A Spline based computational simulations for solving selfadjoint singularly perturbed two-point boundary value problems

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1 ISSN England UK Journal of Informaton and Computng Scence Vol. 7 No. 4 pp A Splne based computatonal smulatons for solvng selfadjont sngularly perturbed two-pont boundary value problems K. Aruna and A. S. V. Rav Kant + Flud Dynamcs Dvson Scool of Advanced Scences VIT Unversty Vellore-634 Inda (Receved July3 accepted October 7 ) Abstract. In ts paper we proposed a splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems. Te orgnal problem s reduced to ts normal form and te reduced boundary value problem s treated by usng dfference approxmatons va cubc splnes n tenson. Te convergence of te metod s analyzed. Some numercal examples are gven to demonstrate te computatonal effcency of te present metod. Keywords: Self-adjont Sngularly Perturbed Problems Dfference Approxmatons Cubc Splne n Tenson.. Introducton We consder te followng self-adjont sngularly perturbed two-pont boundary value problem: Ly ( a( y ) + b( y f ( on [ ] wt y ( ) α y() β (.) were α β are gven constants and s a small postve parameter. We also assume tat te coeffcents a( b( are suffcently smoot functon satsfyng a ξ > a( b( ξ (.) ( > were ξ and ξ are some postve constants. Under tese condtons operator L admts a maxmum prncple []. Tese type of problems arse frequently n flud mecancs aerodynamcs plasma dynamcs magneto ydrodynamcs oceanograpy optmal control cemcal reactons etc. In recent years seekng numercal solutons of sngularly perturbed boundary value problems as been te focus of a number of autors. Njma [ 3] produced a unformly second order accurate dfference scemes were as Mller [4] gave suffcent condtons for te unform frst order convergence of a general tree-pont dfference scemes. Boglayev [] dscussed a varatonal dfference sceme for solvng boundary value problems wt a small parameter n te gest dervatve. Scatz and Walbn [6] used fnte element tecnques for solvng sngularly perturbed reacton dffuson problems n two and one dmenson. In [7] a metod based on splne collocaton was presented for solvng sngularly perturbed boundary value problems. Cubc splne n compresson for second order sngularly perturbed boundary value problems was presented n [8]. Kadalbajoo and Devendra Kumar [9] presented a numercal metod based on fnte dfference metod wt varable mes for solvng second order sngular perturbed self-adjont two-pont boundary value problems. In [] a ftted operator fnte dfference metod va te standard Numerov s metod was presented for solvng self-adjont sngular perturbaton problems. Rordan and Stynes [] presented a unformly accurate fnte element metod for solvng sngularly perturbed one dmensonal reacton-dffuson problem. In [] a splne approxmaton metod was presented for solvng self-adjont sngular perturbaton problem on nonunform grds. Lubuma and Patdar [3] presented unformly convergent non-standard fnte dfference metods for solvng self-adjont sngular perturbaton problems. Msra et.al [4] extended te ntal value + Correspondng autor. Tel.: fax: E-mal address: k.aruna@vt.ac.n. Publsed by World Academc Press World Academc Unon

2 34 K. Aruna et.al.: A Splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems tecnque to self-adjont sngularly perturbed boundary value problems. Recently varable mes fnte dfference metod [] was extended for solvng self-adjont sngularly perturbed two-pont boundary value problems. In order to solve te self-adjont sngularly perturbed problem frst we reduce equaton (.)-(.) to ts normal form and ten te reduced problem s treated by splne metod. In general fndng numercal soluton of a second order boundary value problem wt y term s more dffcult as compared to a second order boundary value problem wt absence of y term. Terefore t s better to convert te second order boundary value problem wt y term to te second order boundary value problem wtout y term.e to ts normal form. In ts paper we ave presented computatonal smulatons of self-adjont sngular perturbed two-pont boundary value problems va splne metod. Convergence of te metod s analyzed and some numercal evdences are ncluded to sow te applcablty and effcency of te metod... Descrpton of te Metod We consder te followng self-adjont sngularly perturbed two-pont boundary value problem: ( a( x )y ) + b( x )y f ( x ) on [ ] wt y( ) α y( ) β (.) were α β are gven constants and s a small postve parameter. We also assume tat te coeffcents a( b( are suffcently smoot functon satsfyng a ( x ) ξ > a( x ) b( x ) ξ (.) > were and ξ are some postve constants. Equaton (.) can be wrtten as ξ a ( x )y ( x ) a ( x )y ( x ) + b( x )y( x ) f ( x ) or a ( x ) b( x ) f ( x ) y ( x ) + y ( x ) y( x ) or a( x ) a( x ) a( x ) y ( x ) + p( x )y ( x ) + q( x )y( x ) r( x ) (.3) were p( a ( a( b( f ( q( and r( a( a( Consder te transformaton y ( x ) U( x )V( x ) (.4) Ten equaton (.3) can be wrtten as ts normal form as V ( x ) + A( x )V( x ) G( x ) (.) y( ) y( ) wt V ( ) γ V( ) γ γ γ R (.6) U( ) U( ) were A( q( p ( ( p( ) 4 JIC emal for contrbuton: edtor@jc.org.uk

3 Journal of Informaton and Computng Scence Vol. 7 () No. 4 pp x G( x ) r( x )exp ψ ψ p( )d (.7) x U( x ) exp ψ ψ p( )d. (.8) Multplyng equaton (.) trougout by wt boundary condtons we get V ( x ) + W( x )V( x ) Z( x ) (.9) V ( ) γ ) γ (.) V( were W( x ) A( x ) Z( x ) G( x ) and w( x ) w >. Now snce x x f ( x ) Z( x ) G( x ) r( x )exp ψ ψ ψ ψ p( )d exp p( )d a( x ) Equaton (.) sows tat Z( s ndependent on.however W( may or may not be depend on. (.) 3. Dervaton of te Sceme We develop a smoot approxmate soluton of (.9) usng splne n tenson. For ts purpose we dscretze te nterval [ ] dvded nto a set of grd ponts x N wt. A functon N S( x τ ) of classc [ ab ] wc nterpolates y( at te mes pont x depends on a parameter τ reduces to cubc splne n [a b] as τ s termed as parametrc cubc-splne functon. Te splne functon S( x τ ) S( x ) satsfyng n x x ] te dfferental equaton [ S ( x ) τs( x ) ( x [ S ( x ) τs( x )] + [ S ( x ) τs( x )] x ) ( x x ) (3.) were S ( x ) and τ > s termed as cubc splne n tenson. Solvng te equaton (3.) and determnng V te arbtrary constants from te nterpolatory condtons wrtng τ we get S ( x ) and S( x ) V. After V + S( x ) M sn ( x x sn ) + M ( x sn x ) ( x x ) M ( x + x ) M V V (3.) Dfferentatng equaton (3.) and usng contnuty condtons wc lead to te trdagonal system. JIC emal for subscrpton: publsng@wau.org.uk

4 36 K. Aruna et.al.: A Splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems ( M + M + M ) V V + V (3.3) were sn M S ( x ) ()N- ( cot ) Te condton (3.3) ensures te contnuty of te frst order dervatves of te splne S ( x τ ) at nteror nodes. We rewrte (.9) n te form M W( x )V( x ) Z( x ) and substtutng nto equaton (3.3) we get te followng tree term recurrence relaton wc gves te approxmatonv V V N of te soluton V( at te ponts x x x. N ( + W )V + ( + W )V + ( + W V + ( Z + Z + Z ) ) N- (3.4) Usng (3.4) wt (.) we get te approxmate soluton of V( at te grd ponts snce U( s known terefore te soluton of te orgnal problem (.) at tese grd ponts wll be obtaned by usng (.4). 4. Convergence Analyss Were Te above system (3.4) can be wrtten n te matrx form as follows BV C (4.) x ( + W ) + W B O + ( + O K K W W ) + + O W W 3 N 3 ( + + K K W W N N ) + ( + W W N N ) [ ] T and F ( + W )V F F...F F ( + W ) V C 3 N N N + k wt F ( Z + Z + Z ) ()N-. Now consder te above system wt te exact soluton Were T T [T... T( N ) ] wt truncatng error V [V...V N ] we get B V C + T (4.) T N T 4 v 6 v V ( x ) + V ( x ) + V ( x ) + O( 36 [ + ( + )] 8 ) Let e V V () N be te dscretzaton error subtractng (4.) and (4.) we ave JIC emal for contrbuton: edtor@jc.org.uk

5 Journal of Informaton and Computng Scence Vol. 7 () No. 4 pp T BE T (4.3) were E [ e...en ]. For W( > we can coose suffcently small so tat te matrx B s rreducble and Monotone [6]. It follows tat B exsts and ts elements are non-negatve. From (4.3) we ave E B. Followng [7] T E B T E B T. Terefore E O( ) for any coce of and wose sum s. Tus we summarze te followng. and E O( ) for Teorem: Te metod gven by (3.4) for solvng te boundary-value problem (.9)-(.) for W( x ) and suffcently small gves a second-order convergent soluton for arbtrary and wt + and a fourt-order convergent soluton for. After knowng te value of V( n te gven doman we can calculate te value of y( usng (.4). 4. Computatonal Results To sow te computatonal competence of proposed numercal metod we mplemented te present metod on tree self-adjont sngularly perturbed problems. Te rate of convergence s determned as gven n [7]. I Rate of convergence log k r k... k I k+ were max k k+ I k y j y j k... j Example : Frst we consder te problem 4 y ( x ) + ( + ( x + ) ) y f ( x ) 4 ( x + ) subject to te boundary condtons y() y() - were f( s cosen suc tat te exact soluton s gven by x ( x+ ) 3 e e 4πx y( x ) cos +. x + e In Table (a)-(d) we ave compared te maxmum absolute errors for te present metod wt te metods n [ 3 ] for dfferent values of and N. Maxmum absolute errors and order of convergence for te present metod are gven n Table (e) and Table (f) respectvely. JIC emal for subscrpton: publsng@wau.org.uk

6 38 K. Aruna et.al.: A Splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems Table (a): Maxmum absolute errors for te Example wt N. N Stynes[] Patdar[] Lubuma Metod n Present metod [3] [] E-.8E- 3.8E-.E-.E-.3E-3 3.3E-.3E- 9.6E-3 4.7E-3.3E- 7.9E- 64.6E-3 3.E-3.4E-4.E-3 3.E-3 4.8E-6 8.3E-3 7.8E-4 6.E-4.6E-4 7.8E-4.9E E-4.9E-4.E-4 6.E-.9E-4.7E-8.. Table (b): Maxmum absolute errors for te Example wt N. N Stynes[] Patdar[] Lubuma Metod n Present metod [3] [] E- 4.8E-.E-.7E- 4.8E-.E-3 3.8E-.E- 6.3E-3 4.E-3.E-.9E E-3 3.E-3.6E-3 9.E-4.9E-3 3.E-6 8.E-3.E-3 3.9E-4.E-4 7.E-4.E-7 6.E-4.9E-4 9.8E-.E-.8E-4 4.E-9. 7 Table (c): Maxmum absolute errors for te Example wt N. N Stynes[] Patdar[] Lubuma [3] Metod n Present metod [] E- 4.8E-.6E-.E- 4.6E- 7.E-4 3.6E-.E- 4.3E-3 3.4E-.E-.9E E-3 3.E-3.E-3 9.3E-4.9E-3.E-6 8.E-3 7.6E-4.7E-4.4E-4 8.3E-4.E-7 6.6E-4.E-4 6.9E- 6.4E-.8E-4 9.E-8 Table (d): Maxmum absolute errors for te Example wt N. N Stynes[] Patdar[] Lubuma [] Metod n Present metod [] E- 4.8E-.4E-.4E- 4.6E-.9E-4 3.7E-.E- 7.9E-3 4.E-3.3E- 9.E E-3 3.4E-3.4E-3.E-3 4.4E-3.3E- 8.3E-3.E-3 6.E-4 3.E-4.9E-3 4.8E E-4 3.E-4.6E-4 9.6E- 8.8E-4 9.9E-7 JIC emal for contrbuton: edtor@jc.org.uk

7 Journal of Informaton and Computng Scence Vol. 7 () No. 4 pp Table (e): Maxmum absolute errors for te present metod of Example N E-4 7.E-8.8E-4 4.E-9 4.4E-.8E- 7.8E-4 3.8E-7.9E-4.4E-8 4.9E-.E-9 6.E-3.E-6.8E-4 9.E-8 6.9E-.7E-9 7.9E-3 4.8E-6 4.8E-4.9E-7.E-4.9E E-3.6E- 8.8E-4 9.9E-7.E-4 6.E-8 Table (f): of convergence for te present metod of Example N86 N Approxmate soluton Exact soluton Example : Next we consder te problem y + y f ( x ) subject to te boundary condtons Fg : Numercal soluton for Example wt 3 and N64. JIC emal for subscrpton: publsng@wau.org.uk

8 3 K. Aruna et.al.: A Splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems y() y() were f( s cosen suc tat te exact soluton of te problem s gven by x x y( x ) e + e x( e + e ) ( x ). We ave compared maxmum absolute errors for te present metod wt te metod [6] n Table (a). Maxmum absolute errors for dfferent values of N and are gven n Table (b)-(c). Table (a): Comparson of Maxmum absolute errors for Example wt 6 N Metod n [6] Metod n [] Present metod order order E-3.E-3 4.E-. 3.8E E-4 9.E- 9.7E-3.4.7E Table (b) Maxmum absolute errors and order of convergence for Example wt usng present metod N N N 4 / e e-9.994e- 4.6 / e-7.964e-8.974e /4.4333e e e-9 / e e-7.963e-8 /6.976e e e Table (c) Maxmum absolute errors and order of convergence for Example wt usng present 6 3 metod N N N 4 /.3488e e e-6. / e e e-. /4 6.43e e e-. /8.9e e e-. / e e e-4. JIC emal for contrbuton: edtor@jc.org.uk

9 Journal of Informaton and Computng Scence Vol. 7 () No. 4 pp Approxmate soluton Exact Soluton Fg: Numercal Soluton for Example wt Example 3: Fnally we consder te problem y + ( + x( x ))y subject to te boundary condtons y() y() were f( s cosen n suc a way tat te exact soluton s gven by x 3 and N64. f ( ( x ) y( x ) + ( x )e xe. We ave compared te maxmum absolute errors for te present metod wt te metod [] n Table 3(a). Maxmum absolute errors presented n Table 3(b)-3(c). In Table 3(d) we ave presented maxmum absolute errors and order of convergence for dfferent coces of +. x ) JIC emal for subscrpton: publsng@wau.org.uk

10 3 K. Aruna et.al.: A Splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems Table 3(a): Maxmum absolute errors for te Example 3 wt N8. Results n [] Results n [] Present metod Unform mes Varable mes Unform mes Varable mes E-.6E-.7E- 4.9E-.7E- 7.47E- 4 3.E- 3.9E- 3.6E-.44E- 3.6E-.94E-9 4.E-.3E- 4.4E-.9E- 4.4E-.47E E- 7.E- 7.84E- 3.74E- 7.84E-.8E-8 7.E-4 9.E-.4E-4 4.8E-.4E-4 6.4E-8 8.8E-4.E-4.74E-4.9E-.7E-4.37E-7 9.3E-4.4E-4.6E-4.47E-.9E E-7.E-3.E-4.E-3.8E-.3E E-6.E-3 3.E-4.99E-3.8E-4.3E-3.3E- Table 3(b): Maxmum absolute errors for te Example 3 wt usng present metod. N 3 N 64 N 8 N 6 N N e-7.9e e e- 3.74e-.83e e e e e e-.783e e e e e-6.996e-7.483e e e- 8.46e e e e e e e- 4.93e e e-4 Table 3(c): Maxmum absolute errors for te Example 3 wt usng present metod. 6 3 N 3 N 64 N 8 N 6 N N e-4.67e e e-6.944e e e-3.e-3.73e e-.7893e e-6 6.8e-.76e- 4.67e e-4.486e e e-.883e- 6.39e-.747e- 3.98e e e-.397e-.68e-.6e e-.6994e- JIC emal for contrbuton: edtor@jc.org.uk

11 Journal of Informaton and Computng Scence Vol. 7 () No. 4 pp Table 3(d): Maxmum absolute errors and order of convergence for Example 3 wt N6 N N e-.43e-.39e e-.99.8e e e-4..e e e e e e e e e e e- 4. usng present metod Approxmate soluton Exact soluton Conclusons Fg: 3 Numercal soluton for Example 3 wt 3 and N64. We ave presented splne based computatonal smulatons for solvng self-adjont sngularly perturbed boundary value problems. We ave analyzed te convergence of te metod and t s found to be fourt order for specfc coce of parameters. Tree examples are gven to demonstrate te effcency of te JIC emal for subscrpton: publsng@wau.org.uk

12 34 K. Aruna et.al.: A Splne based computatonal smulatons for solvng self-adjont sngularly perturbed two-pont boundary value problems proposed metod. It s observed from te tables tat te present metod s more effcent tan te metods n [6 3 ]. Te computatonal result sows tat te present metod s fourt order convergence only for and. Also t s sown tat for any oter coce of + te order of convergence s two. To furter autentcate te applcablty of te present metod te graps ave been 3 plotted between exact soluton and approxmate soluton of all te tree examples for a fxed and N64. It can be seen from Fg. -3 te computed solutons are n very good agreement wt te exact soluton. Acknowledgements Te autors would lke to tank te DRDO Govt. of Inda for provdng te fnancal support under te grant number ERIP/ER/9383/M//8. 7. References [] M. H. Protter H. F. Wenberger Maxmum Prncples n Dfferental Equatons Prentce-Hall New Jersey 967. [] K. Njma On a tree-pont dfference sceme for a sngular perturbaton problem wtout a frst dervatve term. II Mem. Numer. Mat. 7 (98) pp. -7. [3] K. Njma On a tree-pont dfference sceme for a sngular perturbaton problem wtout a frst dervatve term. I Mem. Numer. Mat. 7 (98)-. [4] J. J. H. Mller On te convergence unformly n of dfference scemes for a two-pont boundary sngular perturbaton problem Numercal Analyss of sngular perturbaton problems Academc Press New York 979 pp [] I. P. Boglayev A varatonal dfference sceme for boundary value problems wt a small parameter n te gest dervatve USSR Comput. Mat. Pys. (4) (98) pp [6] A. H. Scatz L. B. Walbn On te fnte element metod for sngularly perturbed reacton dffuson problems n two and one dmenson Mat. Comput. 4 (983) pp [7] K. Surla V. Jerkovc Some possbltes of applyng splne collocatons to sngular perturbaton problems Numer. Metods Approxm. Teory Vol.II Novsad (98) pp. 9-. [8] Tarq Azz Arsad Kan A Splne metod for second-order sngularly perturbed boundary-value problems J. Comp. Appl. Mat. 47 () pp [9] Moan K. Kadalbajoo Devendra Kumar Geometrc mes FDM for self-adjont sngular perturbaton boundary value problems Appl. Mat. Comput. 9( )(7) pp [] Kalas C. Patdar Hg order ftted operator numercal metod for self-adjont sngular perturbaton problems Appl. Mat. Comput. 7() pp [] E.O Rordan and M. Stynes A unformly accurate fnte element metod for a sngularly perturbed one dmensonal reacton-dffuson problem Mat. Comput. 47(986) pp. -7. [] M. K. Kadalbajoo and K. C. Patdar Splne approxmaton metod for solvng self-adjont sngular perturbaton problems on non-unform grds J. Comput. Anal. Appl. (3) pp [3] J. M. S. Lubuma and K. C. Patdar Unformly convergent non-standard fnte dfference metods for selfadjont sngular perturbaton problems J. Comput. Anal. Appl. 9(6) pp [4] H. K. Msra M. Kumar P. Sng Intal-value tecnque for self-adjont sngular perturbaton boundary value problems Comput. Mat. Modellng. () (9) pp [] M. K. Kadalbajoo and Devendra Kumar Varable mes fnte dfference metod for self-adjont sngularly perturbed two-pont boundary value problems J. Comput. Mat. 8()() pp [6] P. Henrc Dscrete Varable Metods n ordnary dfferental Equatons Jon Wley New York 96. [7] E. P. Doolan J. J. H. Mller W. H. A. Sclders Unform Numercal Metods for Problems wt Intal and Boundary Layer Boole Press Dubln Ireland 98. JIC emal for contrbuton: edtor@jc.org.uk