps = 222 i MATLAB JIC for sbscriptio: ifo@jicorg

4 G A Afrozi, et al: Nmerical methods for fidig Mltiple soltios of a sperliear problem Approximatio of for λ = x \ y = Approximatio of for λ = x \ y = 124 Approximatio of for λ = x \ y = 76 Accordig above tables 8 as λ It is well-ow that there mst always exists a soltio for problems sch as (2) betwee a sb-soltio v ad a sper-soltio sch that v for all x (see []) Cosider the bodary vale problem Let, C 2 ( ) satisfy as well as Choose a mber c > sch that x ( ) + f( xx, ( )) = o x ( ) = o (21) ( x) + f( x, ( x)) o (22) vx ( ) + f( xvx, ( )) o (2) c+ > ( x, ) [ v, ] f ( x, ) ad sch that the operator ( c) with Dirichlet bodary coditio has its spectrm strictly cotaied i the ope left-half complex plae The the mappig T : φ w, w= Tφ, φ C 2 ( ), φ( x) [ v, ], x ( 1) JIC for cotribtio: editor@jicorg

5 Joral of Iformatio ad Comptig Sciece, Vol 2 (27) No1, pp 27-1 where wx ( ) is the iqe soltio of the BVP wx ( ) cwx ( ) = [ cφ( x) + f( x, φ( x))] o wx ( ) = o, is mootoe, ie for ay φ1, φ2 satisfyig (1) ad φ1 φ2, we have Tφ1, Tφ 2 satisfies (1), ad Tφ1 Tφ2 o ad Coseqetly, by lettig f ( x, ) = c+ f( x, ), the iteratios yield iteratio v ad satisfyig so that the limits exists i C 2 ( ) We have v ( ) ( ) (i) x x o (ii) c ( x) = ( x) ( c ) + 1( x) = fc( x, ( x)) o, = 12,,, + 1( x) = o, v ( x) = v( x) ( cv ) + 1( x) = fc( xv, ( x)) o, = 12,,, v+ 1( x) = o, v= v v L v L L =, 1 1 ( x) = lim ( x), v ( x) = lim v ( x) ad v are, respectively, stable from above ad below; (iii) if / v ad both ad v are asymptotically stable, the there exists a stable soltio φ C 2 ( ) sch that v φ We se followig algorithm Sb- ad sper-soltio algorithm 1 Fid a sbsoltio ad a spersoltio Choose a mber c > ; v 2 Solve the bodary vale problem for w v ad w =, respectively; = w+ 1( x) cw+ 1( x) = fc( x, w( x)) o x ( ) = o If w + 1 w < ε, otpt ad stop Else go to step 2 We will se the otatio to represet a array of real mbers agreeig with o a grid We will tae the grid to be reglar We cosider the problem ( x) = λf ( x, ( x)) with = [, 1] [, 1] ad 2 f ( ) = a b + c ( a, b, c > ) (27) (24) (25) (26) JIC for sbscriptio: ifo@jicorg

6 2 G A Afrozi, et al: Nmerical methods for fidig Mltiple soltios of a sperliear problem Let a = c = 1, b =, it is clear f ( ) where < 8 So we cosider = as sbsoltio ad v = 9 as spersoltio, it ca be easily show that or sb ad spsoltios satisfy i (22),(2) The obtaied reslts shows there is a array of soltio that before (that we obtai is arod 2 sice before it Or code do ot coverges to positive soltio) is idetically zero ad after it has the orm less tha the horizotal asymptote 85 whe we defie = = sp x [,1] ( x) (see the followig tables) We repeated mootoe iteratio til or sb ad sper soltio coicide For brevity we express jst some of those merical reslts Approximatio of for λ = 21 + λ 1 x \ y = 14 Approximatio of for λ = x \ y = 566 Approximatio of for λ = x \ y = 819 Approximatio of for λ = x \ y λ 1 JIC for cotribtio: editor@jicorg

7 Joral of Iformatio ad Comptig Sciece, Vol 2 (27) No1, pp 27- = 82 So by sig the reslts tables we ca draw the bifrcatio diagram i the plae ( λ, ) Referece Bifrcatio diagram [1] R A Adams Sobolev spaces Pre ad Applied Mathematics Academic Press, New Yor, 1975, 65 [2] Ambrosetti, P H Rabiowitz Dal Variatioal Methods i Critical Poit Theory ad Applicatios JFctioal Aal 197, 14: [] G Che, J Zho ad W M Ni, Algorithms ad Visalizatio for Soltios of Noliear Eqatios It Joral of bifracatio ad chaos 2, : [4] H B Keller, D S cohe Some Positive Problem Sggested by Noliear Geeratio J Math Mech 1967, 16: [5] M Willem Miimax Theorems, Birhaser, Beli, 1996 JIC for sbscriptio: ifo@jicorg