Initial-value Technique For Singularly Perturbed Two Point Boundary Value Problems Via Cubic Spline

Transcription

1 Unversty of Central Florda Electronc Teses and Dssertatons Masters Tess (Open Access) Intal-value Tecnque For Sngularly Perturbed Two Pont Boundary Value Problems Va Cubc Splne 010 Lus G. Negron Unversty of Central Florda Fnd smlar works at: ttp://stars.lbrary.ucf.edu/etd Unversty of Central Florda Lbrares ttp://lbrary.ucf.edu Part of te Matematcs Commons STARS Ctaton Negron, Lus G., "Intal-value Tecnque For Sngularly Perturbed Two Pont Boundary Value Problems Va Cubc Splne" (010). Electronc Teses and Dssertatons ttp://stars.lbrary.ucf.edu/etd/1650 Ts Masters Tess (Open Access) s brougt to you for free and open access by STARS. It as been accepted for ncluson n Electronc Teses and Dssertatons by an autorzed admnstrator of STARS. For more nformaton, please contact lee.dotson@ucf.edu.

2 INITIAL-VALUE TECHNIQUE FOR SINGULARLY PERTURBED TWO POINT BOUNDARY VALUE PROBLEMS VIA CUBIC SPLINE by LUIS G. NEGRON B.S. Unversty of Central Florda, 00 A tess submtted n partal fulfllment of te requrements for te degree of Master of Scence n te Department of Matematcs n te College of Scences at te Unversty of Central Florda Orlando, Florda Fall Term 010

3 010 Lus G. Negron

4 ABSTRACT A recent metod for solvng sngular perturbaton problems s examned. It s desgned for te appled matematcan or engneer wo needs a convenent, useful tool tat requres lttle preparaton and can be readly mplemented usng lttle more tan an ndustry-standard software package for spreadseets. In ts paper, we sall examne sngularly perturbed two pont boundary value problems wt te boundary layer at one end pont. An ntal-value tecnque s used for ts soluton by replacng te problem wt an asymptotcally equvalent frst order problem, wc s, n turn, solved as an ntal value problem by usng cubc splnes. Numercal examples are provded to sow tat te metod presented provdes a fne approxmaton of te exact soluton. Te frst capter provdes some background materal to te cubc splne and boundary value problems. Te works of several autors and a comparson of dfferent soluton metods are also dscussed. Fnally, some background nto te specfc sngularly perturbed boundary value problems s ntroduced. Te second capter contans calculatons and dervatons necessary for te cubc splne and te ntal value tecnque wc are used n te solutons to te boundary value problems. Te trd capter contans some worked numercal examples and te numercal data obtaned along wt most of te tables and fgures tat descrbe te solutons. Te tess concludes wt some reflectons on te results obtaned and some dscusson of te error bounds on te calculated approxmatons to te exact solutons for te numerc examples dscussed.

5 Ts tess s dedcated to my wfe Katuska and to my students, wo ave contnuously remnded me of te value of patence and persstence. It s also dedcated to my famly, wo remnd me tat te most mposng of tasks can be accomplsed one small step at a tme. v

6 ACKNOWLEDGEMENTS Dr. Ram Moapatra as been an deal mentor and tess advsor. Hs sage advce, tougtful crtcsm, and contnual encouragement ave been nstrumental to te completon of ts tess. I would also lke to tank Dr. Davd Rollns wose metculous attenton to detal was greatly needed and deeply apprecated. Fnally, I would tank Rcard Wenser and Dane Vargas wose steadfast support aded n te wrtng of ts tess n nnumerable ways. v

7 TABLE OF CONTENTS LIST OF FIGURES... v LIST OF TABLES... x CHAPTER 1: INTRODUCTION... 1 CHAPTER : INTRODUCTORY DISCUSSION FOR THE METHOD... 7 Te Cubc Splne... 7 Te Intal-Value Tecnque CHAPTER 3: NUMERICAL EXAMPLES... 4 Lnear Sngular Perturbaton Problems wt Left-end Boundary Layer... 4 Example Example... 8 Example Lnear Sngular Perturbaton Problems wt Rgt-end Boundary Layer Example Example Nonlnear Sngular Perturbaton Problems wt Left-end Boundary Layer Example Example CONCLUSION v

8 LIST OF REFRENCES v

9 LIST OF FIGURES Fgure 1: Grap of te soluton y(x) for te dfferental equaton (30) and (31) Fgure : Grap of te splne soluton of Example Fgure 3: Grap of te splne soluton of Example Fgure 4: Grap of te splne soluton of Example Fgure 5: Grap of te splne soluton of Example Fgure 6: Grap of te splne soluton of Example Fgure 7: Grap of te splne soluton of Example Fgure 8: Grap of te splne soluton of Example v

10 LIST OF TABLES Table 1: Numercal results wt = Table : Numercal results of Example 1 wt 10, Table 3: Numercal results of Example 1 wt 10, Table 4: Numercal results of Example 1 wt 10, Table 5: Numercal results of Example wt 10, Table 6: Numercal results of Example wt 10, Table 7: Numercal results of Example wt 10, Table 8: Numercal results of Example 3 wt 10, Table 9: Numercal results of Example 3 wt 10, Table 10: Numercal results of Example 3 wt 10, Table 11: Numercal results of Example 4 wt 10, Table 1: Numercal results of Example 4 wt 10, Table 13: Numercal results of Example 4 wt 10, Table 14: Numercal results of Example 5 wt 10, Table 15: Numercal results of Example 5 wt 10, Table 16: Numercal results of Example 5 wt 10, Table 17: Numercal results of Example 6 wt 10, Table 18: Numercal results of Example 6 wt 10, x

11 4 4 Table 19: Numercal results of Example 6 wt 10, Table 0: Numercal results of Example 7 wt 10, Table 1: Numercal results of Example 7 wt 10, Table : Numercal results of Example 7 wt 10, x

12 CHAPTER 1: INTRODUCTION Te cubc splne nterpolaton s based on te engneerng tool used to draw smoot curves troug a fnte number of ponts. Te engneer s splne conssts of wegts attaced to a flat surface at te ponts to be connected, and a flexble strp s ten bent across eac of te wegts, resultng n a smoot curve; te matematcal splne uses a smlar prncple. Te ponts to be connected represent numercal data, and te wegts are represented by numercal coeffcents for a cubc polynomal tat bend te lne so tat t passes contnuously troug eac of te data ponts [5]. Snce real-world numercal data s often dffcult to analyze, fndng a functon tat specfcally relates te data s usually dffcult to obtan and use. For example, n te study of eat transfer, problems of te deflecton of plates and n a number of oter scentfc applcatons, we fnd a system of dfferental equatons of dfferent order wt dfferent boundary condtons. Many problems are formulated matematcally n boundary value problems for second order dfferental equatons as n eat transfer and deflecton n cables. Instead of tryng to ft one functon as te soluton to te dfferental equatons, we can use te seres of unque cubc polynomals ftted between a set of data ponts. We wll stpulate tat te curve obtaned must be contnuous and smoot, and we can ten use ts cubc splne to nterpolate data and rates of cange over an nterval. A boundary value problem as condtons specfed at te extremes of te ndependent varable. If te problem s dependent on bot space and tme, ten nstead of specfyng te value of te problem at a gven pont for all tme, te data could be gven at a gven tme for all space. For example, te temperature of an ron bar wt one end kept at absolute zero and te 1

13 oter end at te freezng pont of water would be a boundary value problem. Tere are tree types of boundary value problems, namely problems dealng wt Drclet boundary condtons, Neumann boundary condtons, and Caucy boundary condton If te boundary gves a value to te problem ten t s a Drclet boundary condton. For example f one end of an ron rod ad one end eld at absolute zero ten te value of te problem would be known at tat pont n space. A Drclet boundary condton mposed on an ordnary dfferental equaton or a partal dfferental equaton specfes te values a soluton s to take on te boundary of te doman. Te queston of fndng solutons to suc equatons s known as te Drclet problem. If te boundary gves a value to te normal dervatve of te problem ten t s a Neumann boundary condton. For example f one end of an ron rod ad a eater at one end ten energy would be added at a constant rate but te actual temperature would not be known. A Neumann boundary condton mposed on an ordnary dfferental equaton or a partal dfferental equaton specfes te values te dervatve of a soluton s to take on te boundary of te doman. If te boundary as te form of a curve or surface tat gves a value to te normal dervatve and te problem tself ten t s a Caucy boundary condton. A Caucy boundary condton mposed on an ordnary dfferental equaton or a partal dfferental equaton specfes bot te values a soluton of a dfferental equaton s to take on te boundary of te doman and te normal dervatve at te boundary. It bascally corresponds to mposng bot a Drclet and a Neumann boundary condton. Caucy boundary condtons can be understood from te teory

14 of second order ordnary dfferental equatons, were to ave a partcular soluton one as to specfy te value of te functon and te value of te dervatve at a gven ntal or boundary pont. E.A. Al-Sad as solved te system of second-order boundary value problems of te type f ( x), a x c u" g( x) u( x) f ( x) r, c x d f ( x), d x b () wt te Drclet boundary condtons u( a) and u( b) = () 1 assumng te contnuty condtons of u and u ' at c and d, and were f and g are contnuous functons on [ ab, ] and [ cd, ], respectvely. Te parameters r, 1, are real fnte constants. He used a cubc splne functon to develop a numercal metod for computng smoot approxmatons to te soluton and ts dervatves for a system of second-order boundary-value problems of te type () []. A. Kan and T. Azz appled parametrc cubc splne functons to develop a new numercal metod for obtanng smoot approxmatons to te soluton of te system of second-order boundary value problem of te type () avng Drclet boundary condtons [9]. Sraj-ul-Islam and Ikram A. Trmz ave appled non-polynomal splne functons tat ave a polynomal and trgonometrc parts to develop a new numercal metod for obtanng smoot approxmatons to te soluton of te system of second-order boundary value problem of te type () avng Drclet boundary condtons [16]. 3

15 Arsad Kan as derved a unformly convergent unform mes dfference sceme usng parametrc cubc splne for te soluton of te two-pont boundary value problem wt Drclet boundary condtons of te type y"( x) f ( x) y( x) g( x), a x b y( a)=, y( b)=, 0 1 () were f() x and gx ( ) are contnuous functons on [ ab, ] and ab,, 0, 1 are arbtrary real fnte constants. Suc problems arse plate deflecton teory and a number of oter scentfc applcatons [10]. In general t s dffcult to obtan te analytcal soluton of () for arbtrary coces of f() x and gx ( ). Te standard numercal metods for te numercal treatment of () consst of fnte dfference metods dscussed by many autors. Te lterature of numercal analyss contans lttle on te soluton of second order twopont boundary value problems subjected to Neumann boundary condtons. M.A. Ramadan and I.F. Lasen ave used bot polynomal and non-polynomal splne functons to develop numercal metods for obtanng smoot approxmatons for te soluton of te lnear second order two-pont boundary value problem subjected to Neumann boundary condtons [14]. Albasny and Ragavarao solved lnear second order two-pont boundary problem () subjected to Drclet boundary condtons usng cubc polynomal splne [1] [13]. Blue solved ts problem usng quntc polynomal splne [4], wle, Caglar et al. solved ts problem usng cubc B-splne [6]. Sngular perturbaton problems occur commonly n many brances of matematcs. Te governng equatons of varous matematcal models n pyscal, bologcal, economc, or 4

16 engneerng applcatons often nvolve caracterstcs tat make t dffcult to obtan an exact soluton to a problem. Some solutons may ave a closed form, but result n a complcated ntegral soluton, wle solutons to oter models are more easly obtaned but result n an nfnte seres soluton. Wen a large parameter or small parameter occurs wtn te matematcal model n one of tese processes, perturbaton metods are used to construct a seres of smpler equatons wc can be used to approxmate te soluton to te problem. In general, many tecnques used to solve sngularly perturbed problems consst of dvdng te problem nto nner and outer regons, expressng te nner and outer solutons as asymptotc expansons, equatng terms n te expressons developed to determne te constants n te expressons, and untng te nner and outer solutons to obtan te fnal vald soluton to te problem. However, te dffcultes n usng tese tecnques often arse wen matcng te coeffcents n te nner and outer expansons n order to yeld te fnal soluton. Recently, non-asymptotc metods ave been used to solve certan classes of sngularly perturbed problems, replacng sngularly perturbed two-pont boundary value problems by ntal-value tecnques. We sall nvestgate an ntal-value tecnque for sngularly perturbed two-pont boundary value problems va cubc splnes ntroduced by Manoj Kumar, Ptam Sng, and Hradyes Kumar Msra. Te ntal-value tecnque wll be examned and tested an alternate metod to approxmate solutons. Te approxmate solutons wll ten be compared to te exact solutons to see f te metod s successful for solvng ordnary dfferental equatons. Ten te 5

17 metod wll be appled to approxmate solutons to lnear and nonlnear sngularly perturbed boundary value problems. 6

18 CHAPTER : INTRODUCTORY DISCUSSION FOR THE METHOD Te Cubc Splne We want to ft a pecewse functon of te form s1 ( x), x1 x x s ( x), x x x S( x) 3 (1) s ( x), x x x n1 n1 n were s s a trd-degree polynomal defned by s x a x x b x x c x x d () 3 ( ) ( ) ( ) ( ) for 1,,, n 1. Te frst and second dervatves are also necessary to te process and are gven by s x a x x b x x c (3) ( ) 3 ( ) ( ) s ( x) 6 a ( x x ) b (4) for 1,,, n 1. Te cubc splne wll need to conform to te followng four propertes: 1. Te pecewse functon Sx ( ) wll nterpolate all te data ponts; tat s, S( x ) y (5) 7

19 for 1,,, n 1. Snce eac x [ x, x 1], S( x) s( x), and from () we ave y s( x ) y a x x b x x c x x d y 3 ( ) ( ) ( ) d (6) for 1,,, n 1.. Te functon Sx ( ) wll be contnuous on te nterval [ x1, x n]. Ten we must ensure tat eac sub-functon must jon at te data ponts, so for,, n, we ave s ( ) ( ) 1 x s x (7) Ten usng equaton (), we ave s ( x ) d and s ( x ) a ( x x ) b ( x x ) c ( x x ) d (8) so d a ( x x ) b ( x x ) c ( x x ) d (9) for,3,, n 1. If we let x x 1 n equaton (9), we ave d a b c d (10)

20 for,3,, n Te functon S () x wll be contnuous on te nterval [ x1, x n]. In order to ensure tat te curve s smoot across te nterval, te dervatves must be equal at te data ponts; tat s, s ( ) ( ) 1 x s x (11) for,3,, n 1 Ten usng equaton (3), we ave s( x ) c and s ( x ) 3 a ( x x ) b ( x x ) c so c 3 a ( x x ) b ( x x ) c (1) for,3,, n 1. If we let x x 1 n equaton (1), we ave c 3a b c (13) for,3,, n 1. 9

21 4. Te functon S () x wll be contnuous on te nterval [ x1, x n]. In order to ensure tat te curve s contnuous across te nterval, te second dervatves must be equal at te nteror data ponts; tat s, s ( x ) s 1( x ) for 1,,, n 1. From equaton (4) we ave s( x) 6 a ( x x ) b, so s( x) 6 a ( x x ) b, (14) s ( x ) 6 a ( x x ) b s ( x ) b for,3,, n. Snce we must ave s ( x) contnuous across te nterval, s ( x ) s 1( x ) for 1,,, n 1. Ten combnng ts and equaton (14), we ave tat s( x ) 6 a ( x x ) b 1 1 (15) s ( x ) 6 a ( x x ) b (16) If we let x 1 x n equaton (14) and (16), we ave s ( x ) b (17) 10

22 b 1 6a b (18) At ts pont, we could substtute M for s (x) and express te equatons above n terms of M and y to determne te wegts of a, b, c, and d.[7] We wll nstead smplfy tese equatons by makng a substtuton usng equaton (3): s( x ) c m c (19) and expressng te equatons above n terms of m and y. We can ten determne te remanng coeffcent a, b, and d. We want to relate te cubc splne functon to boundary value problems and ntal value problems wt boundary condtons gven n terms of functon values and frst dervatves. We note tat d as already been determned to be d y. (0) We can smlarly use equaton (3) as c 3 a ( x x ) b ( x x ) c m 3 a ( x x ) b ( x x ) m (1) Usng equaton () we can wrte d a ( x x ) b ( x x ) c ( x x ) d y a ( x x ) b ( x x ) m ( x x ) y () 11

23 We can now use equatons (1) and () to determne expressons for a and b. Let = x +1 x and solve te system: m 1 m 3a b ( y y ) m a b 3 1 Multplyng (1) by, () by, and combnng equatons results n te followng equaton: ( m m ) ( y y ) a 1 1 ( m 1m ) ( y 1 y ) a 3. 3 (3) Multplyng (1) by, () by 3, and combnng equatons results n te followng equaton: ( m m ) 3( y y ) b 1 1 3( y 1 y ) ( m 1 m ) b. (4) We can now wrte te functon S(x) strctly n terms of m and y, and we ave ( m 1m ) ( y 1 y ) 3 S( x) ( x x ) 3 3( y 1 y ) ( m 1 m ) ( x x ) m ( x x ) y (5) We wll fnally rewrte te cubc splne functon S(x) n terms of ts frst dervatves by collectng terms for m and y. We ad let = x +1 x for te prevous equatons. Collectng terms for m, we ave 1

24 3 m ( x x ) m ( x x ) m ( ) x x m( x x) ( x x ) ( x x ) ( x x ) and expandng terms n te brackets yelds m( x x) x ( x ) x x m x x x x ( )( 1 ). 1 1 (6) Collectng terms for m +1, we ave m ( x x ) m ( x x ) m 1( x x) [ x x ] and rewrtng n te brackets, yelds m 1( x x ) ( x 1x). (7) Collectng terms for y, we ave y ( x x ) 3 y ( x x ) 3 3 y y 3 3 ( x x ) 3( ) 3 x x 13

25 and expandng terms n te brackets yelds y x ( 1 x) 3 [ x x x x 1 ] y x x x x ( 1 ) [( ) ] 3. (8) Fnally, collectng terms for y +1, we ave y ( x x ) 3 y ( x x ) y x ( x) 3 [ ( x x ) 3 ] and snce = x +1 x, we ave y 1( x x ) [( x 1 x) ] 3. (9) Te cubc splne functon S(x) n terms of ts frst dervatves S (x) s now gven by m ( x x )( x 1 x) m 1( x x ) ( x 1 x) Sx ( ) y ( x 1 x) [( x x ) ] 3 y 1( x x ) [( x 1 x) ] 3 (30) were = x +1 x wll be used as te mes sze for calculatng te splne functon. We wll now dfferentate wt respect to x and smplfy te equaton. 14

26 Dfferentatng (30) wt respect to x, we obtan m ( x 1 x)( x 1 x 3 x) m 1( x x )( x x 1 3 x) S'( x) 6 y ( x 1 x)( x x ) 6 y 1( x x )( x 1 x) 3 3 (31) We dfferentate (31) agan wt respect to x, and we ave m (x 1 x 3 x) m 1( x 1 x 3 x) S''( x) 6 y ( x 1 x x) 6 y 1( x 1 x x) 3 3 (3) wc yelds m (x 1 x 3 x 1) m 1( x 1 x 3 x 1) S''( x 1) 6 y ( x 1 x x 1) 6 y 1( x 1 x x 1) 3 3 Fnal smplfcaton results n m 4m 6y 6y S''( x ) m 4m 6S 6S S''( x ) (33) If we consder an ntal-value problem dy f ( x, y) dx (34) 15

27 y( x ) y 0 0 (35) ten from te can rule, we ave d y f f dy dx x y dx and f f y ''( x 1) ( x 1, y 1) ( x 1, y 1) f ( x 1, y 1) x y f f y ''( x 1) ( x 1, S 1) ( x 1, S 1) f ( x 1, S 1) x y (36) Wen we equate te two expressons (33) and (36), we obtan m 4m 1 6S 6S 1 f f ( x, S ) ( x, y ) f ( x, S ) x y (37) and we wll use ts equaton to compute te values of S and n turn use (30) to get S(x). Let us consder a bref numercal example to llustrate te cubc splne metod of solvng a dfferental equaton. Consder te ntal value problem dy y dx (38) y(0) 1 (39) Equatng (33) and (36) we ave m 4m 6S 6S 1 1 S 1 (40) From (38), we can replace te m n (40) and we ave 16

28 S 4S 6S 6S 1 1 S 1 (41) and solvng for S +1 we ave S ( 6) S ( 4 6) (4) 1 and we can now use te ntal value to calculate next value n te sequence. Te numercal results are gven n Table 1 for = Te grap of te functon y(x) s gven n Fgure 1. Table 1: Numercal results wt =0.01 x y(x) exact soluton We see tat te data ponts are remarkably close te exact soluton for a reasonably small step sze, and te data ponts can be used now to provde a soluton for te dfferental equaton. 17

29 3 y Fgure 1: Grap of te soluton y(x) for te dfferental equaton (30) and (31) 18

30 Te Intal-Value Tecnque In ter paper, Kumar, Sng, and Msra [1] ntroduce an ntal-value tecnque for sngularly perturbed two-pont boundary value problems wt a layer on te left (or rgt) end of te underlyng nterval n wc te orgnal second order problem s replaced by an asymptotcally equvalent tree frst-order ntal-value problems, wc are ten solved va cubc splne. For convenence, we wll call ts metod te ntal-value tecnque. We frst consder a lnear sngularly perturbed two-pont boundary problem of te form: u"( x) p( x) u'( x) q( x) u( x) r( x), x [ a, b] (43) ua () (44) ub () (45) were s a small postve parameter (0 1) and, are known constants. We assume tat p(x), q(x), and r(x) are suffcently contnuously dfferentable functons n [a, b], and furtermore, we assume tat p( x) M 0 trougout te nterval [a, b], were M s some postve constant. Ts assumpton mples tat te soluton of (43), (44), and (45) wll be n te negborood of x = a. Snce sngular perturbaton problems exbt boundary layer beavor of te soluton, te soluton of (43), (44), and (45) s gven by u( x, ) v( x) w( x) e t( x)/ (46) wt 19

31 x t( x) p( ) d a were v( x, ) v ( x) n and n0 n w( x, ) w ( x) n (cf.[17, p.9]), so we ave n0 n u( x, ) v ( x) v ( x) e n n t( x)/ n n n0 n0 (47) wt x t( x) p( ) d. (48) a Dfferentatng (47) wt respect to x yelds n n t( x)/ n t( x)/ px ( ) u '( x, ) vn '( x) wn '( x) e wn ( x) e n0 n0 n0 (49) n n t ( x)/ n t ( x)/ px ( ) u ''( x, ) vn ''( x) wn ''( x) e wn '( x) e n0 n0 n0 n t ( x)/ p x n t ( x)/ wn( x) e wn( x) e n0 n0 ( ) p '( x) (50) We can now substtute (47), (48), and (49) n (43), and we ave 0

32 n1 n1 t( x)/ n t( x)/ vn ''( x) wn ''( x) e wn '( x) e p( x) n0 n0 n0 n1 t ( x)/ n t( x)/ wn( x) e [ p( x)] wn( x) e p '( x) n0 n0 n1 t( x)/ n [ p( x)] wn( x) e q( x) vn( x) n0 n0 q x w x e r x n ( ) n( ) t( x)/ ( ) n0 (51) By restrctng tese seres to ter frst terms, we te get p( x) v '( x) q( x) v ( x) [ p( x) w '( x) p'( x) w ( x) p x w x q x w x e r x t( x)/ ( ) 0'( ) ( ) 0( )] ( ) We terefore ave te followng: p( x) v '( x) q( x) v ( x) r( x) (5) 0 0 and d [ p ( x ) w 0( x )] q ( x ) w 0( x ) (53) dx Te representatons (47) and (48) can be nserted to te boundary condtons (44) and (45), and te boundary condtons become v ( a) w ( a) (54) 0 0 and v ( b) 0 (55) 1

33 were te exponentally small term e tb ( )/ s neglected n order to obtan te boundary condton (55) at x = b. Te dfferental equaton (5) can be solved wt te boundary condton (55) to determne v 0 ( x ), ten w ( ) 0 x s determned by solvng te dfferental equaton (53) wt te boundary condton w 0 ( a) v 0 ( a). x From (48), we ave t( x) p( ) d, so t '( x) p( x) wt ta ( ) 0 a Te tree ntal-value problems correspondng to (43), (44), and (45) are gven by [1] (IVP.I) p( x) v0'( x) q( x) v0( x) r( x) wt v ( b) (56) 0 d p ( x ) w 0( x ) q ( x ) w 0( x ) wt w0( a) v0( a) (57) dw (IVP.II) (IVP.III) t '( x) p( x) wt ta ( ) 0 (58) Tese ntal-value problems are ndependent of te perturbaton parameter and wll be solved by te cubc splne metod. For te problems exbtng rgt-end beavor, we use te tree ntalvalue problems as well. We consder a lnear sngularly perturbed two-pont boundary problem of te form: u"( x) p( x) u'( x) q( x) u( x) r( x), x [ a, b] (59) ua () (60) ub () (61)

34 were s a small postve parameter (0 1) and, are known constants. We assume tat p(x), q(x), and r(x) are suffcently contnuously dfferentable functons n [a, b], and furtermore, we assume tat p( x) M 0 trougout te nterval [a, b], were M s some negatve constant. Ts assumpton mples tat te soluton of (43), (44), and (45) wll be n te negborood of x = b. by[1] Terefore te tree ntal-value problems correspondng to (43), (44), and (45) are gven (IVP.I) p( x) v0'( x) q( x) v0( x) r( x) wt v ( a) (6) 0 d p ( x ) w 0( x ) q ( x ) w 0( x ) wt w0( a) v0( b) (63) dw (IVP.II) (IVP.III) t '( x) p( x) wt tb ( ) 0 (64) Tese ntal-value problems are ndependent of te perturbaton parameter and wll be solved by te cubc splne metod. 3

35 CHAPTER 3: NUMERICAL EXAMPLES Lnear Sngular Perturbaton Problems wt Left-end Boundary Layer Example 1 Frst, consder te followng omogeneous sngular perturbaton problem from Bender and Orzag [3, p.480, Problem 9.17 wt =0] y"( x) y '( x) y( x) 0 x[0,1], y(0) 1, y(1) 1. (65) Here we note tat p( x) 1, q( x) 1, r( x) 0. From (56), we ave v 0 '( x) v 0 ( x) 0 wt v 0 (1) 1, so v0 ''( x) v0 '( x) v0( x) we can set v m and we ave v 1, so te resultng equaton from (37) s n 6v 6v m 4m v v ( 6) v 1 m 4m 1 0 (6 ) v [ 4 6] v 1 0 v [ 4 6] (6 ) v 1 In an effort to make te metod readly accessble, we wll be usng an ndustry-standard software package for spreadseets (Mcrosoft EXCEL) to solve for v 0 ( x ) and ten, w ( x ). 0 4

36 From (57), we ave w 0 '( x) w 0 ( x) 0, w 0 (0) 1 v 0 (0), so w0'( x) w0( x) we can set w mand we ave w 1 v 0 1 (0), so te resultng equaton from (37) s 6w 6w m 4m w w (6 ) w 1 m 4m 1 0 (6 ) w [4 6] w 1 0 w (6 ) w [4 6] 1 From (58) we ave t() x x after ntegratng and applyng te ntal condton. Te numercal 3 results are gven n Table for 10. Table : Numercal results of Example 1 wt 10, 10 x v(x) w(x)

37 Te soluton to (65) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t() x x. Te results are gven n Table 3 and Table 4 for and 10 respectvely, and compared to te exact soluton. Te grap of te cubc splne functon s gven n Fgure. Te exact soluton to (65) s gven by m m1 x m1 mx ( e 1) e (1 e ) e yx ( ), m m1 e e were m1 and m1. Table 3: Numercal results of Example 1 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E

38 Table 4: Numercal results of Example 1 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E y(x) Fgure : Grap of te splne soluton of Example 1 7

39 Example Next, consder te followng non-omogeneous sngular perturbaton problem from flud dynamcs for flud of small vscosty [15, Example ] y"( x) y '( x) 1 x x[0,1], y(0) 0, y(1) 1. (66) Here we note tat p( x) 1, q( x) 0, r( x) 1 x. From (56), we ave v 0 '( x) 1 x wt v 0 (1) 1, so we can set m 1 1 x, so te resultng equaton from (37) s 6v 6v m 4m v 6v (1 x ) 4 (1 x ) 1 1 6v (1 x ) 4 (1 x ) v From (57), we ave w 0 '( x) 0, w 0 (0) 1 v 0 (0), so w ( ) 1 0 x and te resultng equaton from (37) s smply w 1 w From (58) we ave t() x x after ntegratng and applyng te ntal condton. Te numercal 3 results are gven n Table 5 for 10. Te splne functon s graped n Fgure 3. 8

40 Table 5: Numercal results of Example wt 10, 10 x v(x) w(x) Te soluton to (66) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t() x x. Te results are gven n Table 6 for 10 3 soluton [1]. Te exact soluton s gven by 4 and Table 7 for 10 and compared to te exact y( x) x( x 1 ) x/ ( 1)(1 e ) 1/ (1 e ). 9

41 Table 6: Numercal results of Example wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E Table 7: Numercal results of Example wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E

42 y(x) Fgure 3: Grap of te splne soluton of Example Example 3 Now, consder te followng varable coeffcent sngular perturbaton problem from Kevorkan and Cole [8, Eqs. (.3.6) and (.3.7) wt 1/ ] x 1 y"( x) 1 y '( x) y( x) 0 x[0,1], y(0) 0, y(1) 1. (67) x 1 Here we note tat p( x) 1, q( x), r( x) 0. 31

43 Te soluton to (67) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t( x) x x / 4. Te results are gven n Table 8 and Table 9 for 10 3 and n Table 10 for 4 10 and compared to te exact soluton [1]. Te splne functon s graped n Fgure 4. Te exact soluton s gven by 1 1 y x x x x ( ) exp ( / 4) /. Table 8: Numercal results of Example 3 wt 10, 10 x v(x) w(x)

44 Table 9: Numercal results of Example 3 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E Table 10: Numercal results of Example 3 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E

45 y(x) Fgure 4: Grap of te splne soluton of Example 3 Lnear Sngular Perturbaton Problems wt Rgt-end Boundary Layer Example 4 Frst, consder te followng omogeneous sngular perturbaton problem y"( x) y '( x) 0 x[0,1], y(0) 1, y(1) 0. (68) Here we see tat p( x) 1, q( x) 0, r( x) 0. Te soluton to (68) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t( x) x 1. Te results are gven n Table 11 and Table 1 for and Table 13 for 10 and 34

46 compared to te exact soluton [11]. Te splne functon s graped n Fgure 5. Te exact soluton s gven by: e yx ( ) e ( x1)/ 1/ 1 1. Table 11: Numercal results of Example 4 wt 10, 10 x v(x) w(x) Table 1: Numercal results of Example 4 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E

47 Table 13: Numercal results of Example 4 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E y(x) Fgure 5: Grap of te splne soluton of Example 4 36

48 Example 5 Frst, consder te followng omogeneous sngular perturbaton problem y"( x) y '( x) (1 ) y( x) 0 x y e y (1 )/ [0,1], (0) 1, (1) 1 1/. (69) Here we see tat p( x) 1, q( x) (1 ), r( x) 0. Te soluton to (69) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t( x) x 1. Te results are gven n Table 14 and Table 15 for 10 3 and n Table 16 for 10 4, and compared to te exact soluton [11]. Te splne functon s graped n Fgure 6. Te exact soluton s gven by (1 )( x 1)/ y( x) e. Table 14: Numercal results of Example 5 wt 10, 10 x v(x) w(x)

49 Table 15: Numercal results of Example 5 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E Table 16: Numercal results of Example 5 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E

50 y(x) Fgure 6: Grap of te splne soluton of Example 5 Nonlnear Sngular Perturbaton Problems wt Left-end Boundary Layer Example 6 For te nonlnear boundary value problems, we convert te nonlnear sngular perturbaton problem s converted to a sequence of lnear sngular perturbaton problems by usng quaslnearzaton, and ten te outer layer soluton s taken to be te ntal approxmaton. Frst, consder te followng sngular perturbaton problem from Kevorkan and Cole [8 and Cole, p. 56, Eq. (.5.1)] y"( x) y( x) y '( x) y( x) 0 x[0,1], y(0) 1, y(1) (69) 39

51 Te ntal approxmaton can be taken from te problem y( x) y'( x) y( x) 0. (70) Gven te value at x = 0, we must suppose y'( x) 0, and y() x C and furter tat y( x) x.9995 n order to satsfy te condton at x = 1.We can te use te lnear problem y"( x) x.9995 y '( x) x x[0,1], y(0) 1, y(1) (71) as te lnear problem concerned to (69). We can now solve te lnear problem as te approxmaton to te nonlnear problem (69) and we ave p( x) x.9995, q( x) 0, and r( x) x Te soluton to (69) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t( x) x /.9995x. Te results are gven n Table 17 and Table 18 for 10 3 and n Table 19 for 10 4, and compared to te exact soluton [8, p , Eqs. (.5.5), (.5.11), and (.5.14)]. Te splne functon s graped n Fgure 7. Te exact soluton s gven by y( x) x c tan( c ( x / c ) / ) 1 1 were c and c (1/ c1 )log e[( c1 1) /( c1 1)]. For ts example, we ave a boundary layer of wdt O( ) at x = 0 [1] 40

52 Table 17: Numercal results of Example 6 wt 10, 10 x v(x) w(x) Table 18: Numercal results of Example 6 wt 10, x y(x) exact soluton error E E E E E E E E E E E E E+00 41

53 Table 19: Numercal results of Example 6 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E y(x) Fgure 7: Grap of te splne soluton of Example 6 4

54 Example 7 Fnally, let us consder te followng sngular perturbaton problem from Bender and Orszag [3, p. 463, Eq (9.7.1)] y x y x e y( x) "( ) '( ) 0 x[0,1], y(0) 0, y(1) 0. (7) Te ntal approxmaton can be taken from te problem and we can ten use te lnear problem y"( x) y '( x) y( x) ln 1 1 x 1 x 1 x x[0,1], y(0) 0, y(1) 0 (73) [11] as te lnear problem concerned to (7). We can now solve te lnear problem as te approxmaton to te nonlnear problem (7) and we ave p( x), q( x), and 1 x rx ( ) ln 1 1x 1x. Te soluton to (7) usng (46) as te form u( x) v( x) w( x) e t( x)/ wt t( x) x. Te results are gven n Table 0 and Table 1 for 10 3 and n Table for 10 compared to te exact soluton [3, p. 463, Eq. (9.7.6)]. Te exact soluton s gven by 4, and y( x) log (/(1 x)) log () e e e x/ For ts example, we ave a boundary layer of wdt O( ) at x = 0 [3]. Te splne functon s graped n Fgure 8. 43

55 Table 0: Numercal results of Example 7 wt 10, 10 x v(x) w(x) Table 1: Numercal results of Example 7 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E

56 Table : Numercal results of Example 7 wt 10, 10 x y(x) exact soluton error E E E E E E E E E E E E E y(x) Fgure 8: Grap of te splne soluton of Example 7 45

57 CONCLUSION Te ntal-value tecnque descrbed by Kumar, et al. as been examned as a metod for solvng sngularly perturbed two-pont boundary value problems. For eac problem examned, we computed te soluton numercally by solvng tree ntal value problems, wc are deduced from te orgnal problem. In general, te numercal soluton of a boundary value problem wll be more dffcult to calculate tan te numercal solutons of te ntal-value problems. It s generally preferable to convert te second order problem nto frst order problems. Ts tecnque was mplemented usng standard, readly avalable software spreadseet program (Mcrosoft EXCEL), wc makes t easy to mplement on any computer and requres only modest preparaton. No knowledge of dfferental equatons s requred to complete te process, and mnmal knowledge of Calculus s requred for te process. We mplemented te metod on lnear boundary value problems wt bot rgt-end and left-end beavor. Usng quaslnearzaton, we were able to approxmate te solutons to nonlnear boundary value problems wt left-end beavor. We used two specfc values of and used te same value for te mes sze, but te data suggests tat ncreasng te mes sze provdes a proportonally better approxmaton. Te approxmaton became more accurate for smaller values of. Tus te ntal-value tecnque provdes a reasonable approxmaton for te soluton of te problem, and te cubc splne metod was easly mplemented to solve te ntal-value problems. In addton, te step sze used for te calculatons could easly be vared n order to more closely approxmate te values near te boundary ponts. 46

58 By calculatng te frst dervatves along wt te pont values, we were also able to buld a splne functon to grap te solutons and to calculate any values oter tan te nodes. Te numercal results ndcate tat te ntal-value tecnque s accurate and sutable for solvng lnear and nonlnear problems wt tn layers. Te splne s also useful for provdng te addtonal data ponts tat can ten be used to refne te splne near te boundary ponts. Once te splne s calculated usng an equal step sze, anoter calculaton can be used to refne te splne near te boundary values usng a smaller step sze as needed. Te error estmate for te cubc splne metod for te frst-order problems s descrbed n Kumar s papers. For a functon y C 4 [ a, b], te cubc splne metod descrbed provdes a fourt-order approxmaton to te soluton of te ntal-value problems used for te ntal-value tecnque. Also, te error bounds for bot cubc splnes and cubc Hermte splnes ave an error 4 bound tat s alsoo. For te soluton to te nonlnear problems, te convergence s quadratc. 47

59 LIST OF REFRENCES [1] Albasny, E. L., and W. D. Hoskns. "Cubc Splne Soluton to Two-pont Boundary Value Problems." Computer Journal 11 (1968): Prnt. [] Al-Sad, E. A. "Te Use of Cubc Splnes n te Numercal Soluton of a System of Second-order Boundary Value Problem." Computer and Matematcs wt Applcatons 4 (001): Prnt. [3] Bender, Carl M., and Steven A. Orszag. Advanced Matematcal Metods for Scentsts and Engneers. New York: McGraw-Hll, Prnt. [4] Blue, James L. "Splne Functon Metods for Nonlnear Boundary Value Problems." Communcatons of te ACM (1969). Prnt. [5] Burden, Rcard L., and J. Douglas. Fares. Numercal Analyss. Pacfc Grove, CA: Brooks/Cole Pub., Prnt. [6] Caglar, H., N. Caglar, and K. Elfatur. "B-splne Interpolaton Compared wt Fnte Dfference, Fnte Element and Fnte Volume Metods Wc Appled to Two-pont Boundary Value Problems." Appled Matematcs and Computaton (006): Prnt. [7] De, Boor Carl. A Practcal Gude to Splnes: wt 3 Fgures. Vol. 7. New York: Sprnger, 001. Prnt. 48

60 [8] Kevorkan, J., and Julan D. Cole. Perturbaton Metods n Appled Matematcs. New York: Sprnger-Verlag, Prnt. [9] Kan, A., and T. Azz. "Parametrc Cubc Splne Approac to te Soluton of a System of Second-Order Boundary-Value Problems." Journal Of Optmzaton Teory and Applcatons (003): Prnt. [10] Kan, Arsad. "Parametrc Cubc Splne Soluton of Two Pont Boundary Value Problems." Appled Matematcs and Computaton 154 (004): Prnt. [11] Kumar, Manoj, Hradyes Kumar Msra, and P. Sng. "Numercal Treatment of Sngularly Perturbed Two Pont Boundary Value Problems Usng Intal-value Metod." Numercal Treatment of Sngularly Perturbed Two Pont Boundary Value Problems Usng Intal-value Metod nd ser. 9.1 (008): Prnt. [1] Kumar, Manoj, P. Sng, and Hradyes Kumar Msra. "An Intal-Value Tecnque for Sngular Perturbed Boundary Value Problems va Cubc Slne." Internatonal Journal for Computatonal Metods n Engneerng Scence and Mecancs 8.6 (007): Prnt. [13] Ragavarao, C.V., Y.V.S.S. Sanyasraju, and S. Sures. "A Note on Applcaton of Cubc Splnes to Two-pont Boundary Value Problems." Computer and Matematcs wt Applcatons 7.11 (1994): Prnt. 49

61 [14] Ramadan, M. A., I. F. Lasen, and W. K. Zara. "Polynomal and Nonpolynomal Splne Approaces to te Numercal Soluton of Second Order Boundary Value Problems." Appled Matematcs and Computaton 184 (007): Prnt. [15] Renardt, Hans Jurgen. "Sngular Perturbatons of Dfference Metods for Lnear Ordnary Dfferental Equatons." Applcable Analyss 10 (1980): Prnt. [16] Sraj-ul-Islam, and Ikram A. Trmz. "Nonpolynomal Splne Approac to te Soluton of a System of Second-order Boundary-value Problems." Appled Matematcs and Computaton 173 (006). Prnt. [17] Smt, Donald R. Sngular-perturbaton Teory: an Introducton wt Applcatons. Cambrdge [Cambrdgesre: Cambrdge UP, Prnt. 50