Solution for singularly perturbed problems via cubic spline in tension

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1 ISSN England UK Journal of Informaton and Computng Scence Vol. No. 06 pp.6-69 Soluton for sngularly perturbed problems va cubc splne n tenson K. Aruna A. S. V. Rav Kant Flud Dynamcs Dvson Scool of Advanced Scences VIT Unversty Vellore 60 IndaE-mal: k.aruna@vt.ac.n. Department of Matematcs Natonal Insttute of Tecnology Kuruksetra 69 Haryana IndaE-mal: asvravkant@yaoo.com (Receved December 0 accepted June 06) Abstract. Ts paper concerns te soluton for sngularly perturbed va cubc splne n tenson. Te derved sceme leads to a trdagonal system. Te error analyss s proved and te metod s sown to ave a fourt order convergence for te partcular coce of te parameters. Computatonal effcency of te metod s confrmed troug numercal eamples wose results are n good agreement wt teory. Keywords: sngularly Perturbed Problems Cubc Splne n Tenson Boundary Value Problems.. Introducton In ts paper we consder te followng second-order sngularly perturbed boundary value problem y''( ) p( ) y'( ) q( ) y( ) r( ) () subject to te boundary condtons y(0) y() () were p( ) q( ) r( ) are smoot bounded functons. It s well-known tat te problem ()-() ebts p ( ) boundary layer at one or bot ends of te nterval dependng on te propertes of []. Sngular perturbaton problems arse very frequently n flud mecancs quantum mecancs optmal control cemcal-reactor teory aerodynamcs reacton-dffuson process geopyscs and many oter areas n appled scence and engneerng. Numercal treatment of te problem ()-() as been wdespread n recent years for nstance [ -]. In [] a tenson splne metod for te lnear sngularly perturbed problems was presented wc as second and fourt order convergence dependng on te coce of te parameters nvolved n te metod. However Kan and Azz[] clam of fourt order convergence for te problem wt frst dervatve term lacks teoretcal and computatonal support because of two reasons. Te replacement of frst dervatve term wt gven appromatons does not affect te error analyss and no numercal eample s gven to test te competence of te metod nvolvng frst dervatve term. Kan and Azz metod[] gves fourt order convergence only for te problems wt absence of frst dervatve term for some partcular coce of parameters concerned but te order of convergence for te problems wt frst dervatve term ` and ` ` and cannot eceed two for any coce of parameters ` and `. Te proposed sceme s te modfed form of Kan and Azz sceme n wc a new parameter s ntroduced to obtan te desred fourt order convergence for problems wt frst dervatve term.e. equaton of te form () and (). For te partcular value of.e. 0 te proposed sceme reduces to Kan and Azz[] sceme. Te dervaton of te sceme s developed n secton. In secton error analyss s dscussed and t sows convergence of order four s aceved only for a partcular value of parameter.e. along wt ` and 0 `. Also t s sowed tat for any oter coce of parameters te order of convergence s two.. A revew of te researc background ` Publsed by World Academc Press World Academc Unon

2 Journal of Informaton and Computng Scence Vol. (06) No. pp We develop a smoot appromate soluton of () usng cubc splne n tenson. For ts purpose we dscretze te nterval [0] S ( ) of C [ a b] [ ab ] cubc splne n as S( ) S( ) satsfyng n were dvded nto a set of grd ponts wc nterpolates 0 [ ] y () at te mes pont 0... N wt N. A functon depends on a parameter reduces to s termed as parametrc cubc-splne functon. Te splne functon te dfferental equaton ( ) ( ) S ''( ) S( ) [ S ''( ) S( )] [ S ''( ) S( )] () and s termed as cubc splne n tenson. Solvng te equaton () and S( ) y 0 determnng te arbtrary constants from te nterpolatory condtons wrtng we get S( ) y ( ) ( ) S M M sn ( ) [ sn sn ] and S( ) y. After ( ) ( ) [ ( ) M y ( M y )] () Dfferentatng equaton () and usng contnuty condtons wc lead to te trdagonal system ( M M M ) y y y () N () were ` ( ) sn ( cot ) M ( ) ` S. Te condton () ensures te contnuty of te frst order dervatves of te splne S ( ) at nteror nodes. We wrte () n te form M p( ) y'( ) q( ) y( ) r( ) and substtutng nto equaton () and usng te followng appromatons for frst order dervatves of : ' y y y y (6) ' y y y y (7) ' y y ' ' y y y ( f ) f ' q ( p p ) y y ( p p ) y q ( p p ) y ( r r ) (8) We get te followng tree term recurrence relaton wc gves te appromaton y y... yn of te soluton y () at te ponts... N ( p q p ( q ( p p )) p ) y ( p p ( p p ) q p ) y y ( p q p ( q ( p p )) p ) y (( p ) r r ( p ) r )... N (9) Usng (9) wt () we get te appromate soluton of yat () te grd ponts. Remark : For 0 te present sceme reduces to Kan and Azz [] metod. JIC emal for subscrpton: publsng@wau.org.uk

3 6 K. Aruna et al.: Soluton for sngularly perturbed problems va cubc splne n tenson Remark : For and 6 [6] second order metod for unform mes. 0. General overvew of trackng objects proposed metod te present sceme reduces to te Kadalbajoo and Bawa s From (6) (7) and (8) we get ' ' v v ( ) e y '( ) y y '''( ) y ( ) y ( ) 0 ' ' v v ( ) e y '( ) y y '''( ) y ( ) y ( ) 0 ' ' v ( ) e y '( ) y ( ) y '''( ) ( ) y ( ) 6 0 substtutng M p( ) y'( ) q( ) y( ) r( ) n () we obtan (0) () () () () () ' ' y y y p y q y r p y q y r ( ) ( ( ) ( ) ' p y q y r ( )) () usng eact soluton n () we ave ( y( ) y( ) y( )) ( ( p y '( ) q y( ) r ) were For any coce of we get ( p y'( ) q y( ) r ) 6 v ( ) v ( ) T( ) ( ) y ( ) ( 0 ) y ( ) 60 ` and ` ( p y'( ) q y( ) r )) T( ) () () () wose sum s. Subtractng () and () and substtutng e y( ) y ( q ) e ( q ) e ( q ) e ' ' ' ( p e pe p e ) T ( ) (6) Usng (0)-() we get ( q ) e ( q ) e ( q ) e v [ ( p p ) p ( )] y '''( ) ( p p ) y ( ) 6 6 v ( ) v ( ) 6 v ( ) ( p y ( ) p y ( )) p ( ) y ( ) T( ) (7) 0 0 Let ' '' ( ) p p p p ( ) (8) ' '' ( ) p p p p ( ) (9) were () (). Usng (8)(9) and () n (6) we get were ( q ) e ( q ) e ( q ) e T ( ) (0) o JIC emal for contrbuton: edtor@jc.org.uk

4 Journal of Informaton and Computng Scence Vol. (06) No. pp v ( ) 6 To ( ( )) p y '''( ) ( ) y ( ) O( ) () 6 It can be seen easly tat ( ) ( ) To O for any coce of and for any value of and ( ) ( 6 ) To O for and. Let J trd[ ] and 0 D trd[ ] are N N trdagonal matrces and Q [ q q... q ] T N and E [ e e... e ] T N are N component vectors. So equaton (0) can be wrtten n matr vector form as AE T o were ` ` A J DQ () Followng[] t can be sown tat for suffcently small E A T E A T () o Terefore E O( ) for any coce of and E O( ) for and. Tus we summarze te followng. 0 Teorem: Let y( ) C [ a b] ten our metod provdes a second order convergent appromaton for soluton of te boundary value problem ()-() for arbtrary coce of wt and a fourt order convergent soluton for ` ` and. 0 y (). An educatonal process In ts secton we present te numercal smulaton to demonstrate te applcablty of te sceme by consderng two eamples. Mamum absolute errors (.e. ma y( ) y ) at nodal ponts are computed for dfferent values of and N. Eample : Consder te followng omogeneous sngular perturbaton problem y ''( ) y'( ) ( ) y( ) 0 () Subject to te boundary condtons ( ) y(0) e y() () e Te eact soluton s gven by ( )( ) y( ) e e. (6) In Table we ave compared te mamum absolute errors for dfferent values of ` obtaned by te present metod and te ftted fnte dfference metod []. Te Mamum absolute errors and order of convergence obtaned by te proposed metod for dfferent values of N ` and are presented n Table. Te estmated Mamum absolute errors and - unform errors N E usng te proposed metod sown n Table. Eample : Consder te followng omogeneous sngular perturbaton problem y ''( ) ( ) y '( ) ( ) y( ) e [( )( ) ] (7) Subject to te boundary condtons 7 o ` ` y(0) 0 y() e e (8) JIC emal for subscrpton: publsng@wau.org.uk

5 66 K. Aruna et al.: Soluton for sngularly perturbed problems va cubc splne n tenson Te eact soluton s gven by ( y( ) e e We ave compared te mamum errors and te order of convergence obtaned by te present metod and te Kan and Azz metod [] n Table -. Table : Comparson of mamum absolute errors for N = 8 Present Metod Present Metod Metod n [] ` ` 6 (9) e 0.70e e 0.69e 0.6e 07.e 0.68e 0.8e 06.98e 0.689e 0.e e 0 6.7e 0.90e 0.78e Table : Mamum absolute errors and order of convergence for Eample usng present metod N=6 Order N=8 Order N=6 Order ` 6.60e e e e e e e e e e e e e e e 0.00 `.880e e e e e e e 0.0.8e e e 0.06.e e e e 0.0.e 0.0 JIC emal for contrbuton: edtor@jc.org.uk

6 Journal of Informaton and Computng Scence Vol. (06) No. pp Table : Mamum absolute errors and - unform errors for Eample usng present metod N=6 Order N=8 Order N=6 Order ` 6.60e e e e e e e e e e e e e e e 0.00 `.880e e e e e e e 0.0.8e e e 0.06.e e e e 0.0.e 0.0 Table : Mamum absolute errors and order of convergence for Eample N=6 N= Metod n [] Present Metod Metod n [] Present Metod ` 6.666e 0.9e 0.68e 06.98e e 0.90e 0.7e 0.798e e 0.099e 0.00e 0.07e e 0.77e 0.99e 0.6e e 0.907e e e `.9e 0.007e e e e 0.07e 08.76e e E N JIC emal for subscrpton: publsng@wau.org.uk

7 68 K. Aruna et al.: Soluton for sngularly perturbed problems va cubc splne n tenson 6 7.6e 0.9e 07.77e e e 0.8e 06.97e 0.0e e 0.e 0.998e 0.067e Table : Mamum absolute errors for second order metod wt ` for Eample N=6 N= N=0 7.77e 0.9e 0.69e e e 0.66e e 0.9e 0.760e 0 9.e 0.66e 0.00e 0. Results of te postve and negatve features patterns We ave presented numercal smulatons for sngularly perturbed boundary value problems usng cubc splne n tenson. It s observed from te tables tat te present metod s more effcent tan te metods gven n [] []. Te computatonal results sows tat te present metod s fourt order only for a partcular coce of te newly ntroduced parameter.e. along wt ` and 0 `.Also t s sown tat for any oter coce of te parameters te order of convergence s two. 6. References [] E. P. Dolan J. J. H. Mller W. H. A. Sclders Unform Numercal Metods for Problems wt Intal and Boundary Layer. Boole Press Dubln980(n Ireland). [] H. Dragoslav H. Djordje On a fourt order fnte dfference metod for sngularly perturbed boundary value problems. Appled Matematcs and Computaton 0080(): [] P. Henrc Dscrete Varable Metods n Ordnary Dfferental Equatons. Wley New York96. [] I. Kan T. Azz Tenson splne metod for second order sngularly perturbed boundary-value problems. Internatonal Journal of Computer Matematcs 00 8():7-. [] M. K. Kadalbajoo Y. N. Reddy An appromate metod for solvng a class of sngular perturbaton problems. Journal of Matematcal Analyss and Applcaton 988():06-. [6] M. K. Kadalbajoo R. K. Bawa Varable mes dfference sceme for sngularly perturbed boundary value problems usng splnes. Journal of Optmzaton Teory and Applcatons 99690():0-6. [7] M. K. Kadalbajoo D. Kumar Intal value Tecnque for sngularly perturbed two-pont boundary value problems usng an eponentally ftted fnte dfference sceme. Computer Matematcs wt Applcatons 009 7:7-6. JIC emal for contrbuton: edtor@jc.org.uk

8 Journal of Informaton and Computng Scence Vol. (06) No. pp [8] M. K. Kadalbajoo V. Gupta A bref survey on numercal metods for solvng sngularly perturbed problems Appled Matematcs wt Computaton 00 7(8):6-76. [9] B. Kress O. H. Kress Numercal Metods for sngular perturbaton problems. SIAM Journal of Numercal Analyss 98 8():6-76. [0] M. Kumar P.Sng H. K. Msra A recent Survey on Computatonal Tecnques for Solvng Sngularly Perturbed Boundary Value Problems. Internatonal Journal of Computer Matematcs 0078(0):9-6. [] S. M. Roberts A boundary-value tecnque for sngular perturbaton problems. Journal of Matematcal Analyss and Applcatons 98 87(): [] Y. N. Reddy P.P. Cakravarty An ntal-value approac for sngularly perturbed two-pont boundary value problems. Appled Matematcs and Computaton 00():9-0. [] A. Awoke Y. N. Reddy Ftted fourt order trdagonal fnte dfference metod for sngular perturbaton problems. Appled Matematcs and Computaton 0079(): [] J. Vgo-Aguar S. Natesan A parallel boundary value tecnque for sngularly perturbed two-pont boundary value problems. Journal of Supercomputng 00 7:9-06. JIC emal for subscrpton: publsng@wau.org.uk

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