A cubic spline collocation method for solving Bratu s Problem

Transcription

1 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg A cbc splne collocaton method for solvng Brat s Problem Abstract Abdelmajd El hajaj * Khald Hlal El merzgo Mhamed and El ghordaf Jalla Department of Mathematcs Faclty of Scence and Technology Unversty Sltan Molay Slmane Ben-Mellal Morocco Department of Mathematcs Faclty of Scence and Technology Unversty Abdelmalek Essaad Tanger Morocco * E-mal: a_elhajaj@yahoofr Ths paper we develop a nmercal method for solvng a Brat-type eqatons by sng the cbc splne collocaton method (SM) and the generalzed Newton method Ths method converges qadratcally f a relaton-shp between the physcal parameter and the dscretzaton parameter h s satsfed An error estmate between the exact solton and the dscret solton s provded To valdate the theoretcal reslts Nmercal reslts are presented and compared wth other collocaton methods gven n the lteratre Keywords: Brat-type eqatons Bondary vale problems bc splne collocaton method Introdcton The Brat problem appears n a large varety of applcatons sch as: Fel gnton model of thermal combston radatve heat transfer thermal reacton the handrasekhar model of the expanson of the nverse chemcal reactor theory and nanotechnology [5 6] In [6] a smmary of the hstory of the problem s gven Stdes on fel gnton n thermal combston theory have been on the ncrease over the last few years The reason for the ncreased stdy s to ensre the safety of workng envronment especally when workng wth combstble fld n some petro-chemcal engneerng processes ombston problems are generally characterzed by strong nonlnearty and snglarty as sch n most cases exact solton of combston problems are very dffclt to get Therefore researchers workng n ths area have resolved to approxmate soltons by ether analytcal or nmercal method Nonlnear phenomena are of fndamental mportance n varos felds of scence and engneerng The nonlnear models of real-lfe problems are stll dffclt to solve ether nmercally or analytcally There has recently been mch attenton devoted to the search for better and more effcent solton methods for determnng a solton approxmate or exact analytcal or nmercal to nonlnear models [0 4 9] onsder the Lovlle-Brat-Gelfand eqaton [5] 0 on where > 0 and s a bonded doman We consder the classcal Brat s problem [6] gven by the followng bondary-vale problem e on

2 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg (0) e () whch s sed to model a combston problem n a nmercal slab on(0) Several nmercal technqes lke the varatonal teraton method (VIM) [9] non-polynomal splne method [7] the Adoman decomposton method [3] hebyshev wavelets method [5 8] the parametrc cbc splne method [6] and the Laplace transform decomposton method [4] have been mplemented ndependently to handle the Brat model nmercally In ths paper we develop a nmercal method for solvng a one dmentonal Brat s problem by sng the SM and the generalzed Newton method Frst we apply the splne collocaton method to approxmate the solton of a bondary vale problem of second order The dscret problem s formlated as to fnd the cbc splne coeffcents of a nonsmooth system ( Y ) Y where m m :R R 0 In order to solve the nonsmooth eqaton we apply the generalzed Newton method (see [0 ] for nstance) We prove that the SM converges qadratcally provded that a property coplng the parameter and the dscretzaton parameter h s satsfed Nmercal methods to approxmate the solton of bondary vale problems have been consdered by several athors We only menton the papers [3 4] and references theren whch se the splne collocaton method for solvng the bondary vale problems The cbc B-splne collocaton method s wdely sed n practce becase t s comptatonally nexpensve easy to mplement and gves hgh-order accracy In [8 9] the athors solved a degree splnes where they consderer the collocaton ponts as the knots of the cbc splne space () the Brat s problem by sng thrd In or paper we consder a cbc splne space defned by mltple knots n the bondary and we propose a smple and effcent new collocaton method by consderng as collocaton ponts the md-ponts of the knots of the cbc splne space It s observed that the collocaton method developed n ths paper when appled to some examples can mprove the reslts obtaned by the collocaton methods gven n the lteratre (see [ ] for nstance) The present paper s organzed as follows In Secton we present the Brat s Problem In Secton 3 we constrct a cbc splne to approxmate the solton of the bondary problem and we present the generalzed Newton method In Secton 4 we show the convergence of the cbc splne to the solton of the bondary problem and provde an error estmate The obtaned nmercal reslts are compared to the [ ] n Secton 5 Fnally a conclson s gven n Secton 6 The exact solton of the Brat s problem () s gven by cosh[( x /) /] ( x) ln cosh( /4) n whch s the solton of the eqaton cosh( /4) soltons assocated wth eqaton snh( /4) 4 c c > < respectvely where c c Ths Brat problem has zero one or two [9 0] satsfes the In ths paper we shall apply SM to fnd the approxmate analytcal solton of the bondary and ntal vale c

3 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg problem of the Brat-type model omparsons wth the exact solton shall be performed Brat s Problem Applcatons of the Brat type eqaton are employed n the fel gnton Model of the thermal combston theory the model of thermal reacton process the handrasekhar model of the expanson of the nverse qestons n geometry and relatvty abot the handrasekhar model chemcal reacton theory radatve heat transfer and nanotechnology [7] In ths paper we consder the Brat s bondary vale problem n one-dmensonal planar coordnates n the form J( ) on (0) () (0) () 0 wth J e where the physcal parameter > 0 It s easy to see that J s a nonlnear contnos fncton on ; and for any two fnctons and v J satsfes the followng Lpschtz condton: J( x ( x)) J( x v( x)) ( x) v( x) a e on xi (3) Now we defne the followng nterpolaton cbc splne of the solton of the nonlnear second order bondary vale problem (6) 3 bc splne solton In ths secton we constrct a cbc splne whch approxmates the solton of problem (3) wth I s the I ( a b) nterval R Let { a x x x x < x < < x < x x x x } be a sbdvson of 3 0 n n n n n3 b the nterval I Wthot loss of generalty we pt 0 by S I ) { s ( I) ( I ) s( a) s( b) 0 s P( x x )} 4( ( x x ) x a h where 0 n and h ( b a) / n Denote the space of pecewse 0 polynomals of degree 3 over the sbdvson 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 Proposton 3: Let be the solton of problem () Then there exsts a nqe cbc splne nterpolant S S ( I ) of whch satsfes: 4 S( t ) ( t ) 0 n where t 0 x0 t x x n xn n t x t and n n Proof: Usng the Schoenberg-Whtney theorem (see []) t s easy to see that there exts a nqe cbc splne 3

4 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg whch nterpolates If we pt at the ponts n c 3 t 0 n S B then by sng the bondary condtons of problem () we obtan c S(0) (0) 0 and S() () 0 3 c n S Hence n c B d Frthermore snce the nterpolaton wth splnes of degree d gves nform norm errors of order ( h ) d r the nterpolant and of order O( h ) 0 any ( I) ( I ) (see [7]) we have O for for the rth dervatve of the nterpolant (see [] for nstance) then for S( t ) J( t ) O() n (4) The cbc splne collocaton method that we present n ths paper constrcts nmercally a cbc splne n S c B whch satsfes the eqaton () at the ponts t 0 n It s easy to see that 3 and the coeffcents c n c c 3 n 0 satsfy the followng nonlnear system wth n n c B ( t j ) J( t j c B ( t )) Relatons (5) and (6) can be wrtten n the matrx form respectvely as follows j n eqatons: j n (5)  F Ê Â F (6) where F [ J( t ( t)) J( tn ( t n ))] T F [ J( t S( t)) J( tn S( tn ))] T and Ê s a vector where each component s of order () ndependent of h wth the matrx A s nvertble [] Then relaton (7) becomes O It s well known that  A where A s a matrx h 4

5 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg A h F E A h F (7) wth E s a vector where each one of ts components s of order O ( h ) Theorem 3 Assme that the penalty parameter and the dscretzaton parameter h satsfy the followng relaton: h A < (8) Then there exsts a nqe cbc splne whch approxmates the exact solton of problem () Proof: From relaton (7) we have h A F Let : R n R n be a fncton defned by Y) h A F (9) ( Y To prove the exstence of cbc splne collocaton t sffces to prove that admts a nqe fxed pont Indeed let Y and Y be two vectors of n R Then we have Usng relaton (4) and the fact that B we get Then we obtan From relaton (0) we conclde that Then we have wth h A ( Y ) ( Y ) h A F Y F (0) n j Y j J( t SY ( t)) J( t SY ( t)) SY ( t) SY ( t) Y Y FY FY Y Y ( Y ) ( Y ) h A Y Y ( Y ) ( Y ) k Y Y k by relaton (8) Hence the fncton admts a nqe fxed pont In order to calclate the coeffcents of the cbc splne collocaton gven by the nonsmooth system we propose the generalzed Newton method defned by ( ) () 5

6 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg where n ( k) ( k ) ( k ) ( k ) I s the nt matrx of order (see [0 ] for nstance) ( I n n and k V k ) ( ( )) () V s the generalzed Jacoban of the fncton ( ) 4 onvergence of the method Theorem 4 If we assme that the penalty parameter and the dscretzaton parameter h satsfy the followng relaton then the cbc splne O ( h ) h A < (3) S converges to the solton Moreover the error estmate S Proof: From (7) and the matrx A s nvertble [] we have h A ( F F ) A E s of order Snce E s of order ( h ) O then there exsts a constant K sch that E k h Hence we have h A F F K A h (4) On the other hand we have J ( t ( t )) J( t S( t )) ( t ) S( t ) ( t ) S( t ) S( t ) S( t ) Snce S s the cbc splne nterpolaton of then there exsts a constant K sch that (see [7]) S K h (5) Usng the fact that then we obtan n S S B F j j K F h (6) 6

7 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg By sng relaton (4) and assmpton (3) t s easy to see that We have h A ( Kh K) h A (7) A ( K h K ) h S S S S Then from relatons (5) (6) and (7) we dedce that S s of order O ( h ) complete Hence the proof s Remark 4 Theorem 4 provdes a relaton coplng the parameter and the dscretzaton parameter h whch garantees the qadratc convergence of the cbc splne collocaton S to the solton of the Brat s problem () 5 Nmercal examples To llstrate the ablty and relablty of the method for Brat s problem some examples are provded The reslts reveal that the method s very effectve and smple onsder the case for Brat s eqaton as follows when and 35 The maxmm absolte errors n soltons of ths problem are compared wth methods n [ ] for h / 0 and tablated n Tables and 3 The tables show that or reslts are more accrate Table Absolte errors for x Present wavelet[4] Splne[8] B-Splne[9] LGSM[0] DTM[] Laplace[] Decmposton[3] E-3 340E-3 46E-3 693E-3 97E-3 693E-3 46E-3 340E-3 65E-3 4E- 885E- 50E- 09E- 67E- 3E- 37E- 47E- 47E- 577E-0 47E-0 456E- 964E- 46E-0 964E- 456E- 47E-0 577E-0 98E E E E E E E E-06 98E-06 75E-07 0E E-07 54E E-09 54E E-07 0E-06 74E-07-00E-0 00E-0 00E-0 00E-0 00E-0 300E-0 00E-0 00E-0 98E E E E E-06 E-05 6E-05 35E-05 0E-05 69E-03 0E-03 5E-04 0E-03 30E-03 0E-03 5E-04 0E-03 69E-03 7

8 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg Table Absolte errors for x Present wavelet[4] Splne[8] B-Splne[9] LGSM[0] DTM[] Laplace[] Decmposton[3] 0 46E-3 756E- 97E-05 7E E-06 30E-06 3E-03 5E E-3 84E- 4E-08 36E E-06 56E-06 4E-03 47E E-3 35E- 98E E-05 5E E-06 69E E E-3 437E- 4E-08 58E E E E-03 35E E-3 548E- 60E E-05 46E E E E E-3 643E- 4E-08 58E E E-06 09E-0 35E E-3 70E- 98E E-05 59E-06 79E-06 9E-0 589E E-3 777E- 4E-08 36E E-06 77E-06 4E-0 47E E-3 838E- 97E-09 7E-05 40E-06 74E-06 09E-0 5E-0 Table 3 Absolte errors for 35 x Present wavetet[4] Splne[8] B-Splne[9] LGSM[0] E-0 47E-0 345E-0 437E-0 475E-0 437E-0 345E-0 47E-0 55E-0 934E-09 8E-08 56E E E-08 36E-08 70E-08 00E-08 7E-08 66E E-06 69E E-06 73E E-06 69E E-06 66E E-0 748E-0 06E-0 7E-0 35E-0 7E-0 06E-0 748E-0 384E-0 445E-05 7E E E E E-05 70E E-05 44E-05 6 oncldng remarks In ths paper we have consder an approxmaton of a Brat eqaton problem presented n [9 0] Then we have developed a nmercal method for solvng each nonsmooth eqaton based on a cbc 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 nmercal reslts show the convergence of the approxmate soltons to the exact one and confrm the error estmates provded n ths paper The analytcal reslts are llstrated wth two nmercal examples The proposed scheme s smple and comptatonally attractve and shows a very hgh precson comparng wth many other exstng nmercal methods 8

9 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg Acknowledgement We are gratefl to the revewers for ther constrctve comments that helped to mprove the paper References [] de Boor (978) A Practcal gde to Splnes Sprnger Verlag NewYork [] E Mermr A Serghn A El hajaj and K Hlal (0) A bc Splne Method for Solvng a Unlateral Obstacle Problem" Amercan Jornal of omptatonal Mathematcs Vol No 3 pp 7- do: 0436/ajcm0308 [3] Samel O Adesanya E S Babadpe and S A Arekete (03) A New Reslt On Adoman Decomposton Method For Solvng Brat s Problem Mathematcal Theory and Modelng Vol3 No [4] BAl-wardat KAlkhnafes KAl-khaled (03) An effectve nmercal method for solvng nonlnear second-order bondary vale problems J Math ompt Sc 3 No [5] hangqng Yang Janha Ho (03) hebyshev wavelets method for solvng Brat s problem Bondary Vale Problems: 4 do:086/ [6] Mohammad Zarebna Zahra Sarvar (03) New Approach for Nmercal Solton of the One-Dmensonal Brat Eqaton Tha Jornal of Mathematcs ISSN [7] R Jallan (00) Non-polynomal splne method for solvng Brat s problem ompter Physcs ommncatons [8] G Harharan P Prabaharan (03) An Effcent Wavelet Method for Intal Vale Problems of Brat-Type Arsng n Engneerng Appled Mathematcal Scences Vol 7 no [9] B Batha (00) Nmerval solton of Brat-type eqatons by the varatonal teraton method Hacettepe Jornal of Mathematcs and Statstcs Volme 39 () 3-9 [0] X hen (997) A verfcaton method for soltons of nonsmooth eqatons omptng [] X hen Z Nashed L Q (000) Smootng methods and semsmooth methods for nondffentable operator eqatons SIAM J Nmer Anal 38 (4) 00-6 [] MJ Śmetaǹsk (007) Agenerlzd Jacoban based Newton method for semsmooth block-tranglar system of eqatons Jornal of omptatonal and Appled Mathematcs [3] H N Çaglar S N Çaglar E H Twzell (999) The nmercal solton of ffth-order bondary vale problems wth sxth-degree B-splne fncton Appled Mathematcs Letter 5-30 [4] A Lamn H Mrao D Sbbh A Tjn (0) Sextc Splne Solton of Nonlnear Ffth-Order Bondary Vale Problems Inter J ompt Math Do : 0080/ [5] AS Monm BM de Dormale (006) From the fttng technqes to accrate schemes for the Lovlle-Brat-Gelfand problem Nmer Methods Partal Dfferental Eqatons (4) [6] G Brat (94) Sr les eqaton ntegrals non-lneares Bll Math Soc France [7] Bors I Kvasov (000) Methods of Shape-Preservng Splne Approxmaton World Scentfc Pblshng o Pte Ltd ISBN [8] Jallan R (00) Non-polynomal splne method for solvng Brat s problem ompt Phys ommn 8:868 7 [9] aglar H aglar N Özer M Valarstos A Anagnostopolos AN (00) B-splne method for solvng Brat s 9

10 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-05 (Onlne) Vol3 No4 03 wwwsteorg problem Int J ompt Math 87:885 9 [0] Abbasbandy S Hashem MS L S (0) The Le-grop shootng method for solvng the Brat eqaton ommn Nonlnear Sc Nmer Sml 6: [] Hassan IHAH Ertürk VS (007) Applyng dfferental transformaton method to the one-dmensonal planar Brat problem Int J ontemp Math Sc : [] Khr SA (004) A new approach to Brat s problem Appl Math ompt 47:3 6 [3] Deeba E Khr SA Xe S (000) An algorthm for solvng bondary vale problems J ompt Phys 59:5 38 [4] Xaojng L Yohe Zho Xaomn Wang Jzeng Wang (03) A wavelet method for solvng a class of nonlnear bondary vale problems ommn Nonlnear Sc Nmer Smlat

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