An Efficient Method for Sixth-order Sturm-Liouville Problems. Ain, United Arab Emirates

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1 Iteratioal Joural of Sciece & Techolog Volume, No, 9-4, 7 A Efficiet Method for Sith-order Sturm-Liouville Problems Daiel LESNIC ad Basem S. ATTILI Departmet of Applied Mathematics, Uiversit of Leeds, Leeds LS 9JT, UK amt5ld@maths.leeds.ac.u UAEU, Departmet of Mathematics ad Computer Sciece, College of Sciece, P.O. Bo 755, Al- Ai, Uited Arab Emirates B.Attili@uaeu.ac.ae (Received: 6..6; Accepted:..7 Abstract: Followig earlier wors o secod- ad fourth-order problems, we develop a efficiet method based o the Adomia decompositio for computig the eigeelemets of sith-order Sturm-Liouville boudar value problems. Numerical eamples show that the method proposed is eas to implemet ad produces accurate results. Kewords: Decompositio Method, Eigefuctios, Eigevalues, Sturm-Liouville Problems. Seizici Mertebede Strum-Liouville Problemleri İçi Etili Bir Metot Özet: So zamalarda iici ve dördücü mertebede problemler üzerie çalışmalar apılmıştır, seizici mertebede sıır değerli Strum Liouville problemlerii öz elemetlerii hesaplama içi Adomia arışım metodua bağlı olara etili bir metot geliştirdi. Saısal öreler, suula metodu güveilir souçlar ortaa çıardığıı göstermiştir. Aahtar Kelimeler: Arışım metot, öz fosiolar, özdeğerler sturm Liouville Problemleri.. Itroductio Ver recetl [5, 6], the Adomia decompositio method (ADM [] has bee developed for determiig the eigeelemets, i.e. the eigevalues ad the correspodig eigefuctios, for secod-order Sturm-Liouville boudar value problems of the form ( w(, (, ( subject to some two poit specified coditios at the boudar {, } o ad/or, ad for fourth-order Sturm-Liouville boudar value problems of the form ( ( s( ( + ( w( q(, (, subject to some four poit specified coditios at the boudar {, } o,, p ad/or ( p s. I this paper, we eted the ADM aalsis to o-sigular sith-order Sturm-Liouville problems of the form ( ( F(, (,, (, ( iv (, ( q( + ( r( + ( w( s(, ( (, subject to some si poit specified coditios at the boudar {, } o u u, u, v r ( q + ( p, q ( p, v p. (4 This problem has importat applicatios i studies of hdrodamic ad hdromagetic stabilit e.g. i viscous flow betwee rotatig cliders, the ihibitio of covectio b a magetic field, the thermal istabilit of fluid spheres ad spherical shells, etc., for more details see [9]. I equatio (, ad are fiite umbers, p, q, r, s ad w are piecewise cotiuous fuctios ad p, w >. These v

2 Daiel LESNIC ad Basem S. ATTILI coditios impl that equatio ( is regular, i.e. o-sigular. It is well ow [,4] that ( has a ifiite sequece of eigevalues ( which are bouded from below b a costat, i.e with lim ad each eigevalue has multiplicit at most. I geeral, equatio ( is solved subject to separated, self-adjoit boudar coditios o the variables (4; techicall equatio ( should be writte i the form {[ ( p ( + q( ] r( } + s( w ( to be equivalet to the Hamiltoia formulatio. Based o the defiitio (4 oe ca obtai sith-order Sturm-Liouville problems from secod-order oes. This fact will be helpful i costructig suitable sith-order eamples ad for testig the reliabilit of the proposed method. For eample, if is a eigevalue of the secodorder Sturm-Liouville problem l( : ( + s( w( (5 (,,, the is a eigevalue of the sith-order Sturm-Liouville problem L( : l ( l( l( l( w( (, ( (,,, (6 whilst the correspodig eigevalues are the same. Numericall ot much wor was doe o sith-order problems compared to secod- ad fouth-order oes, see the available software codes "SLEIGN" [7], "SLEIGN" [8], ad "SLEGDGE" [] for secod-order problems ad "SLEUTH" [] for fourth-order problems. I the et sectio we develop the ADM for the sith-order ordiar differetial equatio (, ad several test eamples are discussed i Sectio.. The Adomia Decompositio Method Defiig the differetial operator d d D (7 d d equatio ( ca be rewritte as ( iv D F(,,,,. (8 Applig the formal left-iverse operator 4 D to (7, results i + ( + ( p( + ( p( + ( ( ( ( + ( + +! ddd ( v ddd + p( 5 d... d 6 ( ( ( (9 ( iv + D F(,,, (, (,. ( The, usig the ADM, a solutio to ( is sought i the form of the series [] d d d ( iv ( iv (, [, ]. ( The compoets of the series ( are obtaied usig the recursive relatio

3 A Efficiet Method for Sith-order Sturm-Liouville Problems ( ( + +! ddd + ( p( + ( p( + ( ( ( v ( + ( ( + d d + p( d ( ( iv d d d, ( ( iv ( ( + ( D ( A (,,. ( I equatio (, A are the Adomia polomials defied as [4] d F i ( iv i A iµ,..., i µ,! dµ i i µ. (4 Alterative forms of (4 ca be foud i [], whilst the covergece of the ADM, as applied to differetial equatios has bee ivestigated i []. Based o (4, the Adomia polomials A ca be calculated eplicitl for a form of aaltical oliearit F. We also remar that if ( iv F(,,,, + α + β + γ (iv + δ is a liear fuctio, the ( iv A + α + β + γ + δ for all. Whe carrig out the umerical details ad if three coditios are specified at, the solutio obtaied will be a three-parameter series solutio that has the form (, az(, + bz (, + cz(,. (5 To satisf the other three coditios, sa (,,, we eed to solve z (, z (, z (, z ( z (,, z ( z (,, z ( z (,, (6 which is a polomial equatio i givig the eigevalues of the problem, with the correspodig eigefuctios give b (5.. Numerical Results ad Discussio Usig the ADM outlied i the previous sectio, we will solve several bechmar test eamples. At this stage we remar that, ulie the previous methods reported, the ADM does ot require: (i automatic mesh selectio or discretisatio, as with the fiite-differece method; (ii the coefficiets i equatio ( be costat fuctios, as with a aaltical method; (iii the use of Richardsos etrapolatio ad the boudar coditios be self-adjoit, as with the oscillatio method of [4]. Here we report the results o just three test eamples, two of them (Eamples ad previousl ivestigated i [4] for compariso purposes, but we metio that our method ca also be applied to the much loger list of Sturm- Liouville, both self- or o-selfadjoit problems, give i [9]. From a applied poit of view, most researchers are iterested mail i computig the eigevalues of the problem, as ca be easil see from the literature. For this reaso we will give the details ol for the first eample i terms of eigevalues ad the correspodig ormalized eigefuctios, whilst for the remaiig eamples ol the eigevalues are reported. Eample. First, we tae p (,, i.e. s r q, w, ad the the equatio ( becomes ( vi (, (,. (7 Cosider the boudar value problem give b (7 subject to the homogeeous boudar coditios ( iv ( (8 ( iv (. This problem has the eigevalues 6,,,... From ( we have 5 b c ( a + +,! 5! (9

4 Daiel LESNIC ad Basem S. ATTILI where a (, b ( (v ad c (. Applig the ADM give b ( to (7, we obtai ( D (,, i.e a b c + +, 7! 9!! 5 7 a b c + +,! 5! 7! ad, i geeral, (, a a a ( + +, (6! (6! (6 5! for. Usig ( we obtai, (, + (, + (, +, +... ( Usig the first 6 terms i (, we satisf the other three boudar coditios at, ( iv amel ( b solvig 5 (- 6 + (6 +! 5 (- 6 (6! 5 (- 6 (6! 5 (- 6+ (6+! 5 (- 6 + (6 +! 5 (- 6 (6! 5 ( (6+ 5! 5 (- 6 + (6 +! 5 (- 6+ (6+! which is a polomial equatio i. The first eigevalue obtaied is eactl. The et eigevalues are give i Table which are approimatios to the eact eigevalues 6 64, 6 79, ad It ca be see that the first eigevalues are accuratel obtaied if oe cosiders ol 6 terms i (. Taig 9 terms i ( leads to the first 8 eigevalues give i Table. Higher-order terms lead accuratel to the other eigevalues such as , ad From Tables ad it ca be see that for retrievig accuratel the first N eigevalues about N terms eed to be cosidered i the series epasio (. The results of Table are also slightl more accurate tha the umerical results reported i [4]. Further, it is clear, as metioed earlier i Sectio, that the eigevalues icrease as the umber of terms i ( icreases, ad that, i the limit, the ted to ifiit. Table : The eigevalues i, i, 5 for Eample, whe 6 terms are cosidered i the series epasio ( Havig obtaied accuratel, we substitute bac ito (, give b ( to obtai the eigefuctio (, ad to compare it with the aaltical solutio. The ormalized eigefuctio is give b (, (,. (, d Table : The eigevalues i, i, 8 for Eample, whe 9 terms are cosidered i the series epasio ( The umerical results for (, obtaied usig 9 terms i the series epasio (, are give i Figure, i compariso with the eact solutio (, si( /. The figure shows ver good agreemet betwee the umerical ad eact solutios. Fig. : The first ormalized eigefuctio (,, as a fuctio of [, ], obtaied

5 A Efficiet Method for Sith-order Sturm-Liouville Problems usig 9 terms ( i the series epasio (, i compariso with the eact solutio (, si( / show b cotiuous lie. Eample. I this eample we cosider a sith-order ordiar differetial equatios give b the cube of the "harmoic oscillator" operator d / d + α I, where α is a costat ad I is the idetit operator, amel d d + α + ((8α α I 4 ( vi + ( α 6 + (α 4α ( (,5, which correspods to the case 5, p ( w(, q( α, 4 6 r ( α 8α ad s( α 4α i (. Ulie i Eample, the coefficiets i the differetial equatio are o loger costat. We tae the homogeeous boudar coditios ( iv ( ( ( iv 5 5 (5. Applig the ADM give b ( to (, i.e. 6 ( D (( α 4α ( + α + α ( + α 4 ( + (4α α ( iv (, (, ( with the iitial term (9 ields α c 7 α b α (6aα 67c c + 6bα b + 56aα 68cα 9 6bα a As metioed earlier i (5 ad (6, oe ca show that the eigevalues of this problem are the cubes of the eigevalues of the secod-order problem give b + α (,5 (4 subject to ( 5. (5 Usig the first terms i ( results i the first 9 eigevalues give i Table for α.. The first two eigevalues are the cubes of the eigevalues ad of the secod-order problem (4 ad (5. The results obtaied b SLEIGN [8] for the secod-order problem ad are withi -7 error. Table : The eigevalues i, i, 9 for Eample, with α., whe terms are cosidered i the series epasio ( Eample. The ordiar differetial equatio d d ( vi d I d (ν + ( ν + I ( iv ( ν + ν + (, arises i the aalsis of a Berard problem [,p.4]. Equatio (6 correspods to the case,, w(, + ( ν + 4ν + (6 s ( ν + ν +, r( ν + 4ν +, q ( ν + i (. I equatio (6, the costat ν is regarded as a eigeparameter which tpicall ca tae the values [4], ± / ν ( + j ± ( + j, j,,... j The boudar coditios are ( iv ( ( iv (. (7 (8

6 Daiel LESNIC ad Basem S. ATTILI Applig the ADM give b equatios ( ad ( to (6 ad (8 we obtai that ( is give b (9 ad the recurrece ( becomes ( ( iv D ((ν + ( + ( ν + 4ν + ( + ( ν + ν + +,. This ields ( ν + ν + + c 9968 b( ν + ν + + c( ν + 4ν a( ν + ν + + b( ν + 4ν c(ν (9 Usig the first 4 terms i (, we obtai the first three eigevalues give b., ad for ν ( +, ad., ad for ν ( + 4. It is worth otig that [4] gave.5 ad for ν ( +, ad. ad for ν ( + 4, however, beig reported that their code is ot ver accurate o the problem of Eample. 4. Coclusios The Adomia decompositio method (ADM has bee proved ver efficiet for computig the eigeelemets of sith-order Sturm-Liouville boudar value problems. It was show that the method is eas to implemet ad produces accurate results. It competes ver well with the oscillatio method developed i [4], ad moreover, is superior to this oe, see Eamples ad. Furthermore, as described i Sectio, the ADM ca also be applied to oliear boudar value problems. Future stud 4 will cocer developig the ADM for solvig higher-order Sturm-Liouville problems [5]. Refereces. Abbaoui, K., Cherruault, Y., (994, Covergece of Adomias method applied to differetial equatios, Comput. Math. Appl., 8, -9.. Abbaoui, K., Cherruault, Y., Seg, V., (995, Practical formulae for the calculus of multivariate Adomia polomials, Mathl. Comput. Modellig,, Adomia, G., (994, Solvig Frotier Problems of Phsics: The Decompositio Method, Kluwer Academic Publishers, Bosto. 4. Adomia, G., Rach, R., (99, Geeralizatio of Adomia polomials to fuctios of several variables, Comput. Math. Appl., 4, Attili, B.S., (5, The Adomia decompositio method for computig eigeelemets of Sturm-Liouville two poit boudar value problems, Appl. Math. Comput., 68, Attili, B.S., Lesic, D., (6 A efficiet method for computig eigeelemets of Sturm-Liouville fourth-order boudar value problems, Appl. Math. Comput., 8, Bail P., Gordo, M., Shampie, L., (978, Automatic solutio for Sturm-Liouville problems, ACM Tras. Math. Software, 4, Bail P., Everitt, W., Zettl, A., (99, Computig eigevalues of sigular Sturm-Liouville problems, Results Math.,, Chadrasehar, S., (954, O characteristic value problems i high order differetial equatios which arise i studies o hdrodamic ad hdromagetic stabilit i Part: Proc. Smp, Special Topics i Applied Mathematics, Amer. Math. Mothl 6, Drazi, P.G., Reid, W.H., (98, Hdrodamic Stabilit Cambridge Uiversit Press, Cambridge.. Fulto, C., Pruess, S., (99, Mathematical software for Sturm-Liouville problems, NSF Fial report for grats DMS88- ad DMS88-89, Computatioal Mathematics.. Greeberg, L., (99, A Prufer method for calculatig eigevalues of selfadjoit sstems of ordiar differetial equatios, Parts ad, Tech. Rep. TR9-4, Departmet of Mathematics, Uiversit of Marlad, College Par, MD.. Greeberg, L., Marletta, M., (997, Algorithm 775: The code SLEUTH for solvig fourth order Sturm- Liouville problems, ACM Tras. Math. Software,, Greeberg, L., Marletta, M., (998, Oscillatio theor ad umerical solutio of sith order Sturm-Liouville problems, SIAM J. Numer. Aal., 5, Greeberg, L., Marletta, M., (, Numerical methods for higher order Sturm-Liouville problems, J. Comput. Appl. Math., 5, 67-8.

Chapter 2: Numerical Methods

Chapter 2: Numerical Methods Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,