New numerical scheme for solving Troesch s Problem

Transcription

1 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg Abstract New numercal scheme for solvng Troesch s Problem Abdelmajd El hajaj * Khald Hlal El ghordaf Jalla and El merzguou Mhamed Department of Mathematcs Faculty of Scence and Technology Unversty Sultan Moulay Slmane Ben-Mellal Morocco Department of Mathematcs Faculty of Scence and Technology Unversty Abdelmalek Essaad Tanger Morocco * E-mal: a_elhajaj@yahoofr In ths paper we wll manpulate the cubc splne to develop a collocaton method (SM) and the generalzed Newton method for solvng the nonlnear Troesch problem Ths method converges quadratcally f a relaton-shp between the physcal parameter and the dscretzaton parameter h s satsfed An error estmate between the exact soluton and the dscret soluton s provded To valdate the theoretcal results Numercal results are presented and compared wth other collocaton methods gven n the lterature Keywords: Troesch problem Boundary value problems ubc splne collocaton method Introducton onsder a two-pont boundary value problem Troesch s problem as follows: u snh( u) on u() u() = () where s a postve constant Troesch s problem s dscussed by Webel and arses n the nvestgaton of the confnement of a plasma column by radaton pressure [] and also n the theory of gas porous electrodes [3] The closed form soluton to ths problem n terms of the Jacoban ellptc functon has been gven [4] as u( x) = snh u'() sc( x u' ()) 4 () where u '() the dervatve of u at s gven by the expresson u' () = m wth m beng the soluton of the transcendental equaton snh = sc( m) m sn( ) where the Jacoban ellptc functon sc( m) s defned by sc( m) where are related cos( ) by the ntegral d msn 65

2 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg From () t was notced that a pole of u(t) occurs at a pole of sc( x u' ()) 4 6 m the pole occurs at x ln It also has an equvalent defnton gven n terms of a lattce It was also notced that Troesch s problem has been solved by another method M Zarebna et al [4] have ntroduced an snc-galerkn method for solvng ths problem; ther method s based on the modfed homotopy perturbaton technque They have compared ths method wth homotopy perturbaton method (HPM) Laplace method perturbaton method and splne method The dscontnuous Galerkn fnte element (DG) method s appled for solvng Troesch s problem by H Temm [5] Mohamed El-Gamel [6] have ntroduced an effcent algorthm based on the snc-collocaton technque they have compared ths method wth the modfed homotopy perturbaton technque (MHP) the varatonal teraton method and the Adoman decomposton method M Zarebna M Sajjadan [7] have ntroduced an effcent algorthm based on The snc-galerkn method they have compared ths method wth [5 6] In another artcle Mustafa Inc Al Akgül [3] have ntroduced the reproducng kernel Hlbert space method (RKHSM) s appled for solvng Troesch s problem They have compared ths method wth the homotopy perturbaton method (HPM) the Laplace decomposton method (LDM) the perturbaton method (PM) the Adoman decomposton method (ADM) the varatonal teraton method (VIM) the B-splne method and the nonstandard fnte dfference scheme (FDS) In ths paper we develop a numercal method for solvng a one dmentonal Troesch s problem by usng the SM and the generalzed Newton method Frst we apply the splne collocaton method to approxmate the soluton of a boundary value problem of second order The dscret problem s formulated ( Y ) = Y where m m :R R as to fnd the cubc splne coeffcents of a nonsmooth system In order to solve the nonsmooth equaton we apply the generalzed Newton method (see [5 6 7] for nstance) We prove that the SM converges quadratcally provded that a property couplng the parameter and the dscretzaton parameter h s satsfed Numercal methods to approxmate the soluton of boundary value problems have been consdered by several authors We only menton the papers [] and references theren whch use the splne collocaton method for solvng the boundary value problems The cubc B-splne collocaton method s wdely used n practce because t s computatonally nexpensve easy to mplement and gves hgh-order accuracy In [] the authors solved a the Troesch s problem by usng thrd degree splnes where they consderer the collocaton ponts as the knots of the cubc splne space In our paper we consder a cubc splne space defned by multple knots n the boundary and we propose a smple and effcent new collocaton method by consderng as collocaton ponts the md-ponts of the knots of the cubc splne space It s observed that the collocaton method developed n ths paper when appled to some examples can mprove the results obtaned by the collocaton methods gven n the lterature (see [] for nstance) The present paper s organzed as follows In Secton we construct a cubc splne to approxmate the soluton of the boundary problem Secton 3 s devoted to the presentaton of the generalzed Newton method and we show the convergence of the cubc splne to the soluton of the boundary problem and provde an error estmate Fnally some numercal results are gven n Secton 4 to valdate our methodology In ths paper we shall apply SM to fnd the approxmate analytcal soluton of the boundary and ntal value problem of the Troesch problem omparsons wth the exact soluton shall be performed ubc splne collocaton method In ths secton the cubc splne collocaton method s developed and mplemented for solvng the Troesch s problem defned by 66

3 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg u J ( u) on () u() u() = wth J = snh( u) where s a postve constant It s easy to see that J s a nonlnear contnuous functon on u ; and for any two functons u and v J satsfes the followng Lpschtz condton: J( x u( x)) J( x v( x)) u( x) v( x) a e on x I () In order to mplement the cubc splne collocaton method we frst create a subdvson of the nterval () { = x = x = x = x < x < < x < x = x = x = x = } wthout loss of = 3 n n n n n3 generalty we put x = h where n and h = / n _ Denote by S I ) { s ( I) ( I) s() s() s P( x x )} 4( x x () (3) the space of pecewse polynomals of degree 3 over the subdvson and of class everywhere on I and _ class everywhere on I Note that S4( I ) H ( I) Let B = 3 n be the B-splnes of degree 3 assocated wth These B-splnes are postves and form a bass of the space S ( I ) 4 Now we defne the followng nterpolaton cubc splne of the soluton u of the nonlnear second order boundary value problem () Proposton 3: Let u be the soluton of problem () Then there exsts a unque cubc splne nterpolant Sp S ( I ) of u whch satsfes: 4 where t = x t x x = n Sp( t ) = u( t ) = n = n = x n t = x t and n n Proof: Usng the Schoenberg-Whtney theorem (see [8]) t s easy to see that there exts a unque cubc splne whch nterpolates u at the ponts t = n If we put n c = 3 Sp = B then by usng the boundary condtons of problem () we obtan c = Sp() = () = and = Sp() = () = Hence 3 u c n u Sp S = where c n Bn n = and S = c B d Furthermore snce the nterpolaton wth splnes of degree d gves unform norm errors of order O ( h ) d r for the nterpolant and of order O( h ) then for any u ( I) ( I) (see [9]) we have for the rth dervatve of the nterpolant (see [8] for nstance) S( t) = G( t u) O() = n (4) 67

4 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg where G t u) J ( t u) ''( t ) ( The cubc splne collocaton method that we present n ths paper constructs numercally a cubc splne n S = c B = 3 whch satsfes the equaton () at the ponts t = n It s easy to see that c 3 = and c n = and the coeffcents c = n satsfy the followng nonlnear system wth n equatons: n n = c B ( t j ) = G( t j = c B ( t )) Relatons (4) and (5) can be wrtten n the matrx form respectvely as follows ^ ^ A = F E ^ A = F where T F = [ G( t u( t)) G( tn u( t n ))] j j = n (5) (6) F = [ G( t S( t)) G( tn S( tn ))] T and ^ ^ E s a vector where each component s of order O () It s well known that A = A h matrx ndependent of h wth the matrx A s nvertble [] Then relaton (6) becomes where A s a A = h F E A = h F (7) Theorem 3 Assume that the penalty parameter and the dscretzaton parameter h satsfy the followng relaton: h A < (8) Then there exsts a unque cubc splne whch approxmates the exact soluton u of problem () Proof: From relaton (7) we have = h A F Let : R n R n be a functon defned by Y) = h A F (9) ( Y To prove the exstence of cubc splne collocaton t suffces to prove that admts a unque fxed pont Indeed let Y and Y be two vectors of n R Then we have 68

5 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg Usng relaton (4) and the fact that B we get Then we obtan From relaton () we conclude that Then we have ( Y ) ( Y ) h A F Y F () n j= Y j G( t SY ( t )) G( t S ( t )) S ( t ) S ( t ) Y Y Y Y Y FY F Y Y Y ( Y ) ( Y ) h A Y Y ( Y ) ( Y ) k Y Y wth k = h A by relaton (8) Hence the functon admts a unque fxed pont In order to calculate the coeffcents of the cubc splne collocaton gven by the nonsmooth system = ( ) () we propose the generalzed Newton method defned by ( k) ( k ) ( k ) ( k ) = ( I n V k ) ( ( )) () where n I s the unt matrx of order ( ) (see [5 6 7] for nstance) n and V k s the generalzed Jacoban of the functon 3 onvergence of the method Theorem 3 If we assume that the penalty parameter and the dscretzaton parameter h satsfy the followng relaton h A < (3) then the cubc splne S converges to the soluton u Moreover the error estmate u S s of order O ( h ) Proof: From (7) and the matrx A s nvertble [] we have 69

6 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg = h A ( F F ) A E Snce E s of order O ( h ) then there exsts a constant K such that E k h Hence we have h A F F K A h (4) On the other hand we have G( t u( t )) G( t S( t )) u( t) S( t) u( t) S( t) S( t) S( t) Snce S s the cubc splne nterpolaton of u then there exsts a constant K such that (see [9]) u S K h (5) Usng the fact that then we obtan n S S B F j j= K F h (6) By usng relaton (4) and assumpton (3) t s easy to see that We have h A h A A ( K ( K h h K ) h K ) u S u S S S Then from relatons (5) (6) and (7) we deduce that u S s of order O ( h ) Hence the proof s complete Remark 3: Theorem 3 provdes a relaton couplng the parameter and the dscretzaton parameter h whch guarantees the quadratc convergence of the cubc splne collocaton S to the soluton u of the Troesch s problem () (7) 7

7 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg 4 Numercal examples In ths secton we solve Troesch s problem for dfferent values of the parameter usng the computer algebra system Matlab and make a comparson between our results and those ones reported n the lterature to confrm the effcency and accuracy of our method onsder the Troesch s equaton as follows when parameter =5 and The maxmum absolute errors n solutons of ths problem are compared wth methods n [734] for and tabulated n Tables and The tables show that our results are more accurate h =/ Table Absolute errors for =5 x Present RKHSM [3] Wavelet[4] SGME[7] HPME[] LME[] PME[] SME[] E- 38E- 5E- 66E- 78E- 84E- 78E- 54E- 38E- 7E- 34E-9 4E- 76E-9 5E-9 E-9 5E-9 5E- 6E-9 76E-4 5E-3 E-3 7E-3 3E-3 3E-3 3E-3 4E-3 5E-3 767E-4 49E-3 4E-3 66E-3 3E-3 33E-3 96E-3 44E-3 48E-3 77E-4 5E-3 5E-3 67E-3 3E-3 34E-3 97E-3 45E-3 49E-3 77E-4 5E-3 E-3 7E-3 3E-3 3E-3 3E-3 4E-3 5E-3 8E-4 6E-3 3E-3 9E-3 3E-3 34E-3 3E-3 7E-3 6E-3 77E-4 5E-3 E-3 7E-3 3E-3 3E-3 3E-3 4E-3 5E-3 Table Absolute errors for = x Present RKHSM [3] Wavelet[4] SGME[7] HPME[] LME[] PME[] SME[] E-9 676E-9 9E-9 65E-8 387E-8 443E-8 378E-8 93E-8 63E-9 439E-7 94E-8 89E-7 49E-7 34E-7 99E-7 474E-7 345E-7 94E-7 8E-3 56E-3 8E-3 E- E- 3E- 3E- E- 74E-3 86E-3 564E-3 8E-3 4E-3 E-3 3E-3 3E-3 4E-3 739E-3 33E-3 66E-3 896E-3 3E- 3E- 4E- 4E- E- 776E-3 9E-3 59E-3 8E-3 E- E- 3E- 3E- E- 74E-3 36E-3 7E- E- 3E- 6E- 7E- 7E- 5E- 97E-3 8E-3 56E-3 8E-3 E- E- 3E- 3E- E- 74E-3 6 oncludng remarks In ths paper we have consder an approxmaton of a Troesch equaton problem presented n [3] Then we have developed a numercal method for solvng each nonsmooth equaton based on a cubc collocaton splne method and the generalzed Newton method We have shown the convergence of the method provded that the physcal and the dscretzaton parameters satsfy the relaton (3) Moreover we have provded an error estmate of order O ( h ) wth respect to the norm The obtaned numercal results show the convergence of the approxmate solutons to the exact one and confrm the error estmates provded n ths paper The analytcal results are llustrated wth two numercal examples The proposed scheme s smple and computatonally attractve and shows a very hgh precson comparng wth many other exstng numercal methods Acknowledgment We are grateful to the revewers for ther constructve comments that helped to mprove the paper 7

8 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) Vol4 No 4 wwwsteorg References [] ES Webel On the confnement of a plasma by magnetostatc felds Physcs of Fluds () (959) 5 56 [] VS Markn AA hernenko YA hzmadehev YG hrkov Aspects of the theory of gas porous electrodes n: VS Bagotsk YB Vaslev (Eds) Fuel ells: Ther Electrochemcal Knetcs onsultants Bureau New York 966 pp 33 [3] D Gdaspow BS Baker A model for dscharge of storage batteres Journal of The Electrochemcal Socety (973) 5 [4] Xnlong Feng Lquan Me Guolang He An effcent algorthm for solvng Troesch s problem Appled Mathematcs and omputaton 89 (7) 5 57 [5] Shh-Hsang hang A varatonal teraton method for solvng Troesch s problem Journal of omputatonal and Appled Mathematcs 34 () [6] Shh-Hsang hang Numercal soluton of Troesch s problem by smple shootng method Appled Mathematcs and omputaton 6 () [7] M Zarebna M Sajjadan The snc Galerkn method for solvng Troesch s problem Mathematcal and omputer Modellng 56 () 8 8 [8] de Boor A Practcal gude to Splnes Sprnger Verlag NewYork (978) [9] Bors I Kvasov Methods of Shape-Preservng Splne Approxmaton World Scentfc Publshng o Pte Ltd () ISBN [] E Mermr A Serghn A El hajaj and K Hlal "A ubc Splne Method for Solvng a Unlateral Obstacle Problem" Amercan Journal of omputatonal Mathematcs Vol No 3 pp 7- do: 436/ajcm38 [] SH Mrmorada I Hossenpoura S Ghanbarpourb A Barara Applcaton of an approxmate analytcal method to nonlnear Troesch s problem Appled Mathematcal Scences 3 (3) (9) [] SA Khur A Sayfy Troesch s problem: A B-splne collocaton approach Mathematcal and omputer Modellng 54 (9 ) () [3] Mustafa Inc Al Akgül The reproducng kernel Hlbert space method for solvng Troesch s problem Journal of the Assocaton of Arab Unverstes for Basc and Appled Scences (3) [4] A Kazem Nasab Z Pashazadeh Atabakan and A KJlJçman( 3) An Effcent Approach for Solvng nonlnear Troesch s and Bratu s Problems by Wavelet Analyss Method Hndaw Publshng orporaton Mathematcal Problems n Engneerng [5] G Harharan P Prabaharan An Effcent Wavelet Method for Intal Value Problems of Bratu-Type Arsng n Engneerng Appled Mathematcal Scences Vol 7 3 no 43-3 [6] B Batha Numerval soluton of Bratu-type equatons by the varatonal teraton method Hacettepe Journal of Mathematcs and Statstcs Volume 39 () () 3-9 [7] X hen A verfcaton method for solutons of nonsmooth equatons omputng 58 (997) 8-94 [8] X hen Z Nashed L Q Smootng methods and semsmooth methods for nondffentable operator equatons SIAM J Numer Anal 38 (4) () -6 [9] MJ S meta n sk Agenerlzd Jacoban based Newton method for semsmooth block-trangular system of equatons Journal of omputatonal and Appled Mathematcs 5 (7)