A Nonclassical Collocation Method For Solving Two-Point Boundary Value Problems Over Infinite Intervals

Transcription

1 Austrlin Journl of Bsic nd Applied Sciences 59: 45-5 ISS A onclssicl Colloction Method For Solving o-oint Boundry Vlue roblems Over Infinite Intervls M Mlei nd M vssoli Kni Deprtment of Mthemtics Mobreh Brnch Islmic Azd University Isfhn Irn Deprtment of Mthemtics Khorsgn Brnch Islmic Azd University Isfhn Irn Abstrct: A colloction method for solving to-point boundry vlue problems posed on n infinite intervl involving second order liner differentil eqution is proposed By reducing the infinite intervl to finite intervl tht is lrge nd pproimting the vrible using grnge interpoltion the resulting boundry vlue problem is reduced to n lgebric system by using nonclssicl pseudospectrl method he pplictions re demonstrted vi test emples Key ords: Colloction method o-point boundry vlue problem infinite intervl IRODUCIO During the lst fe yers much progress hs been mde in the numericl tretment of initil vlue problems nd boundry vlue problems over infinite intervls see for emple Ibiol ; Rvi Knth 3 nd references therein ypiclly these problems rise very frequently in fluid dynmics erodynmics quntum mechnics electronics nd other domins of science A fe notble emples re the Von Krmn sirling flos entini 98; Mroich 98 combined forced nd free convection over horizontl plte Schneider 979 nd eigenvlue problems for the Schrodinger eqution entini 98 In mny cses the domin of the governing equtions of these problems is infinite or semi-infinite so tht the specil tretment is required for these so clled infinite intervl problems he nlyticl solutions for these problems re not redily ttinble nd thus the problem is brought to the problem of finding efficient computtionl lgorithms for obtining numericl solution In the present pper e consider the liner to-point boundry vlue problem of the from y y y - Q y R ith y b y c or lim y c 3 here Q nd R re continuous functions nd Q We ssume tht -3 hve unique solution y to be determined Before computing the solution e plummet the infinite intervl to finite but lrge one so tht finite point represent infinity his is stndrd pproch of solving such problems tht re posed on infinite intervls nd nmed domin trunction he boundry condition t infinity is replced ith the sme conditions t finite vlue his provides very ccurte results provided tht is sufficiently lrge De Hoog nd Weiss De Hoog FR 979 proposed n nlyticl trnsformtion of the independent vrible tht reduces the originl problem to boundry vlue problem over finite intervl Usully tht produces singulrity of the second ind t the origin nd must be solved by suitble difference methods In this pper e introduce ne nonclssicl colloction method for solving -3 his method consists of reducing the solution of -3 to set of liner lgebric equtions by first epnding y in terms of grnge interpolting polynomils bsed on set of nonclssicl Guss-obtto G nodes hese nodes hich rise from nonclssicl orthogonl polynomils bsed on n rbitrry eight function over intervl [ ] re presented is then collocted t these G colloction points to evlute the unnon coefficients hich re the vlues of the function y t these colloction points his pper is orgnized s follos: the folloing section is devoted to the genertion of G colloction points nd function pproimtion In section 3 e eplin our method nd in section 4 e give our numericl findings by considering to emples to demonstrte the ccurcy nd pplicbility of the proposed technique Corresponding Author: M Mlei Deprtment of Mthemtics Mobreh Brnch Islmic Azd University Isfhn Irn E-mil: mm_mlei5@yhoocom 45

2 Aust J Bsic & Appl Sci 59: onclssicl seudospectrl Method: gl oints nd Weights: In clssicl pseudospectrl methods Elngr 995; Elngr 998 the clssicl Guss-obtto colloction points re bsed on Chebyshev or egendre polynomils nd lie on the closed intervl [-] In the present or e consider the genertion of the G colloction points bsed on nonclssicl orthogonl polynomils ith respect to rbitrry eights in the intervl ] [ et be the number of colloction points nd t be the th-degree nonclssicl orthogonl polynomil ith respect to the eight t hich cn be obtined from the folloing three-term recurrence reltion Shizgl B 98 he recurrence coefficients in re given in Shizgl 996 by d d t d d d he G colloction points nd eights for re obtined by the method outlined by Golub Golub GH 973 he tridigonl Jcobi-obtto mtri of order is defined by 3 J here re the solution of the liner system heorem Golub 973 he Guss-obtto nodes re the eigenvlues of J nd the Guss-obtto eights re given by v here v is the normlized eigenvector of J corresponding to the eigenvlue v v e i nd v its firs component Function Approimtion: In order to interpolte function ] [ f t the point ] [ e use the grnge interpoltion of degree of the form f F f ] [ here re set of G colloction points interpoltion nodes in ] [ nd

3 Aust J Bsic & Appl Sci 59: 45-5 Differentiting the series of to times nd evluting the result t the collo-ction points gives m m d m F [ ] f D f m 3 m d here the coefficients D re entries of n squre differentition mtri m 3 Solution of o-oint Boundry Vlue roblems: In order to solve -3 ith nonclssicl colloction method e first choose n intervl [ is ny positive integer et y y y here y [ y y y ] nd [ ] Differentiting 3 nd using 3 e obtin 3 y y y D 33 y y y D 33 here D [ ] nd D [ ] By substituting 3 33 in e get y y D y D Q y R 34 hich cn be reritten s y y [ D D Q ] R 35 We no collocte 35 t G colloction points s y y [ D D Q ] R 36 he sets of G colloction points re defined on the intervls [ ] In ddition using e get e here e is n vector hose th component is one nd other components re zero Furthermore using 3 e pproimte the boundry conditions in 3 s follos: y y b 37 y y c 38 here is chosen so tht the computed solution pproimtes the ctul solution Using e obtin system of liner equtions Solving this system nd substituting the obtined vlues of vector y in 3 the pproimte solutions cn be obtined 4 Computtionl Results: In this section e hve implnted the present method on to emples he numericl results re compred ith ect results nd other results in the literture here re mny orthogonl eights functions tht cn be used Mlei ; Chen In this pper e use only to eights he cses re summrized in ble ble : Different Cse e 47

4 Aust J Bsic & Appl Sci 59: 45-5 Emple : Consider the boundry vlue problem y y y y e 4 ith y 4 y 43 his problem hs erlier been considered in Kdlboo 984 nd Rvi Knth 3 nd its ect solution is 3 ye e e he boundry vlue problem given by 4-43 hs been solved using nonclssicl colloction method for the cses given in ble he numericl results re presented in bles -3 nd compred ith the method in Rvi Knth 3 nd the ect solutions ote tht in Rvi Knth 3 defines domin trunction [ ] Define E m ye y : here y nd y e re pproimted nd ect solution respectively In tble 4 the mimum bsolute errors E for different vlues of nd re given ble 4 shos tht in this method by chnging the eight function the obtined results cn be improved ble : Computtionl results of y for emple = 8 Ect Method in Rvi Knth resent resent 3 8 h Cse 64 5 Cse ble 3: Computtionl results of y for emple = Ect Method in Rvi Knth resent resent 3 h Cse 64 Cse Emple : A second emple y y y 44 ith y 45 y 46 48

5 Aust J Bsic & Appl Sci 59: 45-5 ble 4: Mimume bsolute error Cse E for emple 8 cse nd cse nd cse nd cse nd his emple hs erlier been considered in Robertson 97 nd lter in Rvi Knth 3 For this problem e consider intervls of the form [ We pplied the method presented in this pper nd solved nd then evluted the different vlues of y hich lso ere evluted in Rvi Knth 3 by using forth order finite difference method he computtionl results re presented in bles 5-6 nd comprison is mde ith the method outlined in Rvi Knth 3 ble 5: Computtionl results of y for emple =6 Method in [] resent resent 5 h 3 Cse 5 Cse ble 6: Computtionl results of y for emple 6 Method in [] 5 h 3 resent Cse E resent Cse

6 Aust J Bsic & Appl Sci 59: 45-5 Conclusion: A nonclssicl colloction method hs been used for the pproimte solution of to-point boundry vlue problems over infinite intervls he orthogonl eight function Wnd the intervl of definition of orthogonl polynomils cn be chosen rbitrrily hich me this method computtionlly very ttrctive becuse vriety rnge of Guss-obtto colloction points cn be utilized Emples sho the efficiency nd ccurcy of this method REFERECES Chen H BD Shizgl A spectrl solution of the Sturm-iouville eqution: comprison of clssicl nd nonclssicl bsis sets J Comput Appl Mth 36: 7-35 De Hoog FR R Weiss 979 he numericl solution of boundry vlue problems ith n essentil singulrity SIAM J umer Anl 6: Elngr G MA Kzemi 998 seudospectrl Chebyshev optiml control of constrined nonliner dynmicl systems Comput Opt Appl : 95-7 Elngr G MA Kzemi M Rzzghi 995 he pseudospectrl egendre method for discretizing optiml control problems IEEE rns Automt Cont 4: Golub GH 973 Some modified mtri eigenvlue problems SIAM Rev 5: Ibiol EA RB Ogunrinde One ne numericl scheme for the solution of Initil vlue problems Ivps in ordinry differentil equtions Aust J Bsic Appl Sci 4: Kdlboo MK KS Rmn 984 Discrete invrint imbedding for the numericl solution of boundry vlue problems over infinity intervls Appl Mth Comput 5: entini M HB Keller 98 Boundry vlue problems over semi-infinite intervls nd their numericl solution SIAM J umer Anl 7: entini M Keller HB 98 he Von Krmn sirling flos SIAM J Appl Mth 38: 5-64 Mlei M M Mshli-Firouzi A numericl solution of problems in clculus of vrition using direct method nd nonclssicl prmeteriztion J Comput Appl Mth 34: Mroich A 98 Asymptotic nlysis of Von Krmn flos SIAM J Appl Mth 4: Rvi Knth ASV Y Reddy 3 A numericl method for solving to-point boundry vlue problems over infinite intervls Appl Mth Comput 44: Robertson 97 he liner to point boundry vlue problems on n infinite intervl Mth Comput 5: Schneider 979 A similrity solution for combined forced nd free convection horizontl plte Int J Het Mss rnsf : 4-46 Shizgl B 98 A Gussin qudrture procedure for use in the solution of the Boltzmn eqution nd relted problems J Comput hys 4: Shizgl B H Chen 996 he qudrture discretiztion method QDM in the solution of the Schrodinger eqution ith nonclssicl bsis functions J Chem hys 4: