Research Article Numerov s Method for a Class of Nonlinear Multipoint Boundary Value Problems

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1 Hndaw Publsng Corporaton Matematcal Problems n Engneerng Volume 2012, Artcle ID , 29 pages do: /2012/ Researc Artcle Numerov s Metod for a Class of Nonlnear Multpont Boundary Value Problems Yuan-Mng Wang, 1, 2 Wen-Ja Wu, 1 and Massmo Scala 3 1 Department of Matematcs, East Cna Normal Unversty, Sanga , Cna 2 Scentfc Computng Key Laboratory of Sanga Unverstes, Dvson of Computatonal Scence, E-Insttute of Sanga Unverstes, Sanga Normal Unversty, Sanga , Cna 3 Department of Matematcs, Unversty of Rome La Sapenza, Pazzale le Aldo Moro 2, Rome, Italy Correspondence sould be addressed to Yuan-Mng Wang, ymwang@mat.ecnu.edu.cn Receved 28 Marc 2011; Revsed 16 May 2011; Accepted 13 June 2011 Academc Edtor: Sengyong Cen Copyrgt q 2012 Yuan-Mng Wang et al. Ts s an open access artcle dstrbuted under te Creatve Commons Attrbuton Lcense, wc permts unrestrcted use, dstrbuton, and reproducton n any medum, provded te orgnal work s properly cted. Te purpose of ts paper s to gve a numercal treatment for a class of nonlnear multpont boundary value problems. Te multpont boundary condton under consderaton ncludes varous commonly dscussed boundary condtons, suc as te tree- or four-pont boundary condton. Te problems are dscretzed by te fourt-order Numerov s metod. Te exstence and unqueness of te numercal soluton are nvestgated by te metod of upper and lower solutons. Te convergence and te fourt-order accuracy of te metod are proved. An accelerated monotone teratve algortm wt te quadratc rate of convergence s developed for solvng te resultng nonlnear dscrete problems. Some applcatons and numercal results are gven to demonstrate te g effcency of te approac. 1. Introducton Multpont boundary value problems arse n varous felds of appled scence. An often dscussed problem s te followng nonlnear second-order multpont boundary value problem: u x f x, u x, 0 <x<1, p p u 0 α u ξ, u 1 β u η,

2 2 Matematcal Problems n Engneerng were f x, u s a contnuous functon of ts arguments and for eac, α, β 0, and ξ, η 0, 1. An applcaton of ts problem appears n te desgn of a large-sze brdge wt multpont supports, were u x denotes te dsplacement of te brdge from te unloaded poston e.g., see 1. For oter applcatons of problem 1.1,wesee 2 4 and te references teren. It s allowed n 1.1 tat α 0orβ 0 for some or all. Ts mples tat te boundary condton n 1.1 ncludes varous commonly dscussed multpont boundary condtons. In partcular, te boundary condton n 1.1 s reduced to u 0 0, u 1 p β u η, 1 1.1a f α 0 for all see 5 14, to te form u 0 p α u ξ, u 1 0, 1.1 b 1 f β 0 for all see 15, to te four-pont boundary condton u 0 αu ξ, u 1 βu η, 1.1 c f p 1andξ 1 ξ, η 1 η see 11, 15 17, and to te two-pont boundary condton u 0 u 1 0, 1.1 d f α 0andβ 0 for all. Condton 1.1 c ncludes te tree-pont boundary condton wen ξ η see 16, 17. Te study of multpont boundary value problems for lnear second-order ordnary dfferental equatons was ntated n 18, 19 by Il n and Moseev. In 20, Gupta studed a tree-pont boundary value problem for nonlnear second-order ordnary dfferental equatons. Snce ten, more general nonlnear second-order multpont boundary value problems n te form 1.1 ave been studed. Most of te dscussons were concerned wt te exstence and multplcty of solutons by usng dfferent metods. Applyng te fxed pont ndex teorem n cones, te works n 5 14 sowed te exstence of one or more solutons to te problem a, wle te works n were devoted to te exstence of solutons for te tree- or four-pont boundary value problem c. For te problem 1.1 wt te more general multpont boundary condtons, some exstence results were obtaned n 21, 22 by usng te fxed pont ndex teory or te topologcal degree teory. Based on te metod of upper and lower solutons, te autors of 17, 23 obtaned some suffcent condtons so tat 1.1 or ts some specal form as at least one soluton. Addtonal works tat deal wt te exstence problem of nonlnear second-order multpont boundary value problems can be found n On te oter and, tere are also some works tat are devoted to numercal metods for te solutons of multpont boundary value problems. Te work n 30 made use of te Cebysev seres for approxmatng solutons of nonlnear frst-order multpont

3 Matematcal Problems n Engneerng 3 boundary value problems, and te work n 31 sowed ow an adaptve fnte dfference tecnque can be developed to produce effcent approxmatons to te solutons of nonlnear multpont boundary value problems for frst-order systems of equatons. Anoter metod for computng te solutons of nonlnear frst-order multpont boundary value problems was descrbed n 32, were a multple sootng tecnque was developed. Some oter works for te computatonal metods of frst-order multpont boundary value problems can be seen n In te autors gave several constructve metods for te solutons of multpont dscrete boundary value problems, ncludng te metod of adjonts, te nvarant embeddng metod, and te sootng-type metod. In te case of second-order multpont boundary value problems, tere are only a few computatonal algortms n te lterature. Te paper 39 set up a reproducng kernel Hlbert space metod for te soluton of a secondorder tree-pont boundary value problem. Based upon te sootng tecnque, a numercal metod was developed n 1 for approxmatng solutons and fold bfurcaton solutons of a class of second-order multpont boundary value problems. As we know, Numerov s metod s one of te well-known dfference metods to solve te second-order ordnary dfferental equaton u f x, u. Because Numerov s metod possesses te fourt-order accuracy and a compact property, t as attracted consderable attenton and as been extensvely appled n practcal computatons cf Altoug many teoretcal nvestgatons ave focused on Numerov s metod for two-pont boundary condtons suc as 1.1 d cf. 40, 41, 43, 44, 47 51, tere s relatvely lttle dscusson on te analyss of Numerov s metod appled to fully multpont boundary condtons n 1.1. Te study presented n ts paper s amed at fllng n suc a gap by consderng Numerov s metod for te numercal soluton of te multpont boundary value problem 1.1 wt te more general boundary condtons, ncludng te boundary condtons 1.1 a, 1.1 b,and 1.1 c.itsnotdffcult to gve a Numerov s dfference approxmaton to 1.1 n te same manner as tat for two-pont boundary value problems. However, a lack of explct nformaton about te boundary value of te soluton n te multpont boundary condtons prevents us from usng te standard analyss process of treatng two-pont boundary value problems, and so we ere develop a dfferent approac for te analyss of Numerov s dfference approxmaton to 1.1. Our specfc goals are 1 to establs te exstence and unqueness of te numercal soluton, 2 to sow te convergence of te numercal soluton to te analytc soluton wt te fourt-order accuracy, and 3 to develop an effcent computatonal algortm for solvng te resultng nonlnear dscrete problems. To aceve te above goals, we use te metod of upper and lower solutons and ts assocated monotone teratons. It sould be mentoned tat te proposed fourt-order Numerov s dscretzaton metodology may be stragtforwardly extended to te followng nonomogeneous multpont boundary condton: u 0 p α u ξ λ 1, 1 u 1 p β u η λ2, were λ 1 and λ 2 are two prescrbed constants. Te outlne of te paper s as follows. In Secton 2, we dscretze 1.1 nto a fnte dfference system by Numerov s tecnque. In Secton 3, we deal wt te exstence and

4 4 Matematcal Problems n Engneerng unqueness of te numercal soluton by usng te metod of upper and lower solutons. Te convergence of te numercal soluton and te fourt-order accuracy of te metod are proved n Secton 4. Secton 5 s devoted to an accelerated monotone teratve algortm for solvng te resultng nonlnear dscrete problem. Usng an upper soluton and a lower soluton as ntal teratons, te teratve algortm yelds two sequences tat converge monotoncally from above and below, respectvely, to a unque soluton of te resultng nonlnear dscrete problem. It s sown tat te rate of convergence for te sum of te two produced sequences s quadratc te error metrc s te sum of te nfnty norm of te error between te mt-teraton of te upper soluton and te true soluton wt te nfnty norm of te error between te mt-teraton of te lower soluton and te true soluton and under an addtonal requrement, te quadratc rate of convergence s attaned for one of tese two sequences. In Secton 6, we gve some applcatons to tree model problems and present some numercal results demonstratng te monotone and rapd convergence of te teratve sequences and te fourt-order accuracy of te metod. We also compare our metod wt te standard fnte dfference metod and sow ts advantages. Te fnal secton contans some concludng remarks. 2. Numerov s Metod Let 1/L be te mes sze, and let x 0 L be te mes ponts n 0, 1. Assume tat for all 1 p, te ponts ξ and η n te boundary condton of 1.1 serve as mes ponts. Ts assumpton s always satsfed by a proper coce of mes sze. For convenence, we use te followng notatons: S α u ξ p α u ξ, 1 S β u η p β u η and ntroduce te fnte dfference operators δ 2 and P as follows: δ 2 u x u x 1 2u x u x 1, 1 L 1, P u x 2 12 u x 1 10u x u x 1, 1 L Usng te followng Numerov s formula cf. 52, 53 : δ 2 u x P u x O 6, 1 L 1, 2.3 we ave from 1.1 and 2.1 tat δ 2 u x P f x,u x O 6, 1 L 1, u 0 S α u ξ, u 1 S β u η. 2.4

5 Matematcal Problems n Engneerng 5 After droppng te O 6 term, we derve a Numerov s dfference approxmaton to 1.1 as follows: δ 2 u x P f x,u x, 1 L 1, u 0 S α u ξ, u 1 S β u η, 2.5 were u x represents te approxmaton of u x. For two constants M and M satsfyng M M > π 2, we defne 12 M, M > 8, M>0, 1, M > 8, M 0, { M, M 12 mn M, 12 1 M }, M 8, M>0, π 2 π Mπ, M 8, M 0. π A fundamental and useful property of te operators δ 2 and P s stated below. Lemma 2.1 See Lemma 3.1 of 50. Let M, M, and M be some constants satsfyng π 2 <M M M, 0 L. 2.7 If δ 2 u x P M u x 0, 1 L 1, u 0 0, u 1 0, 2.8 and < M, M,tenu x 0 for all 0 L. Te followng results are also useful for our fortcomng dscussons. Ter proofs wll be gven n te appendx. Lemma 2.2. Assume { p } p σ max α, β < Let M, M, and M be te gven constants suc tat 8 1 σ <M M M, 0 L. 2.10

6 6 Matematcal Problems n Engneerng If δ 2 u x P M u x 0, 1 L 1, u 0 S α u ξ, u 1 S β u η, 2.11 and < M, M,tenu x 0 for all 0 L. Lemma 2.3. Let te condton 2.9 be satsfed, and let M, M, and M be te gven constants satsfyng Assume tat te functons u x and g x satsfy δ 2 u x P M u x g x, 1 L 1, u 0 S α u ξ, u 1 S β u η Ten wen < M, M, u g 8 1 σ mn M, 0 2, 2.13 were u max 1 L 1 u x denotes dscrete nfnty norm for any mes functon u x. Remark 2.4. It s clear from Lemma 2.1 tat f σ 0 ten te condton 2.10 n Lemma 2.2 can be replaced by te weaker condton 2.7. Lemmas 2.1 and 2.2 guarantee tat te lnear problems based on 2.8 and 2.11 wt te nequalty relaton replaced by te equalty relaton are well posed. 3. Te Exstence and Unqueness of te Soluton To nvestgate te exstence and unqueness of te soluton of 2.5, weusetemetod of upper and lower solutons. Te defnton of te upper and lower solutons s gven as follows. Defnton 3.1. A functon ũ x s called an upper soluton of 2.5 f δ 2 ũ x P f x, ũ x, 1 L 1, ũ 0 S α ũ ξ, ũ 1 S β ũ η. 3.1 Smlarly, a functon û x s called a lower soluton of 2.5 f t satsfes te above nequaltes n te reversed order. A par of upper and lower solutons ũ x and û x are sad to be ordered f ũ x û x for all 0 L.

7 Matematcal Problems n Engneerng 7 It s clear tat every soluton of 2.5 s an upper soluton as well as a lower soluton. For a gven par of ordered upper and lower solutons ũ x and û x,weset û, ũ {u ; û x u x ũ x 0 L }, û x, ũ x {u R; û x u ũ x } 3.2 and make te followng basc ypoteses: H 1 For eac 0 L, tere exsts a constant M suc tat M > π 2 and f x,v f x,v M v v 3.3 wenever û x v v ũ x ; H 2 < M, M, were M max 0 L M and M mn 0 L M. Te exstence of te constant M n H 1 s trval f f x,u s a C 1 -functon of u û x, ũ x. In fact, M may be taken as any nonnegatve constant satsfyng M max { f u x,u ; u û x, ũ x }. 3.4 Teorem 3.2. Let ũ x and û x be a par of ordered upper and lower solutons of 2.5, and let ypoteses H 1 and H 2 be satsfed. Ten system 2.5 as a maxmal soluton u x and a mnmal soluton u x n û, ũ. Here, te maxmal property of u x means tat for any soluton u x of 2.5 n û, ũ, one ase u x u x for all 0 L. Te mnmal property of u x s smlarly understood. Proof. Te proof s constructve. Usng te ntal teratons u 0 x ũ x and u 0 x û x we construct two sequences {u m x } and {u m x }, respectvely, from te followng teratve sceme: δ 2 u m x P M u m x P M u m 1 x f x,u m 1 x, 1 L 1, ] ] u m 0 S α u m 1 ξ, u m 1 S β u m 1 η, 3.5 were M s te constant n H 1.ByLemma 2.1, tese two sequences are well defned. We sall frst prove tat for all m 0, 1,..., u m x u m 1 x u m 1 x u m x, 0 L. 3.6 Let w 0 x u 0 x u 1 x. Ten by 3.1 and 3.5, δ 2 w 0 x P M w 0 x 0, 1 L 1, 3.7 w 0 0 0, w

8 8 Matematcal Problems n Engneerng It follows from Lemma 2.1 tat w 0 x 0, tat s, u 0 x u 1 x for all 0 L. x u 0 x for all x. We ave from 3.3 and 3.5 tat A smlar argument usng te property of a lower soluton gves u 1 0 L. Letw 1 x u 1 x u 1 δ 2 w 1 x P M w 1 x 0, 1 L 1, 3.8 w 1 0 0, w Agan by Lemma 2.1, w 1 x 0, tat s, u 1 x u 1 x for all 0 L. Ts proves 3.6 for m 0. Fnally, an nducton argument leads to te desred result 3.6 for all m 0, 1,... In vew of 3.6, te lmts lm m u m x u x, lm m u m x u x, 0 L 3.9 exst and satsfy u m x u m 1 x u x u x u m 1 x u m x, 0 L, m 0, 1, Lettng m n 3.5 sows tat bot u x and u x are solutons of 2.5. Now, f u x s a soluton of 2.5 n û, ũ, ten te par u x and û x are also a par of ordered upper and lower solutons of 2.5. Te above arguments mply tat u x u x for all 0 L. Smlarly, we ave u x u x for all 0 L. Ts sows tat u x and u x are te maxmal and te mnmal solutons of 2.5 n û, ũ, respectvely. Te proof s completed. Teorem 3.2 sows tat te system 2.5 as a maxmal soluton u x and a mnmal soluton u x n û, ũ.ifu x u x for all 0 L, ten u x or u x s a unque soluton of 2.5 n û, ũ. In general, tese two solutons do not concde. Consder, for example, te case p α 1 p β If tere exst two dfferent constants c and c suc tat f x, c f x, c 0 for all x 0, 1 ten bot c and c are solutons of 2.5. Hence to sow te unqueness of a soluton t s necessary to mpose some addtonal condtons on α, β and f. Assume tat tere exsts a constant M u suc tat f x,v f x,v Mu v v, 0 L 3.12

9 Matematcal Problems n Engneerng 9 wenever û x v v ũ x. Ts condton s trvally satsfed f f x,u s a C 1 - functon of u û x, ũ x for all 0 L. In fact, M u may be taken as M u mn 0 L mn{ f u x,u ; u û x, ũ x } Te followng teorem gves a suffcent condton for te unqueness of a soluton. Teorem 3.3. Let te condtons n Teorem 3.2 old. If, n addton, te condtons 2.9 and 3.12 old and eter 8 1 σ <M u 0 or M u > 0,< 12 M u, 3.14 ten te system 2.5 as a unque soluton u x n û, ũ. Moreover, te relaton 3.10 olds wt u x u x u x for all 0 L. Proof. It suffces to sow u x u x for all 0 L, were u x and u x are te lmts n 3.9.Letw x u x u x. Ten w x 0, and by 2.5, δ 2 w x P f x, u x f x,u x, 1 L 1, w 0 S α w ξ, w 1 S β w η Terefore, we ave from 3.12 tat δ 2 w x P Mu w x 0, 1 L 1, w 0 S α w ξ, w 1 S β w η By Lemma 2.2, w x 0 for all 0 L. Ts proves u x u x for all 0 L. To gve anoter suffcent condton, we assume tat tere exsts a nonnegatve constant M u suc tat f x,v f x,v M u v v, 0 L 3.17 wenever û x v v ũ x.iff x,u s a C 1 -functon of u û x, ũ x for all 0 L, te above condton s clearly satsfed by M u max 0 L max{ fu x,u ; u û x, ũ x } Teorem 3.4. Let te condtons n Teorem 3.2 old. If, n addton, te condtons 2.9 and 3.17 old and M u < 8 1 σ, 3.19 ten te conclusons of Teorem 3.3 are also vald.

10 10 Matematcal Problems n Engneerng Proof. Applyng Lemma 2.3 wt M 0to 3.15 leads to w g 8 1 σ 2, 3.20 were g x P f x, u x f x,u x. By 3.17, weobtan g 2 M u w. Consequently, w M u w 8 1 σ Ts togeter wt 3.19 mples w x 0, tat s, u x u x for all 0 L. It s seen from te proofs of Teorems tat te teratve sceme 3.5 not only leads to te exstence and unqueness of te soluton of 2.5 but also provdes a monotone teratve algortm for computng te soluton. However, te rate of convergence of te teratve sceme 3.5 s only of lnear order because t s of Pcard type. A more effcent monotone teratve algortm wt te quadratc rate of convergence wll be developed n Secton Convergence of Numerov s Metod In ts secton, we deal wt te convergence of te numercal soluton and sow te fourtorder accuracy of Numerov s sceme 2.5. Trougout ts secton, we assume tat te functon f x, u and te soluton u x of 1.1 are suffcently smoot. Let u x be te value of te soluton of 1.1 attemespontx,andletu x be te soluton of 2.5. We consder te error e x u x u x. In fact, we ave from 2.4 and 2.5 tat δ 2 e x P f x,u x f x,u x O 6, 1 L 1, e 0 S α e ξ, e 1 S β e η. 4.1 Teorem 4.1. Let te condton 2.9 old, and let u,,u be an nterval n R suc tat u x,u x u,,u. Assume tat max max{ f u x,u ; u u,,u ]} < 8 1 σ L Ten for suffcently small, u u C 4, 4.3 were C s a postve constant ndependent of.

11 Matematcal Problems n Engneerng 11 Proof. Applyng te mean value teorem to te frst equalty of 4.1, we ave δ 2 e x P M e x O 6, 1 L 1, e 0 S α e ξ, e 1 S β e η, 4.4 were M f u x,θ and θ u,,u.letm mn M and M mn M. Ten by 4.2, 8 1 σ <M M M. We, terefore, obtan from Lemma 2.3 tat wen < M, M, e C 1 O 6, were C 1 s a postve constant ndependent of. Fnally, te error estmate 4.3 follows from O 6 C 2 6 for some postve constant ndependent of. Teorem 4.1 sows tat Numerov s sceme 2.5 possesses te fourt-order accuracy under te condtons of te teorem. 5. An Accelerated Monotone Iteratve Algortm Te teratve sceme 3.5 gves an algortm for solvng te system 2.5. However, as already mentoned n Secton 3, ts rate of convergence s only of lnear order because t s of Pcard type. To rase te rate of convergence wle mantanng te monotone convergence of te sequence, we propose an accelerated monotone teratve algortm. An advantage of ts algortm s tat ts rate of convergence for te sum of te two produced sequences s quadratc n te sense mentoned n Secton 1 wt only te usual dfferentablty requrement on te functon f,u. If te functon f u,u possesses a monotone property n u, ts algortm s reduced to Newton s metod, and one of te two produced sequences converges quadratcally Monotone Iteratve Algortm Let ũ x and û x be a par of ordered upper and lower solutons of 2.5 and assume tat f,u s a C 1 -functon of u û, ũ. It follows from Teorems tat 2.5 as a unque soluton u x n û, ũ under te condtons of te teorems. To compute ts soluton, we use te followng teratve sceme: δ 2 u m x P P M m 1 u m M m 1 u m 1 u m 0 S α x x f x,u m 1 x, 1 L 1, ] ξ, u m 1 S β u ], m η u m 5.1

12 12 Matematcal Problems n Engneerng were u 0 x s eter ũ x or û x, and for eac, M m { max f u x,u ; u u m ]} x, u m x. 5.2 Te functons u m x ũ x and u 0 u 0 x and u m x n te defnton of M m are obtaned from 5.1 wt x û x, respectvely. It s clear from 5.2 tat f x,v f x,v m M v v, 0 L, 5.3 wenever u m x v v u m x. Moreover, M m f u x, u m x, f f u x,u s monotone nonncreasng n u f u x,u m x, f f u x,u s monotone nondecreasng n u u m u m ] x, u m x, ] x, u m x. 5.4 Hence, f f u x,u s monotone nonncreasng/nondecreasng n u u m x, u m x for all 0 L, ten te teratve sceme 5.1 for {u m x }/{u m x } s reduced to Newton s form: δ 2 u m x P f u P f u x,u m 1 u m 0 S α x,u m 1 x u m x x u m 1 x f x,u m 1 ] u m ξ, u m 1 S β x, 1 L 1, u m η ]. 5.5 To sow tat te sequences gven by 5.1 are well-defned and monotone for an arbtrary C 1 -functon f,u,weletm u be gven by 3.13 and let M u max 0 L max{ f u x,u ; u û x, ũ x }. 5.6 Lemma 5.1. Let te condton 2.9 old, and let ũ x and û x be a par of ordered upper and lower solutons of 2.5. Assume tat M u > 8 1 σ and < M u, M u. Ten te sequences {u m x }, {u m x }, and {M m } gven by 5.1 and 5.2 wt u 0 x ũ x and u 0 x û x are all well defned and possess te monotone property u m x u m 1 x u m 1 x u m x, 0 L, m 0, 1,

13 Matematcal Problems n Engneerng 13 Proof. Snce M 0 max{ f u x,u ; u û x, ũ x }, 8 1 σ <M u M 0 x and u 1 < M u, M u, we ave from Lemma 2.2 tat te frst teratons u 1 defned. Let w 0 x u 0 x u 1 x. Ten, by 3.1 and 5.1, M u and x are well δ 2 w 0 x P w 0 0 S α M 0 w 0 x 0, 1 L 1, ] ξ, w 0 1 S β w ]. 0 η w We ave from Lemma 2.2 tat w 0 x 0, tat s, u 0 x u 1 x u 0 Smlarly by te property of a lower soluton, u 1 x u 1 x u 1 x. Ten by 5.1 and 5.3, w 1 x for every 0 L. x for every 0 L. Let δ 2 w 1 x P w 1 0 S α M 0 w 1 x 0, 1 L 1, ] ξ, w 1 1 S β w ]. 1 η w It follows from Lemma 2.2 tat w 1 x 0, tat s, u 1 x u 1 x for every 0 L.Ts proves te monotone property 5.7 for m 0. Assume, by nducton, tat tere exsts some nteger m 0 0 suc tat for all 0 m m 0, te teratons u m x, u m 1 x, u m x,andu m 1 x are well-defned and satsfy 5.7. Ten M m 0 1 s well defned and 8 1 σ <M u M m 0 1 we ave from Lemma 2.2 tat te teratons u m 0 2 x u m 0 1 x u m 0 2 x. Snce w m 0 1 x and u m 0 2 M u. Snce < M u, M u, x exst unquely. Let M m 0 1 w m 0 1 M m 0 1 x M m 0 u m 0 1 x M m 0 u m 0 1 x M m 0 1 u m 0 2 x, 5.10 te teratve sceme 5.1 mples tat δ 2 w m 0 1 x P M m 0 1 w m 0 1 x P M m 0 x u m 0 1 x f u m 0 w m S α w m 0 1 x, u m 0 ] x f x, u m 0 1 ξ, w m S β w m 0 1 x, 1 L 1, ] η. 5.11

14 14 Matematcal Problems n Engneerng Usng te relaton 5.3 yelds δ 2 w m 0 1 x P w m S α w m 0 1 x 0, 1 L 1, ] ] ξ, w m S β w m 0 1 η. M m 0 1 w m By Lemma 2.2, w m 0 1 x 0, tat s, u m 0 1 x u m 0 2 x for every 0 L. Smlarly we ave u m 0 2 x u m 0 1 x for every 0 L.Letw m 0 2 x u m 0 2 x u m 0 2 x. Ten by 5.1 and 5.3, w m 0 2 x satsfes 5.12 wt w m 0 1 x replaced by w m 0 2 x. Terefore, by Lemma 2.2, w m 0 2 x 0, tat s, u m 0 2 x u m 0 2 x for every 0 L. Ts sows tat te monotone property 5.7 s also true for m m 0 1. Fnally, te concluson of te lemma follows from te prncple of nducton. We next sow monotone convergence of te sequences {u m x } and {u m x }. Teorem 5.2. Let te ypotess n Lemma 5.1 old. Ten te sequences {u m x } and {u m x } gven by 5.1 converge monotoncally to te unque soluton u x of 2.5 n û, ũ, respectvely. Moreover, u m x u m 1 x u x u m 1 x u m x, 0 L, m 0, 1, Proof. It follows from te monotone property 5.7 tat te lmts lm m u m x u x, lm m u m x u x, 0 L 5.14 exst and tey satsfy Snce te sequence {M m } s monotone nonncreasng and s bounded from below by M u gven n 3.13, t converges as m. Lettng m n 5.1 sows tat bot u x and u x are solutons of 2.5 n û, ũ. Snce M u > 8 1 σ and < M u, M u, te condton 3.14 of Teorem 3.3 s satsfed. Tus by Teorem 3.3, u x u x u x and u x s te unque soluton of 2.5 n û, ũ. Te monotone property 5.13 follows from Wen f u x,u s monotone nonncreasng/nondecreasng n u u m x, u m x for all 0 L, te teratve sceme 5.1 for {u m x }/{u m x } s reduced to Newton teraton 5.5. As a consequence of Teorem 5.2, we ave te followng concluson. Corollary 5.3. Let te ypotess n Lemma 5.1 be satsfed. If f u x,u s monotone nonncreasng n u u m x, u m x for all 0 L, te sequence {u m x } gven by 5.5 wt u 0 x ũ x converges monotoncally from above to te unque soluton u x of 2.5 n û, ũ. Oterwse, f f u x,u s monotone nondecreasng n u u m x, u m x for all 0 L, te sequence {u m x } gven by 5.5 wt u 0 x û x converges monotoncally from below to u x.

15 Matematcal Problems n Engneerng Rate of Convergence We now sow te quadratc rate of convergence of te sequences gven by 5.1. Assume tat tere exsts a nonnegatve constant M suc tat fu x,v f u x,v M v v v,v û x, ũ x, 0 L Clearly, ts assumpton s satsfed f f,u s a C 2 -functon of u. Teorem 5.4. Let te ypoteses n Lemma 5.1 and 5.15 old. Also let {u m x } and {u m x } be te sequences gven by 5.1 and let u x be te unque soluton of 2.5 n û, ũ. Ten tere exsts a constant ρ, ndependent of m, suc tat u m u u m u u m 1 ρ u u m 1 u 2, m 1, 2, Proof. Let w m x u m x u x. Subtractng 2.5 from 5.1 gves δ 2 w m x P P M m 1 w m M m 1 w m 1 x f w m 0 S α x x, u m 1 ] ξ w m x f x,u x, 1 L 1,, w m 1 S β w m η ] By te ntermedate value teorem, M m 1 f u x,θ m 1, 5.18 were θ m 1 u m 1 x, u m 1 x, and by te mean value teorem, f x, u m 1 x f x,u x f u x,γ m 1 w m 1 x, 5.19 were γ m 1 u x, u m 1 x.let g m 1 f u x,γ m 1 f u x,θ m 1 w m 1 x Ten we ave from 5.17 tat δ 2 w m x P M m 1 w m ] w m 0 S α w m ξ, w m x P g m 1, 1 L 1, 1 S β w ]. m η 5.21

16 16 Matematcal Problems n Engneerng Snce 8 1 σ <M u M m 1 M u and < M u, M u, t follows from Lemma 2.3 tat tere exsts a constant ρ 1, ndependent of m, suc tat w m P ρ 1 g m To estmate g m 1, we observe from 5.15 tat g m 1 M m 1 γ θ m 1 w m 1 x Snce bot γ m 1 and θ m 1 g m 1 are n u m 1 x, u m 1 x, te above estmate mples tat M m 1 u x u m 1 x w m 1 x Usng ts estmate n 5.22, weobtan w m ρ 1 M m 1 u u m 1 w m 1, 5.25 or u m u ρ 1 M m 1 u u m 1 u m 1 u Smlarly, we ave u m u ρ 1 M m 1 u u m 1 u m 1 u Addton of 5.26 and 5.27 gves u m u u m ρ 1 M u m 1 u u m 1 u m 1 u u m 1 u Ten te estmate 5.16 follows mmedately. Teorem 5.4 gves a quadratc convergence for te sum of te sequences {u m x } and {u m x } n te sense of 5.16.Iff u x,u s monotone nonncreasng/nondecreasng n u u m x, u m x for all 0 L, te sequence {u m x }/{u m x } as te quadratc convergence. Ts result s stated as follows.

17 Matematcal Problems n Engneerng 17 Teorem 5.5. Let te condtons n Teorem 5.4 old. Ten tere exsts a constant ρ, ndependent of m, suc tat u m u ρ u m 1 u 2, m 1, 2, f f u x,u s monotone nonncreasng n u u m u m u ρ u m 1 x, u m x for all 0 L and u 2, m 1, 2, f f u x,u s monotone nondecreasng n u u m x, u m x for all 0 L. Proof. Consder te monotone nonncreasng case. In ts case, te sequence {u m x } s gven by 5.5 wt u 0 x ũ x.tsmplestatθ m 1 u m 1 x, were θ m 1 s te ntermedate value n Snce γ m 1 γ m 1 θ m 1 n 5.19 s n u x, u m 1 u m 1 x,weseetat x u x Tus, 5.24 s now reduced to g m 1 M m 1 w x Te argument n te proof of Teorem 5.4 sows tat 5.29 olds wt ρ ρ 1 M, were ρ 1 s te constant n Te proof of 5.30 s smlar. 6. Applcatons and Numercal Results In ts secton, we gve some applcatons of te results n te prevous sectons to tree model problems. We present some numercal results to demonstrate te monotone and rapd convergence of te sequence from 5.1 and to sow te fourt-order accuracy of Numerov s sceme 2.5, as predcted n te analyss. In order to mplement te monotone teratve algortm 5.1, t s necessary to fnd a par of ordered upper and lower solutons of 2.5. Te constructon of ts par depends manly on te functon f,u, and muc dscusson on te subject can be found n 54 for contnuous problems. To demonstrate some tecnques for te constructon of ordered upper and lower solutons of 2.5, we assume tat f x, 0 0 for all x 0, 1 and tere exsts a nonnegatve constant C suc tat f x, C 0, x 0,

18 18 Matematcal Problems n Engneerng Ten δ 2 C 0 P f x,c for all 1 L 1. Ts mples tat ũ x C and û x 0 are a par of ordered upper and lower solutons of 2.5 f, n addton, te condton 2.9 olds. On te oter and, assume tat tere exst nonnegatve constants a, b wt a<8 1 σ suc tat f x, u au b for x 0, 1, u 0, 6.2 were σ<1 s defned by 2.9. We ave from Lemma 2.2 tat te soluton ũ x of te lnear problem δ 2 ũ x ap ũ x 2 b, 1 L 1, ũ 0 S α ũ ξ, ũ 1 S β ũ η 6.3 exsts unquely and s nonnegatve. Clearly by 6.2, ts soluton s a nonnegatve upper soluton of 2.5. As applcatons of te above constructon of upper and lower solutons, we next consder tree specfc examples. In eac of tese examples, te analytc soluton u x of 1.1 s explctly known, aganst wc we can compare te numercal soluton u x of te sceme 2.5 to demonstrate te fourt-order accuracy of te sceme. Te order of accuracy s calculated by error u u, error order log error /2 All computatons are carred out by usng a MATLAB subroutne on a Pentum 4 computer wt 2G memory, and te termnaton crteron of teratons for 5.1 s gven by u m u m < Example 6.1. Consder te four-pont boundary value problem: u x θu x 1 u x q x, 0 <x<1, u u, u u, were θ s a postve constant and q x s a nonnegatve contnuous functon. Clearly, problem 6.6 s a specal case of 1.1 wt f x, u θu 1 u q x. 6.7

19 Matematcal Problems n Engneerng 19 Iteratons Fgure 1: Te monotone convergence of {u m x }, {u m x } at x 0.5 forexample 6.1. m To obtan an explct analytc soluton of 6.6, we coose π 2 q x 8 sn 2πx θz x 1 z x, 2 z x 4x 1 x 1 sn 2πx Ten te functon u x z x s a soluton of 6.6. Moreover, q x 0n 0, 1 f θ 32 2π 2. For problem 6.6, te correspondng Numerov sceme 2.5 s now reduced to δ 2 u x P f x,u x, 1 L 1, u u, u u To fnd a par of ordered upper and lower solutons of 6.9, we observe from 6.7 tat f x, 0 q x 0 for all x 0, 1, and, terefore, û x 0 s a lower soluton. Snce q x 14, we ave from 6.7 tat te condton 6.2 s satsfed for te present functon f wt a θ and b 14. Terefore, te soluton ũ x of 6.3 correspondng to 6.9 wt a θ and b 14 s a nonnegatve upper soluton f θ < 7. Ts mples tat ũ x and û x 0 are a par of ordered upper and lower solutons of 6.9. Let θ π/2. Usng u 0 x ũ x and u 0 x 0, we compute te sequences {u m x } and {u m x } from te teratve sceme 5.1 for 6.9 and varous values of. In all te numercal computatons, te basc feature of monotone convergence of te sequences was observed. Let 1/32. In Fgure 1, we present some numercal results of tese sequences at x 0.5, were te sold lne denotes te sequence {u m x } and te dased-dotted lne stands for te sequence {u m x }. As descrbed n Teorem 5.2, te sequences converge to te same lmt as m, and ter common lmt u x s te unque soluton of 6.9 n 0, ũ. Besdes, tese sequences converge rapdly n fve teratons. More numercal results

20 20 Matematcal Problems n Engneerng Table 1: Solutons u x and u x of Example 6.1. x u x u x 1/ / / / / / / / Table 2: Te accuracy of te numercal soluton u x of Example 6.1. Sceme 6.9 SFD sceme error order error order 1/ e e / e e / e e / e e / e e / e e / e e / e e 06 of u x at varous x are explctly gven n Table 1. We also lst te values of te analytc soluton u x. Clearly, te numercal soluton u x meets te analytc soluton u x closely. To furter demonstrate te accuracy of te numercal soluton u x, we lst te maxmum error error and te order order n te frst tree columns of Table 2 for varous values of. Te data demonstrate tat te numercal soluton u x as te fourtorder accuracy. Ts concdes wt te analyss very well. For comparson, we also solve 6.6 by te standard fnte dfference SFD metod. Ts metod leads to a dfference sceme n te form 6.9 wt P I an dentcal operator. Tus, a smlar teratve sceme as 5.1 can be used n actual computatons. Te correspondng maxmum error error and te order order are lsted n te last two columns of Table 2. We see tat te standard fnte dfference metod possesses only te second-order accuracy. Example 6.2. Our second example s for te followng fve-pont boundary value problem: u u u x θu x q x, 0 <x<1, 1 u x u, u u u 3 4, 6.10

21 Matematcal Problems n Engneerng 21 Iteratons m Fgure 2: Te monotone convergence of {u m x }, {u m x } at x 0.5 forexample 6.2. were θ s a postve constant and q x s a nonnegatve contnuous functon. Te correspondng Numerov s sceme 2.5 for ts example s gven by u u δ 2 u x P f x,u x, 1 L 1, u 1, u u u 3, were f x,u x θu x 1 u x q x, 0 L Let q x κ 2 θ 1 sn κx π/6 sn κx π, κ 2π Ten q x 0n 0, 1 f θ 3κ 2 /2, and u x sn κx π/6 s a soluton of Clearly, û x 0 s a lower soluton of On te oter and, te condton 6.2 s satsfed for te present problem wt a θ and b κ 2. Terefore, te soluton ũ x of 6.3 correspondng to 6.11 wt a θ and b κ 2 s a nonnegatve upper soluton of 6.11 f θ< /3. Let θ κ. Usngu 0 x ũ x and u 0 x 0, we compute te sequences {u m x } and {u m x } from te teratve sceme 5.1 for Let 1/32. Some numercal results of tese sequences at x 0.5 are plotted n Fgure 2, were te sold lne denotes te sequence {u m x } and te dased-dotted lne stands for te sequence {u m x }. We see tat te sequences possess te monotone convergence gven n

22 22 Matematcal Problems n Engneerng Table 3: Te accuracy of te numercal soluton u x of Example 6.2. Sceme 6.11 SFD sceme error order error order 1/ e e / e e / e e / e e / e e / e e / e e / e e 06 Teorem 5.2 and converge rapdly n fve teratons to te unque soluton u x of 6.11 n 0, ũ. Te maxmum error error and te order order of te numercal soluton u x by te sceme 6.11 and te SFD sceme are presented n Table 3. Te numercal results clearly ndcate tat te proposed sceme 6.11 s more effcent tan te SFD sceme. Example 6.3. Our last example s gven by u x θ q 4 x u 4 x, 0 <x<1, 2 1 u 0 8 u u 1 12 u u u u, u, were θ s a postve constant and q x s a contnuous functon. For ts example, te correspondng Numerov sceme 2.5 s reduced to δ 2 u x P f x,u x, 1 L 1, u 0 8 u 8 12 u u u 1 12 u u 2 12 u 4,, 6.15 were f x,u x θ q 4 x u 4 x, 0 L. 6.16

23 Matematcal Problems n Engneerng 23 Iteratons Fgure 3: Te monotone convergence of {u m x }, {u m x } at x 0.5 forexample 6.3 m Table 4: Table 1 Te accuracy of te numercal soluton u x of Example 6.3. Sceme 6.15 SFD sceme error order error order 1/ e e / e e / e e / e e / e e / e e / e e 07 To accommodate te analytcal soluton of u x sn κx π/6 were κ 2π/3, we let κ 2 q x θ sn κx π 6 sn 4 κx π 1/ As n te prevous examples, û x 0 s a lower soluton of 6.15 and te soluton ũ x of 6.3 correspondng to 6.15 wt a 0andb κ 2 θ s a nonnegatve upper soluton. Let θ π 2 /2. We compute te correspondng sequences {u m x } and {u m x } from te teratve sceme 5.1 wt te ntal teratons u 0 x ũ x and u 0 x 0. Let 1/32. Fgure 3 sows te monotone and rapd convergence of tese sequences at x 0.5, were te sold lne denotes te sequence {u m x } and te dased-dotted lne stands for te sequence {u m x } as before. Te data n Table 4 sow te maxmum error error and te order order of te numercal soluton u x by te sceme 6.15 and te SFD sceme for varous values of. Te fourt-order accuracy of te numercal soluton u x by te present Numerov sceme s demonstrated n ts table.

24 24 Matematcal Problems n Engneerng 7. Conclusons In ts paper, we ave gven a numercal treatment for a class of nonlnear multpont boundary value problems by te fourt-order Numerov metod. Te exstence and unqueness of te numercal soluton and te convergence of te metod ave been dscussed. An accelerated monotone teratve algortm wt te quadratc rate of convergence as been developed for solvng te resultng nonlnear dscrete problem. Te proposed Numerov metod s more attractve due to ts fourt-order accuracy, compared to te standard fnte dfference metod. In ts work, we ave generalzed te metod of upper and lower solutons to nonlnear multpont boundary value problems. We ave also developed a tecnque for desgnng and analyzng compact and monotone fnte dfference scemes wt g accuracy. Tey are very useful for accurate numercal smulatons of many oter nonlnear problems, suc as tose related to ntegrodfferental equatons e.g., 55, 56 and tose n nformaton modelng e.g., Appendx A. Proofs of Lemmas 2.2 and 2.3 Lemmas 2.2 and 2.3 are te specal cases of Lemmas 2.2 and 2.3 n 61. We nclude ter proofs ere n order to make te paper self-contaned. Defne α, x ξ α for some, 0, oterwse, β, x η β for some, 0, oterwse, 1 L 1. A.1 Let A a,j, B b,j,andd d,j be te L 1 t-order matrces wt a,j 2δ,j δ,j 1 δ,j 1, b,j 5 6 δ,j 1 12 δ,j δ,j 1, d,j δ,1 α j δ,l 1β j, A.2 were δ,j 1f j and δ,j 0f / j. Lemma A.1. Let te condton 2.9 be satsfed. Ten te nverse A D 1 > 0, and A D σ. 2 A.3 Proof. It can be cecked by Corollary 3.20 on Page 91 of 62 tat te nverse A D 1 > 0. Let E 1, 1,...,1 T R L 1 and let S A D 1 E. Ten A D 1 S. It s known

25 Matematcal Problems n Engneerng 25 tat te nverse A 1 J,j exsts, and ts elements J,j are gven by L j, j, J,j L L j, > j. L A.4 A smple calculaton sows tat A 1 L 2 /8 1/ 8 2 and J,1 J,L 1 1 for eac 1 L 1. Ts mples S A 1 E A 1 DS A 1 1 E σ S E E σ S 8 2 E. A.5 Tus te estmate A.3 follows mmedately. Proof of Lemma 2.2. Defne te followng L 1 t-order matrces or vectors: U u x 1,u x 2,...,u x L 1 T, M dag M 1,M 2,...,M L 1, M b dag M 0, 0,...,0,M L, T G b M 0 u 0, 0,...,0, M L u 1. A.6 Usng te matrces A and B defned by A.2, we ave from 2.11 tat A 2 BM U G b. A.7 Snce M > 8 1 σ and < M, M, t s easy to ceck tat 1 2 /12 M 0 0,L. Tus by te boundary condton n 2.11, G b DU 2 12 M bdu, A.8 were D s te L 1 t-order matrx defned by A.2. Ts leads to A D 2 BM 2 12 M bd U 0. A.9 Let Q A D 2 BM 2 /12 M b D. To prove u x 0 for all 0 L,tsuffces to sow tat te nverse of Q exsts and s nonnegatve. Case 1 M 0. In ts case, te matrx Q satsfes te condton of Corollary 3.20 on Page 91 of 62, and, terefore, ts nverse Q 1 exsts and s postve.

26 26 Matematcal Problems n Engneerng Case 2 0 >M> 8 1 σ. For ts case, we defne M dag M 1,M 2,...,M L 1, M max{m, 0}, M M M. A.10 Te matrces M b and M b can be smlarly defned. Let Q A D 2 BM 2 /12 M b D. We know from Case 1 tat Q 1 exsts and s postve. Snce Q Q 2 BM 2 12 M b D Q I 2 Q 1 BM 1 12 M b D, A.11 we need only to prove tat te nverse I 2 Q 1 BM 1/12 M b D 1 exsts and s nonnegatve. By Teorem 3 on Page 298 of 63, tsstruef 2 Q 1 BM 1 D 12 M b < 1. A.12 Snce Q A D wc mples 0 Q 1 A D 1, we ave from Lemma A.1 tat Q 1 A D σ 2. A.13 It s clear tat B 1/12 D 1, M M and M b M. Tus, we ave 2 Q 1 BM 1 D M 12 M b 8 1 σ. A.14 Te estmate A.12 follows from M > 8 1 σ. Proof of Lemma 2.3. Usng te same notaton as before, te system 2.12 can be wrtten as QU G, A.15 were G g x 1,g x 2,...,g x L 1 T. Case 1 M 0. Snce te nverse Q 1 exsts and s postve, we ave 0 <Q 1 A D 1. Ts sows Q 1 A D σ 2. A.16 Tus, by A.15, U G / 8 1 σ 2 wc mples te desred estmate 2.13.

27 Matematcal Problems n Engneerng 27 Case 2 0 >M> 8 1 σ. It follows from A.11 tat By A.13 and A.14, Q 1 Q 1 I 2 Q 1 BM 1 D 1 12 M b Q σ 2. A σ 8 1 σ M 1. A σ M 2 Ts togeter wt A.15 leads to te estmate Acknowledgments Te autors would lke to tank te edtor and te referees for ter valuable comments and suggestons wc mproved te presentaton of te paper. Ts work was supported n part by te Natonal Natural Scence Foundaton of Cna No , E-Insttutes of Sanga Muncpal Educaton Commsson No. E03004, te Natural Scence Foundaton of Sanga No. 10ZR , and Sanga Leadng Academc Dscplne Project No. B407. References 1 Y. Zou, Q. Hu, and R. Zang, On numercal studes of mult-pont boundary value problem and ts fold bfurcaton, Appled Matematcs and Computaton, vol. 185, no. 1, pp , J. Adem and M. Mosnsky, Self-adjontness of a certan type of vectoral boundary value problems, Boletín de la Socedad Matemátca Mexcana, vol. 7, pp. 1 17, M. S. Berger and L. E. Fraenkel, Nonlnear desngularzaton n certan free-boundary problems, Communcatons n Matematcal Pyscs, vol. 77, no. 2, pp , M. Greguš, F. Neuman, and F. M. Arscott, Tree-pont boundary value problems n dfferental equatons, Journal of te London Matematcal Socety, vol. 2, pp , C.-Z. Ba and J.-X. Fang, Exstence of multple postve solutons for nonlnear m-pont boundaryvalue problems, Appled Matematcs and Computaton, vol. 140, no. 2-3, pp , P. W. Eloe and Y. Gao, Te metod of quaslnearzaton and a tree-pont boundary value problem, Journal of te Korean Matematcal Socety, vol. 39, no. 2, pp , W. Feng and J. R. L. Webb, Solvablty of m-pont boundary value problems wt nonlnear growt, Journal of Matematcal Analyss and Applcatons, vol. 212, no. 2, pp , Y. Guo, W. San, and W. Ge, Postve solutons for second-order m-pont boundary value problems, Journal of Computatonal and Appled Matematcs, vol. 151, no. 2, pp , C. P. Gupta, A Drclet type mult-pont boundary value problem for second order ordnary dfferental equatons, Nonlnear Analyss, vol. 26, no. 5, pp , R. Ma, Postve solutons of a nonlnear m-pont boundary value problem, Computers & Matematcs wt Applcatons, vol. 42, no. 6-7, pp , B. Lu, Solvablty of mult-pont boundary value problem at resonance II, Appled Matematcs and Computaton, vol. 136, no. 2-3, pp , X. Lu, J. Qu, and Y. Guo, Tree postve solutons for second-order m-pont boundary value problems, Appled Matematcs and Computaton, vol. 156, no. 3, pp , R. Ma, Multplcty results for a tree-pont boundary value problem at resonance, Nonlnear Analyss, vol. 53, no. 6, pp , R. Ma, Postve solutons for nonomogeneous m-pont boundary value problems, Computers & Matematcs wt Applcatons, vol. 47, no. 4-5, pp , 2004.

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