Singular Two Points Boundary Value Problem

Transcription

1 An-aja atonal Unverst Facult of Graduate Studes Sngular Two Ponts Boundar Value Problem B Sawsan Moammad Hamdan Supervsed b Dr. Samr Matar Ts tess s submtted n partal fulfllment of te requrements for te Degree of Master n Computatonal Matematcs Facult of Graduate Studes at An-aja atonal Unverst ablus Palestne 00

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3 III Dedcaton To M Parents M Husband M cldren Ragad Saf Al-deen Maar M Ssters Broters and to te Soul of te martr Isan and to all wo elped me to fulfll ts tess I dedcate t.

4 IV Acknowledgments Frst m greatest tanks for alla for elpng me fns ts work as good as I ope. Ten m all tanks and wses for m supervsor Dr. Samr Matar wo drected me to fns ts researc successfull. Dr. Moammad. Ass'ad wo supported and elped me to be one of researcer students and granted me s trust. Fnall m all tanks and wses for m moter fater and usband especall for ter elp and encouragement and to m frends for all knds of support and concern.

5 V Sngular Two Ponts Boundar Value Problem Declaraton Te work provded n ts tess unless oterwse referenced s te researcer's own work and as not been submtted elsewere for an oter degree or qualfcaton. Student's name: Sgnature: Date:

6 Declaraton Lst Of Tables Lst of Fgures Abstract Capter : Introducton VI Table Of COTETS V VIII IX X. Introducton. Dfferental Equaton 3.3 Ordnar Dfferental equatons 4.4 Intal Value Problems 5.5 Boundar Value Problems BVPs 5.6 Sngular BVPs 7.7 Prevous Works 8 Capter : Some umercal Metods for Solvng Boundar Value Problems. Introducton. General Forms For Te Dfferental Equatons 3.3 General Forms For Te Boundar Condtons 5.4 Tpe of Boundar Condtons 7.5 Lnear Second-Order BVPs 9.6 Sootng Metod 9.6. Sootng For Lnear Problems.6. Sootng For on-lnear Problems 3.7 Fnte Dfference Metods 44

7 VII.7. Smple One-Step Scemes For Lnear Sstems Fnte Dfference Metod For Lnear Problems eumann Boundar Condtons Fnte Dfference Metod For onlnear Problems 59 Capter 3: Sngular Two-Ponts BVP 3. Introducton Regular Sngular Pont Sngulartes of Te Frst Knd Irregular Sngular Pont Oter Sngular Problem Fnte Dfference Pade Based Metod A umercal Metod Based on te 0 Pade Appromant A umercal Metod Based on te 30 Pade Appromant 83 Append 99 References 36 ب الملخص

8 VIII Lst of Tables Table page Table. : Te appromate and eact soluton for eample.. 6 Table. : Te appromate and eact soluton for eample.. 9 Table.3 : Te appromate and eact soluton for eample Table.4 : Te appromate and eact soluton for eample.4. 4 Table.5 : Te appromate and eact soluton for eample Table.6 : Te appromate and eact soluton for eample Table.7 : Te appromate and eact soluton for eample Table.8 : Te appromate and eact soluton for eample Table 3. : Te appromate and eact soluton for eample Table 3. : Te appromate and eact soluton for eample Table 3.3 : Te appromate and eact soluton for eample Table 3.4 : Te appromate and eact soluton for eample Table 3.5 : Te appromate and eact soluton for eample

9 IX Lst of Fgures Fgures Fgure : sows te appromate and te eact soluton for eample. tat was solved b sootng metod. Fgure : sows te appromate and te eact soluton for eample. tat was solved b sootng metod. Page 7 30 Fgure 3 : sows te appromate and te eact soluton for eample.3 tat was solved b sootng metod. 40 Fgure 4: sows te appromate and te eact soluton for eample.4 tat was solved b sootng metod. Fgure 5: sows te appromate and te eact soluton for eample.5 tat was solved b fnte dfference metod. Fgure 6 : sows te appromate and te eact soluton for eample.6 tat was solved b fnte dfference metod. Fgure 7: sows te appromate and te eact soluton for eample.7 tat was solved b fnte dfference metod. Fgure 8 : sows te appromate and te eact soluton for eample.8 tat was solved b fnte dfference metod. Fgure9: sows te appromate and te eact soluton for eample 3. tat was solved b fnte dfference metod. Fgure0: sows te appromate and te eact soluton for eample 3. tat was solved b sootng metod Fgure : Emden problem BVP wt sngular term. 96

10 X Sngular Two Ponts Boundar Value Problem B Sawsan Moammad Hamdan Supervsed b Dr. Samr Matar Abstract A sngular Two ponts boundar value problem occur frequentl n matematcal modelng of man practcal problems. To solve sngular two ponts boundar value problem for certan ordnar dfferental equatons avng sngular coeffcents. Man numercal metod suc as sootng metod fnte dfference metod and pade appromaton metods ave been studed and analsed.

11 Capter one

12 Introducton. Introducton Matematcs s te bod of knowledge centered on suc concepts as quantt structure space and cange and also te academc dscplne tat studes tem. Benjamn Perce called t " te scence tat draws necessar conclusons". Oter practtoners of matematcs mantan tat matematcs s te scence of pattern and tat matematcans seek out patterns weter found n numbers space scence computers magnar abstractons or elsewere. Matematcans eplore suc concepts amng to formulate new conjectures and establs ter fndngs b rgorous deducton from appropratel cosen aoms and defntons. Toug te use of abstracton and logcal reasonng matematcs evolved from countng calculaton measurement and te sstematc stud of te sapes and motons of pscal objects. Knowledge and use of basc matematcs ave alwas been an nerent and ntegral part of ndvdual and group lfe. Refnements of te basc deas are vsble n matematcal tets orgnatng n te ancent Egptan Mesopotaman Indan Cnese Greek and Islamc worlds. Rgorous arguments frst appeared n Greek matematcs most notabl n Eucld's elements. Te development contnued n ftful bursts untl te renassance perod of te 6 t centur wen matematcal nnovatons nteracted wt new scentfc dscoveres leadng to an acceleraton n researc tat contnues to te present da.

13 3 Toda matematcs s used trougout te world n man felds ncludng natural scence engneerng medcne and te socal scences suc as economcs. Appled matematcs te applcaton of matematcs to suc felds nspres and makes use of new matematcal dscoveres and sometmes leads to te development of entrel new dscplnes. See [3]. Dfferental Equaton A dfferental equaton s a matematcal equaton for an unknown functon of one or several varables tat relates te values of te functon tself and of ts dervatves of varous orders. Dfferental equatons pla a promnent role n engneerng pscs economcs and oter dscplnes. Dfferental equatons arse n man areas of scence and tecnolog wenever a determnstc relatonsp nvolvng some contnuousl cangng quanttes modeled b functons and ter rates of cange epressed as dervatves s known or postulated. Ts s well llustrated b classcal mecancs were te moton of a bod s descrbed b ts poston and veloct as te tme vares. ewton's laws allow one to relate te poston veloct acceleraton and varous forces actng on te bod and state ts relaton as a dfferental equaton for te unknown poston of te bod as a functon of tme. In man cases ts dfferental equaton ma be solved eplctl eldng te law of moton. See [3]&[3]

14 .3 Ordnar Dfferental Equatons 4 In matematcs an ordnar dfferental equaton or ODE s a relaton tat contans functons of onl one ndependent varable and one or more of te functon's dervatves wt respect to tat ndependent varable. A smple eample s ewton's second law of moton wc leads to te dfferental equaton t d m F t. dt For te moton of a partcle of mass m. In general te force F depends upon te poston of te partcle t at tme t and tus te unknown functon t and ts dervatves appears on bot sdes of te dfferental equaton. Ordnar dfferental equatons are to be dstngused from partal dfferental equatons were tere are several ndependent varables nvolvng partal dervatves. Ordnar dfferental equatons arse n man dfferent contets ncludng geometr mecancs astronom and populaton modelng. Man famous matematcans ave studed dfferental equatons and contrbuted to te feld ncludng ewton te Bernoull faml Reccat Claraut and Euler. Man studes as been devoted to te soluton of ordnar dfferental equatons. In te case were te equaton s lnear t can be solved b analtcal metods but te most of te nterestng dfferental equatons are non-lnear and can t be solved eactl. umercal metods tat appromate solutons can be establsed b usng computer. See[3]&[3]

15 5.4 Intal Value Problems In matematcs n te feld of dfferental equatons an ntal value problem IVP s an ordnar dfferental equaton togeter wt specfed values called te ntal condtons of te unknown functon at a gven pont n te doman of te soluton. In pscs or oter scences modelng a sstem frequentl amounts to solvng an ntal value problem te dfferental equaton s an evoluton equaton specfng ow gven ntal condtons. A smple form of ntal value problem IVP s a dfferental equaton ' t = f t t. wt ntal condton t0 0. A soluton to an ntal value problem s a functon tat s a soluton to te t0. dfferental equaton and satsfes te ntal condton 0.5 Boundar Value Problems A boundar value problems BVP s a dfferental equaton togeter wt a set of addtonal restrctons on te boundares called te boundar condtons. A soluton to te boundar value problem s a soluton to te dfferental equaton wc also satsfes te boundar condtons. Boundar value problems arse n several brances of scence. For eample n pscal dfferental equaton for some problems nvolvng te wave equaton suc as te determnaton of normal modes are often stated as boundar value problems.

16 6 To be useful n applcatons a boundar value problem sould be well-posed ts means tat gven te nput to te problem tere ests a unque soluton wc depends contnuousl on te nput. Muc teoretcal work n te feld of partal dfferental equaton s devoted to provng tat boundar value problems arsng from scentfc and engneerng applcatons are n fact well-posed. For a boundar value problem nformaton about a soluton to te dfferental equatons ma be generall specfed at more tan one pont. Often tere are two ponts wc correspond pscall to te boundares of some regon so tat t s a two-ponts boundar value problem. A smple and common form for a two-ponts boundar value problem nvolve a secondorder dfferental equaton s: " = f ' a b.3 togeter wt te boundar condtons a and b were α and β are known constants and te known endponts a and b ma be fnte or nfnte. See []&[3] A more matematcal wa to pcture te dfference between an ntal value problem and a two-ponts boundar value problem s tat IVP as all of te condtons specfed at te same value of te ndependent varable n te equaton and tat value s at te lower boundar of te doman tus te term "ntal value". On te oter and a two-ponts boundar value problem as condtons specfed at te etremes of te ndependent varable.

17 7 For eample f te ndependent varable s tme t over te doman [0] an ntal value problem would specf a value of t and / or ' t at tme t = 0 wle a two-ponts boundar value problem specf values for t or ' t at bot pont s t = 0 and t =. See [3].6 Sngular BVPs Man problems n vared felds as termodnamcs electrostatcs pscs and statstcs gve rse to ordnar dfferental equatons of te form - p + q = w f On some nterval of te real lne wt some boundar condtons. Ver often sngulartes are encountered at one or more ponts n tat nterval. Sngular two-ponts boundar value problem occur frequentl n matematcal modelng of man practcal problems. Sngular pont of a dfferental equaton a pont at wc te coeffcents are not epandable n a Talor seres. We menton ere tree eamples to llustrate te pont. Te equaton [ sn ] 0 sn [0 ] Appears wen separaton of varables s attempted on te eat equaton n a sold spere or te electrostatc potental n te spere. Te source of te sngulart ere s te vansng of te functon p at te endponts.

18 8 Te equaton u f [ ] Represents te stead state temperature dstrbuton n a bar etendng from - to f te termal conductvt s.te same tpe of sngulart occurs ere also. See [5] 3 An eample of a class of sngular BVP s s: '' f.4 0 < 0 = A = B In wc 0 < α and A B are fnte constants. We assume also tat f for 0 < < te real-valued functon f s contnuous ests f and s contnuous and tat 0. See [0] Te obvous dffcult of te equaton above s te beavor of te term near = 0..7 Prevous Works Man prevous works ave been done on studed numercal metods for solvng sngular BVPs Gustafsson used some numercal metods tat treated onl scalar problems not sstems and does not deal at all wt estence or unqueness of solutons. atterer as treated sstems usng a projecton metod and as get O [ln ] r accurac. He also as dealt

19 wt estence and unqueness of solutons but as used unnatural lookng 9 boundar condtons and as not state wen te problem wll ave a soluton onl wen te operator s Fredolm wt nde zero not wen te operator's nverse ests. Jamet also as treated onl scalar equatons and as used tree-pont fnte dfference scemes wc for a model problem wt O accurate solutons 0 s a parameter of te problem. Sampne as dealt wt a class of nonlnear second order scalar equatons all wt te same lnear dfferental operator. He as proved estence and unqueness of solutons of ts equaton for certan boundar value problems and te convergence of collocaton and fnte dfference metods. See [] [0] Twzell 988 as developed numercal metods for ts class of BVPs.4. Twzell's metods gave more accurate numercal results tan tose prevousl avalable suc as tose of Cawla and Katt 98. Te are also more economcal and easer to mplement. See [0] In ts tess we ave eplored some numercal metods for solvng sngular two-ponts boundar value problem and we ave wrtten some codes n matlab. Ts tess contans tree capters. Capter contans te general forms for te dfferental equaton and te tpe of boundar condtons ten we dscuss a numercal metods to solve BVP Sootng metod and Fnte Dfference metod for lnear and nonlnear BVP.

20 0 Capter 3 s devoted to sngular two-ponts BVP. We dscuss regular sngular pont sngulartes of te frst knd rregular sngular pont nfnte nterval problem and oter sngular problem. Ten some numercal metods were used to solve sngular two-ponts BVP. In ts work some numercal metods for solvng tese problems ave been studed and analsed. MATLAB s used as a computatonal tool durng te development of ts tess.

21 00 Capter Two

22 0 Some umercal metods for Solvng Boundar Value Problems. Introducton A sstem of ordnar dfferental equatons ma ave man solutons. Commonl a soluton of nterest s determned b specfng te values of all ts components at a sngle pont = a. Ts pont and a drecton of ntegraton defne an ntal value problem IVP. In man applcatons te soluton of nterest s determned n a more complcated wa. A boundar value problem BVP specfes values or equatons for soluton components at more tan one pont n te range of te ndependent varable. Generall IVP as a unque soluton but ts s not true for BVPs. Lke a sstem of lnear algebrac equatons a BVP ma not ave a soluton at all or ma ave a unque soluton or ma ave more tan one soluton. Because tere mgt be more tan one soluton BVP solvers requre an estmate guess for te soluton of nterest. Often tere are parameters tat must be determned n order for te BVP to ave a soluton. Assocated wt a soluton tere mgt be just one set of parameters a fnte number of possble sets or an nfnte number of possble sets. See [9]

23 . General Forms for te Dfferental Equatons For a second order non-lnear BVPs we ave te general form " = f ' a b and te partcular form tat can be derved from te general one 03 " = f a b Tese dfferental equatons vald n some nterval [ a b ] togeter wt boundar condtons mposed on te dependent varable and / or ts frst dervatve at te two ponts = a and = b gve rse to te second order general and specal boundar value problems respectvel. For a lnear boundar value problem wc as te form " = p ' + q + r a b wt boundar condtons a = α b = β were p q and r contnuous functons on te nterval [a b]. Usuall one assumes tat a general ordnar dfferental equaton can be wrtten as a frst-order sstem ' = f a < < b. were R n T... s te unknown vector functon and n T f f f fn s te generall nonlnear rgt-and sde. Te nterval ends a and b are fnte or nfnte constants. For a lnear problem te ODE smplfes to ' =A + q a < < b.

24 04 were te matr A and te vector q are functons of A R n n and q R n. Te lnear sstem. s called omogeneous f q = 0 and t s non-omogeneous oterwse. Hg-order ODEs can normall be converted to te frst-order form. Gven an scalar dfferental equaton u n = f u u '...u n- a < < b.3 let =.. n T be defned b = u X = u'.4.. n =u n- Ten te ODE can be converted to te equvalent frst-order Form ' = ' = 3.. n- ' = n n ' =f n Ts s n te form..

25 05.3 General Forms for te Boundar Condtons A frst-order sstem of ODEs lke. as normall n boundar condtons BCs g a b = 0.5 were g = g... g n T s a generall nonlnear vector functon and 0 s a vector of n zeros. Te smplest nstance of g s te case for an IVP. Ten te soluton s gven at te ntal pont; tat s a = α.6 were α = α... α n T R n s a known vector of ntal condtons wc unquel determnes near a. Te general form of lnear two-pont BC for a frst- order sstem or for a ger-order ODE s Here B a and B b R n n and β R n. B a a + B b b = β.7 we see tat for te lnear BVP. and.7 to ave a unque soluton t s necessar but not suffcent tat tese BCs be lnearl ndependent; tat s te matr B a B b ave n lnearl ndependent columns or smpl rank B a B b = n. BC of te general form.7 are called non- separated BC snce eac nvolve nformaton about at bot endponts. However t frequentl

26 06 appens tat rank B a < n or rank B b < n or bot. If eter olds we call te boundar condton partall separated. In te case rank B b = q < n te BVP can be transformed to one were te BC ave te form B a a = β B a a + B b b = β.8 were B al R p X n p := n - q B a and B b R q X n β R p and β R q. Te BC are called separated f te smplf furter to B a a = β B b b = β.9 Te nonlnear BC.5 can also occur n partall separated or separated form. Tus te boundar condtons are separated f te are of te form g a = 0 g b = 0.0 were g 0 R p and g 0 R q wt n = p + q. In fact a sgnfcant porton of te currentl avalable software for BVPs assumes tat te BC are separated. See []

27 07.4 Tpes of Boundar Condtons For lnear boundar value problems tere are tree tpes of condtons:. Functonal boundar condtons.e. a = A and b = B are gven.. Dervatve boundar condtons.e. ' a = α and ' b = β are gven. 3. Med boundar condton.e. condtons n te form p 0 a + q 0 ' a = r 0 p b + q ' b = r All tree tpes of lnear boundar condtons ma be epressed n vector matr form as : q0 p0 ' a 0 0 ' b r 0 0 a q p b r 0 so tat q 0 gves tpe p 0 q0 p0 gves tpe and tpe 3 occurs wen all four constants are non-zero. Tpe 3 boundar condtons can be wrtten n te vector form.7. Teorem. Suppose te functon f n te boundar-value problem wc as te form " = f ' a b were a = α b = β s contnuous on te set D = { ' a b - < < - < ' < } and tat te partal dervatves f and f ' are also contnuous on D. If

28 08 f ' > 0 For all ' D and a constant M ests wt f ' ' M for all ' D Ten te boundar-value problem as a unque soluton. See[] * ote tat teorem. gves te condtons under wc te general BVP wt tpe boundar condton as unque soluton estence and unqueness. Wen f ' as te form f ' = p ' + q + r te dfferental equaton.3 s sad to be lnear. Teorem. can be replaced b te followng teorem: Teorem. If te lnear boundar value problem: " = p ' + q + r. a b a = α b = β satsfes: p q and r are contnuous on [a b] q > 0 on [a b] ten. as a unque soluton. See []

29 .5 Lnear Second-Order BVP s 09 Consder te lnear second order BVP.. let o = = ' ten. can be wrtten as te sstem of frst order dfferental equatons: 0' ' q p r Wc can be wrtten n vector-matr form as: D = Q + P. Wt boundar condtons 0 0 = A = B. See [0] umercal Metods To Solve BVP..6 Sootng Metod: Te smplest ntal value metod for BVPs s te sngle sootng metod t's one of te more successful numercal tecnques for solvng te general BVP wt tpe boundar condtons based on te dea of reformulatng te problem as a sequence of IVPs of te form. 3 wt a = A ' a z = 0..3 To do ts all condtons must be specfed at one pont. Suppose we coose to mpose some ntal condton at t = a were tere are some boundar condtons are alread known. We guess te remanng boundar condtons at ts pont and for te moment gnore te known boundar condtons at t = b. We now ave an IVP wc can be solved usng Range Kutta or an oter approprate

30 metod to obtan a numercal soluton at t = b. Tese numercal values are ten compared wt te known boundar condton at t = b. If te guessed ntal condtons are correct tere wll be no dscrepances wt te known boundar condtons at t = b and te soluton to te IVP wll be te soluton to BVP. If not we need to modf te guessed ntal condtons at t = a. Ts s called te sootng metod for obvous reasons. See [3] It s probabl clear to te reader tat 'sootng metods' are so-called because of te analog of frng mssles at a statonar target. Startng wt te parameter z 0 wc determnes te ntal elevaton at wc te mssle s dscarged from te pont = a A. Te trajector of te mssle s computed b solvng te ntal value problem gven b.3 and.3 wt > 0. If te pont of landng = b b z s not suffcentl close 0 to b B te appromaton s corrected b coosng anoter elevaton z and so on untl b z k s acceptabl close to te 'target' b = B. See [] Defnton : A functon f t s sad to satsf a Lpsctz condton n te varable on a set Wenever for f. D R f a constant L > 0 ests wt f t L t t D f t. Te constant L s called a Lpsctz constant

31 0.6. Sootng For Lnear Problems Consder te ntal-value problems p q r a b.4 a.5 a 0.6 And p q a b.7 a 0.8 a.9 If p q r contnuous and q > 0 on [ab] ten te Lpsctz condton ests and.4 to.9 ave unque solutons. soluton of.4 to.6 and Take.9 and take soluton of.7 to b b b 0.0 Were s te appromated soluton for.4 to.6 at = b b and s te appromated soluton for.7 to.9 at = b. b Can be cecked to be unque soluton of BVP. and b b.

32 b b. So. r q p r b b q b b p q p b b r q p Te sootng metod for lnear equatons s based on te replacement of te lnear boundar-value problem b two ntal-value problems. See [3]&[].

33 3 Algortm. Lnear sootng To appromate te soluton of te boundar-value problem " p ' q r 0. a b. a b : IPUT :endponts a b; boundar condtons ; number of subntervals. OUTPUT : appromatons w to ; to ' for eac = w 0... Step Set = b-a / : u u v v ; 0; 0;. Step For = 0..- do steps 3 and 4. Te Runge-Kutta metod for sstems s used n steps 3 and 4. Step 3 Set = a +. Step 4 Set k k k k k 3 u ; [ p u [ u q u ]; q / u [ u k k [ p / u ]; r ]; k k r / ];

34 4 ]. [ ]; [ ]; [ ]; [ ]; / / [ ]; [ ]; / / [ ]; [ ]; [ ; ]; [ ]; [ ]; [ ]; [ ]; / / / [ k k k k v v k k k k v v k v q k v p k k v k k v q k v p k k v k k v q k v p k k v k v q v p k v k k k k k u u k k k k u u r k u q k u p k k u k r k u q k u p k Step 5 Set v u w w 0 0 ; ; OUTPUT 0 0 w w Step 6 For =. Set ; ; 0 0 v w u W v w u W = a +; OUTPUT WW Output s w w.

35 5 Step 7 STOP. Te process s complete.see [3] Eample. Te lnear boundar-value problem snln = = as te eact soluton c c 3 snln cosln 0 0 Were c [8 snln 4cosln] And c c Applng sootng metod to ts problem requres appromatng te solutons to te ntal-value problems snln 0 and 0

36 6 Algortm. uses te fourt-order Runge-Kutta tecnque to fnd te appromaton to and. Te results of te calculatons wt =0 and = 0. are gven n table.. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton.. See [3] Program Table. : Te appromate and eact soluton for eample.. w ee w Te mamum error s

37 Fgure : sows te appromate and te eact soluton for eample. tat was solved b sootng metod. 7

38 8 Eample. Te boundar-value problem = 0 = as te eact soluton e e 4 e e Applng te sootng metod to ts problem requres appromatng te solutons to te ntal-value problems Algortm. uses te fourt-order Runge-Kutta tecnque to fnd te appromaton to and wc can be found n page. Te results of te calculatons wt = 0 and =/0 are gven n table.. Te value lsted as w appromates and s te eact soluton 0 0 and ee s te error between te eact soluton and te appromate soluton.. See [3] program

39 Table. : Te appromate and eact soluton for eample.. 9 w ee Te mamum error s

40 Fgure : sows te appromate and te eact soluton for eample. tat was solved b sootng metod. 3

41 30.6. Sootng For on-lnear Problems Te sootng prncple etends to nonlnear problems. Consder te followng ver smple model of a cemcal reacton u u" e u 0 u 0.4 As an ntal value problem. Wt u 0 = 0 and u' 0 t te problem.3 as a unque soluton. For eac real t; denoted b u t. ow f we fnd te correct "angle of sootng " t* suc tat u ; t* = 0 ten te soluton of te IVP also solves te BVP.3.4. Fnd t = t* wc satsfes te equaton u ; t = 0 Ts latter problem can be solved numercall b an teratve sceme. ote tat eac functon evaluaton n ts teratve sceme nvolves te numercal soluton of an IVP. See [] Te sootng tecnque for te nonlnear second-order BVP.3 s smlar to te lnear tecnque ecept tat te soluton to a nonlnear problem cannot be epressed as a lnear combnaton to two ntal value problems. Instead te soluton to te boundar value problem s appromate b usng te soluton to a sequence of ntal value problem nvolvng a parameter t. Tese problems ave te form " = f ' a b a = α 'a = t.5

42 3 We do ts b coosng te parameters t = t k n manner to ensure tat lm b t b k k Were t k denotes te soluton to te ntal value problem.5 wt t = t k and denotes te soluton to te boundar value problem.3. Start wt a parameter t 0 tat determnes te ntal elevaton at wc te object s fred from te pont a α and along te curve descrbed b te soluton to te ntal value problem: " = f ' a b a = α ' a = t 0. If b t 0 s not suffcentl close to β we correct our appromaton b coosng elevatons t t and so on untl b t k s suffcentl close to β. To determne te parameters t k suppose a boundar value problem of te form.3 satsfes teorem... If t denotes te soluton to te ntal value problem.5 we net determne t wt b t β = 0.6 Ts s a nonlnear equaton tat can be solved b ewton's metod wc use to generate te sequence { t k } onl one ntal appromaton t 0 s needed. Te teraton as te form

43 33 k k k k t b dt d t b t t.7 and t requres te knowledge of d/ dt b t k-. Ts presents a dffcult snce an eplct representaton for b t s not known; we know onl te values b t 0 b t. b t k-. Suppose we rewrte te ntal value problem.5 empaszng tat te soluton depends on bot and te parameter t: " t = f t ' ta b a t = α ' a t = t.8 We ave retaned te prme notaton to ndcate dfferentaton wt respect to. Snce we need to determne d/ dt b t wen t = t k- we frst take te partal dervatve of.8 wt respect to t. Ts mples tat. ' ' ' ' ' ' " t t t t f t t t t f t t t f t t t f t dt d Snce and t are ndependent 0 / t and ' ' ' ' " t t t t f t t t t f t t.9 For a b. Ts ntal condtons gve ' 0 t a t and t a t

44 34 If we smplf te notaton b usng z t to denote / t t and assume tat te order of dfferentaton of and t can be reversed.9 wt te ntal condtons becomes te ntal value problem f f z" t ' z t ' z' t.30 ' a b z a t = 0 z a t = ewton's metod terefore requres tat two ntal value problems be solved for eac teraton.8 and.30. Ten from.7 t k t k b tk z b t k.3 Of course none of tese ntal value problems are solved eactl; te soluton are appromated. See [3] & [6] Algortm. onlnear Sootng wt ewton s Metod To appromate te soluton of te nonlnear boundar-value problem f. a b a b : IPUT : endponts a b; boundar condtons ; number of subntervals ; tolerance TOL; mamum number of teratons M.

45 35 OUTPUT : appromatons w to ; w to ' for eac = 0 or a massage tat te mamum number of teratons was eceeded. Step Set = b-a / ; K = ; TK = - / b - a. note: TK could also be nput. Step Wle k M do steps 3-0. Step 3 Set w w u u 0 0 TK ; 0;. Step 4 For =. do steps 5 and 6. Te Runge-Kutta metod for sstems s used n steps 5and 6. Step 5 Set = a +-. Step 6 Set

46 36

47 37 Step 7 If w TOL ten do steps 8 and 9. Step 8 For = 0 set = a + ; OUTPUT w w. Step 9 Te procedure s complete. STOP. Step 0 Set w TK TK u ewton s metod s used to compute TK. k = k+. Step OUTPUT Mamum number of teratons eceeded ; Te procedure was unsuccessful. STOP. See[3]

48 38 Eample.3 Consder te boundar-value problem as te eact soluton 6 Applng sootng metod to ts problem requres appromatng te solutons to te ntal-value problems t k z z 8 z 3 z 0 z Algortm. uses te Runge-Kutta metod of order four to appromate bot solutons requred b ewton s metod. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton. Te results of te calculatons wt = 0 are gven n table.3 See [3] program 3

49 Table.3 : Te appromate and eact soluton for eample w ee w Te mamum error s

50 Fgure 3 : sows te appromate and te eact soluton for eample.3 tat was solved b sootng metod. 4

51 40 Eample.4 Consder te non-lnear boundar-value problem 3 as te eact soluton 3 Applng sootng metod to ts problem requres appromatng te solutons to te ntal-value problems 3 t k z z z z 0 z Algortm. uses te Runge-Kutta metod of order four to appromate bot solutons requred b ewton s metod page 35. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton. Te results of te calculatons wt = 0 are gven n table.4. See [3] program 3

52 Table.4 :Te appromate and eact soluton for eample.4. 4 w ee w Te mamum error s

53 Fgure 4: sows te appromate and te eact soluton for eample.4 tat was solved b sootng metod. 43

54 44.7 Fnte Dfference Metods In tese metods no ntal value problems are eplctl ntegrated. Rater an appromate soluton representaton s sougt over te entre nterval of nterest. Tus tese metods are sometmes referred to as global metods. Te basc steps of a fnte dfference metod are outlned as follows we coose a mes Ω to te nterval [ab] were a.. b ten appromate soluton values are ten sougt at tese mes ponts for = 3 n Form a set of algebrac equatons for te appromate soluton values b replacng dervatves wt dfference quotents n te dfferental equatons and boundar condtons tat te eact soluton satsfes. Fnall solve te resultng sstem of equatons for te appromate soluton ts gves a set of dscrete soluton values. Fnte dfference metods proceed b replacng te dervatves n te dfferental equatons b fnte dfference appromatons. Ts gves a large algebrac sstem of equatons to be solved n place of dfferental equaton. To appromate ' we can use one-sded appromaton D ' or D '

55 Or we can use centered appromaton : 45 D 0 ' Wc s te average of te two one-sded appromatons. It s clear tat D o gves a better appromaton tan eter of te one-sded appromatons also t gves us a second order accurate appromaton. We can use fnte dfference to solve a dfferental equaton consder te second order dfferental equaton " = f 0 < <.3 0 = α = β Te functon f s specfed and we ws to determne n te nterval 0 < < l. Ts problem s called two ponts boundar value problem. Snce boundar condtons are gven at te two dstnct ponts 0 and..7. Smple One-Step Scemes for Lnear frst-order Sstems Consder now te lnear frst-order sstem ' = A + f [a b] R n.33 B a a + B b b = β.34 and we seek numercal metods wc work equall well for non-unform meses. Ts naturall leads to one-step scemes scemes wc defne te dfference operator based onl on values related to one subnterval

56 46 of te mes Ω at a tme. Te two smplest suc fnte dfference scemes are te mdpont and te trapezodal scemes. For a generall non-unform mes Ω a dscrete numercal soluton... T s sougt Were s to appromate component-wse te eact soluton at = 3.. Te numercal soluton n all metods based on one-step scemes s requred to satsf te boundar condton.34. For te nteror mes ponts two dfference scemes are presented. For eac subnterval =3..- of Ω te dervatve n.33 s replaced b. Ts appromaton s centered at / : wt : at te mddle of te subnterval. Ten A + f s appromated b a centered appromaton eldng second-order accurac. Te trapezodal sceme s defned b: A A f f.35 and te mdpont sceme s defned b: A f / /.36

57 47 Te latter sceme.36 s also known as te bo sceme. In vector-matr form bot of tese metods can be wrtten as F A.37 and n detal b a f f f B B R S R S R S Were R S are n n matrces. For te trapezodal sceme f f f A I A I R S.39 wle for te mdpont sceme f f A I A I / / / R S ;.40 See [].7. Fnte-Dfference Metods for Lnear Problems Consder more general lnear equaton r q p Togeter wt two boundar condtons "Drclet condtons "

58 48 a = α b = β Let a = 0 + and Ts equaton can be dscretzed to second order b: r q p I = Were for eample q q p p and r r ts gves te lnear sstem A Y = F were A s te tr-dagonal matr. / / / / / / q p p q p p q p p q A / /... / /... p r r r p r F Y Y Y Y Y and Ts lnear sstem can be solved wt standard tecnques assumng te matr s nonsngular. A sngular matr would be a sgn tat te dscrete sstem does not ave a unque soluton wc ma occur f te orgnal problem or nearb problem s not well posed.

59 49 Te dscretzaton used above wle second order accurate ma not be te best dscretzaton to use for certan problems of ts tpe. Often te pscal problem as certan propertes tat we would lke to preserve wt our dscretzaton and t s mportant to understand te underlng problem and be aware of ts matematcal propertes before blndl applng numercal metod. See [8].7.3 eumann Boundar Condtons Consder we ave one or more eumann boundar condtons nstead of Drclet boundar condtons meanng tat a boundar condton on te dervatve ' s gven rater tan a condton on te value of tself. We mgt ave eat flu at a specfed rate gvng ' = α at ts boundar. Consder te equaton.3 wt boundar condtons: ' 0 = α = β.4 to solve ts problem numercall we need to ntroduce one more unknown tan we prevousl ad: Y o at te pont o = 0 snce ts s now an unknown value. We also need to augment te sstem.37 wt one more equaton tat models te boundar condtons.4. As a frst tr we mgt use a one-sded epresson for '0 suc as: 0.4

60 5 If we append ts equaton to te sstem.37 we obtan te followng sstem of equaton for te unknowns Y o Y Y / f f f f Y Y Y Y Y.43 Solvng ts sstem of equatons does gve an appromaton to te true soluton. To obtan a second-order accurate metod we sould use a centered appromaton to '0 = α nstead of te one-sded appromaton.4. We can ntroduce anoter unknown Y - and nstead of te sngle equaton.4 use te followng two equatons: Y -- Y 0 +Y = f 0.44 Y -Y - = α Ts results n a sstem of + equaton. Introducng te unknown Y - outsde te nterval [0] were te orgnal problem s posed ma seem unsatsfactor. We can avod ts b elmnatng te unknown Y - from te two equaton.44 resultng n a sngle equaton tat can be wrtten as: -Y0 +Y = α + f 0.45

61 50 We ave now reduced te sstem to one wt onl + equaton for te unknowns Y 0 Y Y. Te matr s eactl te same as te matr n.43 wc came from te one-sded appromaton te onl dfference n te lnear sstem s tat te frst element n te rgt and sde of.43 s now canged from α to α + f 0 we can vew te left and sde of.45 as a centered appromaton to ' 0 + and te rgt and sde as te frst two terms n te Talor seres epanson of ts value ' 0 + = ' 0 + ''0 + = α + f 0 + Algortm.3 Lnear Fnte-Dfference To appromate te soluton of te boundar-value problem p q r a b a b : IPUT : endponts a b; boundar condtons ; nteger and elements of matr A and te rgt and sde. OUTPUT : appromatons w to for eac = 0. +.

62 5 Step Set. / ; / ; ; ; / p r d p b q a a a b Step For = - Set. ; / ; / ; ; r d p c p b q a a Step 3 Set. / ; / ; ; p r d p c q a b Step 4 Set a l ; steps 4-8 solve a trdagonal lnear sstem algortm.. / ; / l d z a b u

63 53 Step 5 For = - set l z c d z l b u c u a l / / Step 6 Set. / ; l z c d z u c a l Step 7 Set. ; ; 0 z w w w Step 8 For = - set. w u z w Step 9 For = 0. + set = a + ; OUTPUT w. Step 0 STOP. Te procedure s complete.

64 54 Eample.5 Algortm.3 wll be used to appromate te soluton to te boundarvalue problem snln = = wt = 0. wc was also appromated b te sootng metod n eample. gves te results lsted n table.5. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton. program 4 Table.5 : Te appromate and eact soluton for eample.5. w ee w Te mamum error s

65 Fgure 5: sows te appromate and te eact soluton for eample.5 tat was solved b fnte dfference metod. 55

66 56 Eample.6 Algortm.3 wll be used to appromate te soluton to te boundarvalue problem = 0 = wt = /0 wc was also appromated b te sootng metod n eample. gves te results lsted n table.6. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton. program 5

67 Table.6 : Te appromate and eact soluton for eample ee Te mamum error s

68 Fgure 6 : sows te appromate and te eact soluton for eample.6 tat was solved b fnte dfference metod. 58

69 .7.4 Fnte-Dfference Metod for onlnear Problems 59 For te general nonlnear boundar-value problem " = f ' a b f te functon f satsfes te followng condtons:. f and te partal dervatves f and f are all contnuous on D = { ' a b - < < - < ' < };. f ' δ on D for some δ > 0; 3. Constants k and L est wt k ma f ' and L f ' ma ' D ' D '. Ts ensures b Teorem. page 7 tat a unque soluton ests. Dscretzng te nterval a b nto + subntervals eac of wdt so tat + = b a a numercal metod determnes a vector Y = [. ] T were s an appromaton to. Te dervatves '' and ' wll be replaced b ter second-order central dfference appromants to te equaton '' = f ' for eac = ts gves f Were 0 = α and + = β. Te soluton s tus found b solvng te nonlnear sstem

70 6 + f α / = α f 3 / = f / = f β - / = β We use ewton s metod for nonlnear sstems to appromate te soluton to ts sstem. A sequence of terates { k k. k t } s generated tat ma converges to te soluton of te sstem.48 provded tat te ntal appromaton t s suffcentl close to te soluton. t and tat te Jacoban matr for te sstem s nonsngular. For sstem.48 te Jacoban matr J. s a trdagonal wt j-t entr ' ' j and j for f j and j for f j and j for f J j ewton s metod for nonlnear sstem requres tat at eac teraton te lnear sstem

71 60 J. v. v t = t f f f f Be solved for v v. v snce k = k- + v for eac =. See [3] & [0] Algortm.4 onlnear Fnte-Dfference To appromate te soluton to te nonlnear boundar-value problem :. b a b a f IPUT : endponts a b; boundar condtons ; nteger tolerance TOL; mamum number of teratons M. OUTPUT : appromatons w to for eac = 0 + or a message tat te mamum number of teratons was eceeded. Step Set. ; ; / 0 w w a b Step For = set. a b w

72 6 Step 3 Set k =. Step 4 Wle K M do steps 5-6. Step 5 Set. ; / ; ; / ; t w f w w d t w f b t w f a w t a Step 6 For = - set. ; / ; / ; ; / ; t w f w w w d t w f c t w f b t w f a w w t a Step 7 Set. ; / ; ; / ; t w f w w d t w f c t w f a w t b Step 8 Set a l ; steps 8- solve a trdagonal lnear sstem.. / ; / l d z a b u

73 63 Step 9 For = - set. / ; / ; l z c d z l b u c u a l Step 0 Set. / ; l z c d z u c a l Step Set. ; v w w z v Step For = - set. ; v w w u v z v Step 3 If TOL v Ten do steps 4 and 5. Step 4 For = 0 + set = a + ; OUTPUT w. Step 5 STOP. Te procedure was successful. Step 6 Set k = k +. Step 7 OUTPUT Mamum number of teratons eceeded ; Te procedure was unsuccessful. STOP.

74 64 Eample.7 Consder te boundar-value problem as te eact soluton 6 Applng fnte dfference metod to ts problem te solutons results n table.7. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton. program 6

75 Table.7 : Te appromate and eact soluton for eample.7. w 65 ee w Te mamum error s

76 Fgure 7: sows te appromate and te eact soluton for eample.7 tat was solved b fnte dfference metod. 66

77 67 Eample.8 Consder te boundar-value problem as te eact soluton Applng fnte dfference metod to ts problem te solutons results n table.8. Te value lsted as w appromates and s te eact soluton and ee s te error between te eact soluton and te appromate soluton. program 7 Table.8 : Te appromate and eact soluton for eample.8. w ee w Te mamum error s

78 Fgure 8 : sows te appromate and te eact soluton for eample.8 tat was solved b fnte dfference metod. 68

79 69 Capter Tree

80 7 Sngular Two-Ponts BVP 3. Introducton Sngular two-ponts boundar value problem occur frequentl n matematcal modelng of man practcal problems. We consder frst a sstem of lnear ordnar dfferental equatons on a fnte nterval wt a sngulart of te frst knd at one endpont. We treat te same problem wt sngulartes at bot endponts and wt a sngulart on te nteror of te nterval. Consder a class of sngular BVPs: '' f 0 0 A B 3. In wc 0 and A B are fnte constants we assume also tat for df 0 < < te real-valued functon f s contnuous ests and s d df contnuous and tat d > 0. See [] 3. Regular Sngular Pont Sngulartes of Te Frst Knd Consder te ODE ' - f = 0 0 < < 3. If we assume ere tat α = n 3.. Te assumptons on te regulart of a soluton of 3. mpl tat lm ests as decreases to 0.

81 70 Ts s needed n order to make te BVP for 3. meanngful and s reasonable n most applcatons. Tese assumptons furter eld tat f 0 0 = wc must be compatble wt te prescrbed BC. In fact te requrement 3.3 s often used to determne part of te BC. To be more specfc let us consder now te lnear BVP wc as a sngular pont of te frst knd ' = A + q 0 < < 3.4 ~ were A R A 3.5 ere q are n component vec n n matrces. R s a constant matr and q C 0]. For an soluton of 3.4 we requre C¹0] we also mpose a lnear sstem of two-ponts boundar condtons wrtten as 0 B B lm note tat we cannot merel wrte B0 0 B 3.7 because s not even necessarl defned at = 0. otce also 3.7 mples tat lm B 0 0 s bounded.

82 Let te fundamental soluton matr Y for te omogeneous equaton for 3.4. Tat s Y satsfes 7 Y' = A Y 0] Y 0 = I 0 0] 3.8 Ten ever soluton to 3.4 can be wrtten cy P 0] 3.9 were s an partcular soluton of 3.4 and were c s a constant vector. were te partcular soluton p satsfes p Were 0 < δ < See [] & [7] Te smootness of f n 3. [or of A X and q n 3.4] does not mpl correspondng smootness of near = 0. For eample te IVP p ' 0 0 as te soluton wc as an unbounded frst dervatve at = 0. However were often te soluton s noneteless smoot near te sngulart. Te performance of numercal metods for problems wt sngulartes of te frst knd were te soluton s smoot at te sngulartes.

83 73 Te stuaton s muc less stragtforward for some of te ntal value. Ts s because not all fundamental soluton components of 3.4 ma be epected to be as smoot near te sngulart as te soluton s. For eample te IVP ' 0 = as te soluton = and a fundamental soluton.terefore a specal treatment near = 0 s often requred before a code based on an ntal value approac can be used. Suc a specal treatment ma consst of power epanson of a fundamental soluton n te vcnt of = 0 followed b use of an ntal value code wen we are suffcentl far awa from te sngulart. Once a fundamental soluton 0 as been found n ts wa an approprate partcular soluton can be found as well and te boundar condton ~ B B 3. can be constructed to replace 3.6. Te locaton of te jont 0 as to be small enoug so tat te power seres epanson for Y on [0δ] can be easl and effcentl constructed and at te same tme large enoug so tat

84 74 te BVP on [] can be solved b a standard ntal value metod wtout dffcult. See [] 3.3 Irregular Sngular Pont Tere s at present no teoretcal work justfng numercal metods for solvng problems wt rregular sngular ponts. Te man practcal occurrence of suc problems seems to be tose formulated on nfnte ntervals and we eamne some smple eamples ere. See [7] Suppose tat we ave te ODE ' f A a 0 3. Ten a transformaton t = a 3.3 reformulates 3. as an ODE defned on te nterval 0] namel t d dt a af 3.4 t n wc we recognze an ODE wt a sngulart of te second knd. In 3.3 we ave assumed tat a > 0. If a 0 ten te transformaton t a and reformulates 3. as an ODE defned on te nterval 0] namel. t d dt f a t

85 75 Snce te formulaton 3. s more natural and snce t turns out to be usuall preferable for numercal dscretzaton as well. Of course wen t comes to numercal dscretzaton te nfnte nterval [a] as to be replaced b a fnte one sa [a b ] were b s "large". See [] 3.4 Oter Sngular Problem We now consder tree cases of sngulartes. Te frst of tree s te case of an equaton wt a sngulart at bot ends of te nterval [0]: ~ R 0 R A b Were R0 and R are constant n n matrces and b C0. We use te boundar condtons lm B0 lm B Substtutng te form of 3.9 & 3.0 nto ts boundar condton we ave lm [ B0cY B0 p ] lm [ BcY B 0 p ] 3.7 R ~ ~ For A 0 ten A ma ave sngulartes wc are weaker tan 0 R.

86 76 Te second case s te case of a sngulart n te nteror of te nterval. Te equaton s te same as our orgnal equaton 3.4 but on te nterval [ 0 0]. ~ R A b [ 0 0] 3.8 We use a sstem of boundar condtons at - and : B B β 3.9 B a soluton to we mean an of te functons cy p [ 0 0] 3.0 Wc satsfes 3.9. In satsfng 3.9 we must ave [ B Y BY ] c B p B p 3. Snce Y-Y and est wt no sngulartes. Ten ere p p te sngulart nde s zero so tat f a soluton s requred for ever β B Y BY must be nonsngular. If B Y B s sngular ten B B must le n ts range. p p Y Te trd and fnal case s smpl treatng te case of a regular dfferental equaton on an nfnte nterval. We wll llustrate ts case for a sem-nfnte nterval treatng te problem

87 77 a A b [0 b lm B B If we make te cange of varable t or 3.3 t We map [ 0 nto t [0. Lettng ˆ t Aˆ t t A bˆ t t b t te problem s ten transformed nto ˆ t ˆ t ˆ a t A tˆ t b t t [0 b lm B ˆ t B ˆ 0 t Ten a necessar and suffcent condton for 3.4 to ave at most a sngulart of te frst knd at t = 0 s and A = 0. Ts statement mples tat f A R o as wll ave eactl a sngulart of te frst knd f R s not te zero matr. See []

88 Fnte Dfference Pade Based Metods We now descrbe scemes of fnte dfference metods based on pade ratonal appromaton. Tese metods are based on ratonal appromants to te eponental functon. Pade appromants are defned as follows: Let f z z C be functon n a regon of te comple plane contanng te orgn z = 0. A pade appromant. R z to te functon f z s defned as b: f z P z were P z Q z and Q z are polnomals of degrees κ and μ respectvel. For te functon z f z e te polnomals z P and z Q are gven eplctl as: P z j0! j! j!! z j! j And Q z j0! j! j!! z j! j

89 P z If e z T z ten te remander T z Q z s gven b: T z zu u u z! e du Q z 0 79 Te Pade appromants for z f z e for =34. and κ =34 Can be generated from te above equatons. Eample : Wen κ = 0 & μ = we wll ave e z z z See Append for more functon appromatons.

90 3.5. A umercal Metod Based on Te 0 Pade Appromant 8 Consder te lnear second order BVP.. let o = = ' ten. can be wrtten as te sstem of frst order dfferental equatons: 0' ' q p r Wc can be wrtten n vector-matr form as: D = Q + P Wt boundar condtons 0 0 = A = B. See [0] To solve ts class of sngular BVP ' ' f 0 0 A In wc 0 < α and A B are fnte constants. B 3.6 '' ' f Ts problem can be wrtten n vector-matr form as D Q p 3.7 Wt specal case tat s p 0.

91 80 Te boundar condtons become 0 0 A B. Usng te D relaton e and replacng te eponental term b ts 0 Pade appromant we get [I D D ] I D DD o 3 Usng 3.7 and ts second dervatve and applng te resultng equaton to te dscrete pont of Ω were Ω s te grd a b obtaned b dscretzng te nterval [a b] nto + subntervals eac of wdt b a z * leads to te fnte-dfference formula: A B k k k k k 0 Were B k I I s te dentt matr and te T k [k 0k ] elements of te matr A K are A K a a k k a a k k Suc tat