Generalized Jacobi polynomials/functions and their applications

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1 Applied uerical Matheatics ) Geeralized Jacobi polyoials/fuctios ad their applicatios Be-Yu Guo a,,jieshe b,,2, Li-Lia Wag c,3 a Departet of Matheatics, Shaghai oral Uiversity, Shaghai, , PR Chia b Departet of Matheatics, Purdue Uiversity, West Lafayette, I 47907, USA c Divisio of Matheatical Scieces, School of Physical ad Matheatical Scieces, ayag Techological Uiversity TU), 63737, Sigapore Available olie 26 April 2008 Abstract We itroduce a faily of geeralized Jacobi polyoials/fuctios with idexes α, β R which are utually orthogoal with respect to the correspodig Jacobi weights ad which iherit selected iportat properties of the classical Jacobi polyoials. We establish their basic approxiatio properties i suitably weighted Sobolev spaces. As a exaple of their applicatios, we show that the geeralized Jacobi polyoials/fuctios, with idexes correspodig to the uber of hoogeeous boudary coditios i a give partial differetial equatio, are the atural basis fuctios for the spectral approxiatio of this partial differetial equatio. Moreover, the use of geeralized Jacobi polyoials/fuctios leads to uch siplified aalysis, ore precise error estiates ad well coditioed algoriths IMACS. Published by Elsevier B.V. All rights reserved. MSC: 6535; 6522; 65F05; 35J05 Keywords: Jacobi polyoials; Spectral approxiatio; Error estiate; High-order differetial equatios. Itroductio The classical Jacobi polyoials have bee used extesively i atheatical aalysis ad practical applicatios cf. [35,2,36,3]). I particular, the Legedre ad Chebyshev polyoials have played a iportat role i spectral ethods for partial differetial equatios cf. [20,3,9,2,22] ad the refereces therei). Recetly, there have bee reewed iterests i usig the Jacobi polyoials i spectral approxiatios, especially for probles with degeerated or sigular coefficiets. For istace, Berardi ad Maday [9] cosidered spectral approxiatios usig the ultra-spherical polyoials i weighted Sobolev spaces. Guo [23,2,24] developed Jacobi approxiatios i certai Hilbert spaces with their applicatios to sigular differetial equatios ad soe probles o ifiite itervals. * Correspodig author. E-ail addresses: she@ath.purdue.edu J. She), lilia@tu.edu.sg L.-L. Wag). The wor of this author is supported partially by SF of Chia, , SF of Shaghai. 04JC4062, The Fud of Chiese Educatio Miistry , The Shaghai Leadig Acadeic Disciplie Project. T040, ad the Fud for E-istitutes of Shaghai Uiversities. E The wor of this author is partially supported by FS grat DMS The wor of this author is partially supported by the Startup Grat of TU, Sigapore MOE grat T207B2202, ad Sigapore RF2007IDM- IDM /$ IMACS. Published by Elsevier B.V. All rights reserved. doi:0.06/j.apu

2 02 B.-Y. Guo et al. / Applied uerical Matheatics ) The Jacobi approxiatios were also used to obtai optial error estiates for p-versio of fiite eleet ethods cf. [3,4]). Recetly, She [33] itroduced a efficiet spectral dual-petrov Galeri ethod for third ad higher odd-order differetial equatios, ad poited out that the basis fuctios used i [33], which are copact cobiatios of Legedre polyoials, ca be viewed as geeralized Jacobi polyoials with egative iteger idexes, ad their use ot oly siplified the uerical aalysis for the spectral approxiatios of higher odd-order differetial equatios, but also led to very efficiet uerical algoriths. More precisely, the resultig liear systes are well coditioed, ad sparse for probles with costat coefficiets. I fact, the basis fuctios used i [32], which are copact cobiatios of Legedre polyoials, ca also be viewed as geeralized Jacobi polyoials with idexes α, β. Furtherore, the special cases with α, β) =, 0),, ) have also bee studied i [6,2,24]. Hece, istead of developig approxiatio results for each particular pair of idexes, it would be very useful to carry out a systeatic study o Jacobi polyoials with idexes α, β which ca the be directly applied to other applicatios. I [25], we defied the geeralized Jacobi polyoials with idexes beig egative itegers, ad preseted soe approxiatio results ad applicatios. However, i ay situatios, it is helpful to defie ad use geeralized Jacobi polyoials with arbitrary o-iteger idexes. For exaple, whe developig ad aalyzig Chebyshev spectral ethods for boudary value probles, it becoes coveiet to use geeralized Jacobi polyoials with idexes /2, /2 l) cf. [34]). Aother exaple is the study of differetial equatios with sigular coefficiets of the for x) α + x) β. The ai purpose of this paper is to geeralize the defiitio of the Jacobi polyoials to arbitrary idexes α, β R, ad to establish their fudaetal approxiatio results, which iclude, as special cases, those aouced i [25] but ot proved due to the page liitatio of [25] as a coferece proceedig paper. The ai criteria that we use to defie the geeralized Jacobi polyoials/fuctios are: i) they are utually orthogoal with respect to the Jacobi weight, ad ii) they iherit soe iportat properties to be specified later) of the classical Jacobi polyoials which are essetial for spectral approxiatios. As a exaple of applicatios, we cosider approxiatios of high-order differetial equatios with suitable boudary coditios. Matheatical odelig of soe physical systes ofte leads to high-order differetial equatios. For exaple, high eve-order differetial equatios ofte appear i astrophysics, structural echaics ad geophysics see, e.g., [,]); high odd-order differetial equatios, such as third-order Korteweg de-vries KdV) ad fifth-order KdV-type equatios, are routiely used i o-liear wave ad o-liear optics theory see, e.g., [37,28,0,6,30]). While it is usually cubersoe to desig a accurate ad stable uerical algoriths usig fiite differece/fiite eleet ethods due to the ay boudary coditios ivolved or usig a spectral-collocatio ethod for which special quadratures ivolvig derivatives at the ed poits have to be developed cf. [7,27,29]) or fictitious poits have to be itroduced [8], the spectral approxiatios usig geeralized Jacobi polyoials/fuctios lead to straightforward ad well-coditioed ipleetatios, ad ca be aalyzed with a uified approach leadig to ore precise error estiates. This paper is orgaized as follows. I the ext sectio, we defie the geeralized Jacobi polyoials/fuctios ad aalyze the approxiatio properties of the orthogoal projectio i suitably weighted Sobolev spaces. The geeralized Jacobi polyoials/fuctios ad their approxiatio results are used i Sectio 3 to costruct ad aalyze spectral-galeri ethod for soe high-order odel equatios. Soe cocludig rears are give i the fial sectio. 2. Geeralized Jacobi polyoials/fuctios I this sectio, we defie the geeralized Jacobi polyoials/fuctios GJP/Fs), ad ivestigate their basic properties. We first itroduce soe otatios. Let ωx) be a weight fuctio i I :=, ). Oe usually requires that ω L I). However, we shall aily cocer with the cases ω L I). We shall use the weighted Sobolev spaces Hω r I) r = 0,, 2,...), whose ier products, ors ad sei-ors are deoted by, ) r,ω, r,ω ad r,ω,respectively. For real r>0, we defie the space Hω r I) by space iterpolatio. I particular, the or ad ier product

3 B.-Y. Guo et al. / Applied uerical Matheatics ) of L 2 ω I) = H ω 0I) are deoted by ω ad, ) ω, respectively. To accout for hoogeeous boudary coditios, we defie H0,ω I) = { v Hω I): v±) = xv±) = = x v±) = 0 }, =, 2,..., where x = d,. The subscript ω will be oitted fro the otatios i case of ω. dx We deote by R ad the sets of all real ubers ad o-egative itegers, respectively. For ay, let P be the set of all algebraic polyoials of degree. We deote by c a geeric positive costat idepedet of ay fuctio ad, ad use the expressio A B to ea that there exists a geeric positive costat c such that A cb. We recall that the classical Jacobi polyoials J x) 0) are defied by x) α + x) β J x) = ) d { x) +α 2! dx + x) +β}, x I. 2.) Let ω x) = x) α + x) β be the Jacobi weight fuctio. For α, β >, the Jacobi polyoials are utually orthogoal i L 2 I), i.e., ω I J x)j x)ω x) dx = γ δ,, 2.2) where δ, is the Kroecer fuctio, ad γ 2 α+β+ Ɣ + α + )Ɣ + β + ) = 2 + α + β + )Ɣ + )Ɣ + α + β + ). 2.3) The restrictio α, β > was iposed to esure that ω L I). Soe other properties of the Jacobi polyoials to be used i this paper are listed i Appedix A. I fact, Szegö etioed i [35] that oe ca defie the Jacobi polyoial with idexes α or β, based o the Rodrigues forula 2.), which is a polyoial of degree, except for + α + β + = 0, 0 l a reductio of the degree i this case). However, the so defied Jacobi polyoials do ot satisfy soe iportat properties which hold for α, β >, e.g., they are ot utually orthogoal i L 2 for all α, β. Hece, they are ot quite suitable ω for uerical coputatios. We shall defie below geeralized Jacobi polyoials/fuctios which iherit selected iportat properties of classical Jacobi polyoials) that play essetial roles i a spectral approxiatio. 2.. Defiitio of the GJP/Fs For otatioal coveiece, we itroduce the followig separable idex sets i R 2 : ℵ = { α, β): α, β }, ℵ 2 = { α, β): α,β > }, ℵ 3 = { α, β): α>,β }, ℵ 4 = { α, β): α, β > }. For ay α, β R, we defie { α, α, ˆα := 0, α>, { α, α, ᾱ := α, α > 2.4) liewise for ˆβ ad β). Throughout the paper, ˆα, ˆβ ad ᾱ, β are always defied fro α, β as above. The sybol [α] represets the largest iteger α, ad let 0 := 0 := [ ˆα]+[ˆβ], := := ) The GJP/Fs are defied by j x) = ω ˆα, ˆβ x)j ᾱ, β x), α, β R, 0,x I. 2.6) We ephasize that {j } are oly defied for 0. This fact is iplicitly assued hereafter.

4 04 B.-Y. Guo et al. / Applied uerical Matheatics ) We ca rewrite 2.6) i a ore explicit fro: x) = j x) α + x) β J α, β x), α, β) ℵ, = [ α] [ β], x) α J x), α, β) ℵ 2, = [ α], + x) β J α, β x), α, β) ℵ 3, = [ β], J x), α, β) ℵ 4. We see that the GJP/Fs are geerated fro the classical Jacobi polyoials. I fact, as etioed i Szegö [35], it is also possible to use the Rodrigues forula 2.) to defie the Jacobi polyoial J with idexes α or β. A particular case is ) ) ) + β x l J l,β x) = J l,β l x), l. 2.8) l l 2 However, there are very few discussios i [35] about the properties of the so-defied Jacobi polyoials with idexes α or β Basic properties of the GJP/Fs The GJP/Fs have the followig properties: The GJP/F j x) is a polyoial of degree if i) α, β) ℵ 4, or ii) α ad β are egative itegers. I these cases, it coicides, apart fro a costat, with the defiitio i Szegö [35]. Uder the coditio ii), x = x) resp. ˆβ). Hece, the GJP/Fs are suitable as base fuctios to approxiate solutios of high-order differetial equatios with a correspodig set of hoogeeous boudary coditios see Sectio 3 below). We fid fro 2.2), 2.6) ad 2.7) that the GJP/Fs are utually L 2 I)-orthogoal, i.e., ω resp. x = ) is the zero of ultiplicity of ˆα for the polyoial j I j x)j x)ω x) dx = η δ,, with η = γ ᾱ, β. 2.9) Here, we used the fact ᾱ = 2 ˆα + α ad β = 2 ˆβ + β. ote that polyoials of the for x) + x) l J x) with α, β > ) have bee frequetly used as basis fuctios to ipose boudary coditios, but they do ot satisfy the orthogoality relatio 2.9). We ca also view {j, l } as the orthogoalizatio of { x) + x) l J } i L 2. ω They satisfy the Stur Liouville equatio see Appedix B.):, l where x x) α+ + x) β+ x j λ = + ) α β), ) ℵ, α + β + ) αβ + ), α, β) ℵ 2, + α β + ) βα + ), α, β) ℵ 3, + α + β + ), α, β) ℵ 4, ad = 0 = [ˆα] [ˆβ] 0. The defiitio 2.7) esures that ω α+,β+ x)j x) x j sides of 2.0) ad itegratig by parts, we derive fro 2.9) that I x j x) xj We ifer fro A.3) see Appedix A) ad 2.6) that 2.7) x) ) + λ x)α + x) β j x) = 0, 2.0) x) 0as x. So ultiplyig j 2.) o both x)ω α+,β+ x) dx = λ η δ,. 2.2) j x) = ) j β,α x), x I. 2.3)

5 B.-Y. Guo et al. / Applied uerical Matheatics ) We ext study the derivative relatios of the GJP/Fs. Let us recall that for α, β >, x j x) = x J x) = α+,β+ + α + β + )j 2 x),. 2.4) Ufortuately, the GJP/Fs do ot satisfy a siilar derivative recurrece relatio for all α, β R. evertheless, soe useful derivative recurrece relatios ca be derived. Lea 2.. If oe of the followig coditios holds the i) 2; ii) α=, β 2; iii) α 2, β= ; iv) α= β =, 2.5) x j x) = 2 [ α] [ β]+ ) j α+,β+ x). 2.6) O the other had, if oe of the followig coditios holds the i) α 2, β> ; ii) α=, β>, 2.7) x j x) = [ α] α ) j α+,β+ x). 2.8) Siilarly, if oe of the followig coditios holds the i)α>, β 2; ii)α>, β=, 2.9) x j x) = [ β] β ) j α+,β+ x). 2.20) The proof of this lea is give i Appedix B.2. Applyig the forulas i Lea 2. repeatedly, we obtai the followig geeral derivative recurrece relatios: Lea 2.2. Let,l, ad,l,. We have i) If β>, the x j,β x) = D,,,β j +,β+ x), ax, ), 2.2) where D,β, = ) i=0 i),, ) Ɣ + + β + ) 2 Ɣ + β + ) i), >. i=0 2.22) ii) If α>, the x j α, x) = ) μ D,,,α α+, + x), ax, ), 2.23) where μ = for ad μ = for >. iii) If l, the x j, l x) = E,,,l j +, l+ x), ax + l,), 2.24)

6 06 B.-Y. Guo et al. / Applied uerical Matheatics ) where E,l, = 2) l + i), l, i= l ) 2 l l + i) i= ) l i=0 l ) Ɣ + l + ) 2 l l + i) Ɣ l + ) i= ) l i), l <, ) l i=0 ) l i), l <. 2.25) iv) If l, the x j, l x) = ) μ E,, l, j +, l+ x), 2.26) where μ = 0,,l for the cases l, < l ad <l, respectively. The proof of this lea is give i Appedix B Approxiatio properties of the GJP/Fs We shall aalyze below the approxiatio properties of geeralized Jacobi orthogoal projectios, which are useful i the error aalysis of spectral-galeri ethods. Sice {j } fors a coplete orthogoal syste i L 2 I), we defie ω Q := spa{ j 0,j } 0 +,...,j, 2.27) ad cosider the orthogoal projectio π : L2 I) Q ω defied by u π u, v )ω = 0, v Q. 2.28) We shall estiate the projectio errors i two differet ways. The first approach is based o the Stur Liouville equatio 2.0). The secod oe is based o the derivative relatios give i Lea 2.2. We start with the Stur Liouville operator defied by A φx):= x) α + x) β { x x) α+ + x) β+ x φx) }. 2.29) We recall that j x) are the eigefuctios of the Stur Liouville operator A with the correspodig eigevalues λ cf. 2.0)), ad defie the followig Sobolev-type spaces associated with the Stur Liouville operator: D A r ) { = u: u L 2 ω I) ad A q u L2 ω I), 0 q r }, r, D A r+/2 ) { ) = u: u D A r ad x A r u L2 ω α+,β+ I) }, r, 2.30) equipped with the ors u DA r ) = A r u ω, Usig the idetity A j u DA r ) = where û u DA r+/2 ) = =0 λ u r+/2 DA ) = x A r u ω α+,β+. = λ j repeatedly, we fid fro 2.9) ad 2.2) that for r, λ =0 ) 2rη û ) 2r+η ) /2 2, û ) /2 2, 2.3) = η ) u, j ) ω. For real r>0, we defie the space DA r ) by space iterpolatio as i [5].

7 B.-Y. Guo et al. / Applied uerical Matheatics ) Before we preset oe of our ai results, we ae the followig observatio: For ay v DA ),ifα resp. β ), the vx) 0asx resp. x ), ad by the defiitio of j,wehave ) α, β α β + α + β)x J x) + x 2 ) x J α, β x), if α, β ; ω α+,β+ x) x j + x) x) = β+{ αj x) + x) x J x) }, if α,β > ; x) α+{ βj α, β x) + + x) x J α, β x) }, if α>,β ; x) α+ + x) β+ x J x), if α, β >, where = 0 0. Thus, for ay v DA ),wehave ω α+,β+ x)vx) x j x) 0, as x, α, β) R ) Usig the idetity A j v,j )ω = λ = λ j ) v,a j agai ad itegratig by parts, we fid that for ay v DA ), )ω = λ ) A v,j ) ω. 2.33) Theore 2.. For ay u DA r/2 ), r ad 0 μ r, π u u μ/2 DA ) μ r u r/2 DA ). 2.34) Proof. The proof follows a siilar procedure used for the classical Jacobi projectios cf. [4,9,23]). We first cosider eve itegers, i.e., r = 2q for q = 0,,... We derive fro 2.3) that for μ, π u u 2 DA μ/2 ) = = =+ =+ λ λ Usig repeatedly the idetities A j u, j )ω = λ Hece, by 2.9) ad 2.), π u u 2 DA μ/2 ) = ) q A q =+ λ μ 2q + u, j λ =+ ) μη ) μ η = λ ) û ) 2 ) ) u, j 2 ω. 2.35) j ad the relatio 2.33), we derive that ω. 2.36) ) μ η η ) ) u, j 2 ω ) q ) A 2 u, j ω 2μ 2q) A q u 2 ω 2μ r) u 2 DA r/2 ). ext, we cosider odd itegers, i.e., r = 2q + forq = 0,,... We observe fro 2.0), 2.32) ad 2.36) that u, j )ω = λ ) q q A u, j )ω = λ ) q q x A u), x j ) ω α+,β+. Therefore, by 2.) ad 2.2), π u u 2 DA μ/2 ) = =+ λ μ 2q + λ ) μ η =+ λ ) ) u, j 2 ω ) ) η q x A u), x j ) 2 ω α+,β+

8 08 B.-Y. Guo et al. / Applied uerical Matheatics ) μ 2q ) =+ λ ) ) η q x A u), x j ) 2 ω α+,β+ 2μ 2q ) q x A u) 2 ω α+,β+ 2μ r) u 2 DA r/2 ). Fially the desired result with real μ follows fro the previous results ad space iterpolatio. Theore 2. provides a geeral approxiatio result for all α, β R. However, the ors used i Theore 2. are expressed by the frequecies of u i ters of j, whose relatio to the derivatives of u is ot straightforward. ext, we derive soe approxiatio results which are expressed i ters of derivatives of u. We itroduce the space B r ω I) := { u: u is easurable o I ad u r,ω < }, r, 2.37) equipped with the or ad sei-or r ) /2 u B r = ω x u 2 ω α+,β+, =0 u B r ω = r x u ω α+r,β+r. For real r>0, we defie the space H r ω I) by space iterpolatio as i [5]. Theore 2.2. Let,l ad,l. If oe of the followig coditios holds: i) α=, β > ; ii)α>, β= l; iii) α=, β = l, 2.38) the for ay u B r I), r,r ad 0 μ r, ω π u u B μ ω μ r r x u ω α+r,β+r. 2.39) Proof. We first prove 2.39) with i). I this case, π,β ux) ux) = I =+ û j,β x), with û = As a cosequece of 2.9) ad 2.2) 2.22), we have the orthogoality: x j,β x) x j,β Thas to 2.4), we deduce fro 2.40) that for μ, where x μ,β π u u ) 2 ω +μ,β+μ = C,β,μ,r = ax > { D,β,β u, j ) ω,β j,β. 2.40) 2 ω,β x)ω +,β+ x) dx = D,,β ) 2η +,β+ δ,. 2.4) =+ C,β μ,) 2 η +μ,β+μ μ D,β r, ) 2 η +r,β+r r D,β μ,,μ,r =+ }. ) 2û2 η +μ,β+μ μ D,β r, ) 2û2 η +r,β+r r C,β,μ,r We ow estiate the upper boud of C,β,μ,r. By usig the Stirlig forula cf. [5]), Ɣs + ) = 2πss s e s + Os /5 ) ), s, x r u 2 ω +r,β+r,

9 B.-Y. Guo et al. / Applied uerical Matheatics ) we derive fro 2.3), 2.9) ad 2.22) that for give ᾱ, β,,μ,γ, γ ᾱ, β, η +μ,β+μ μ η +r,β+r r, Dμ,,β Dr,,β μ r,. The above facts lead to C,μ,r 2μ 2r. This copletes the proof of 2.39) with i). The other two cases ca be proved siilarly. Rear 2.. The results for the classical Jacobi polyoials with α, β > were proved i [26]. The sae results for μ = 0orα = β were also give i [9,3], respectively. The results for the case α = ad β = l were aouced without proof) i [25]. 3. Applicatios A iportat applicatio of GJP/Fs is that they for atural basis fuctios for spectral-galeri approxiatios of differetial equatios. For exaple, oe ca verify see Appedix B) that j, x) = j 2, x) = j, 2 x) = j 2, 2 x) = 2 ) L 2 x) L x) ), 3.) 2 2 2) L 3 x) L 2x) L x) ) 2 L x), 3.2) 2 2) L 3 x) L 2x) L x) 2 3 ) 2 L x), 3.3) 4 2) 3) 22 3) L 4 x) 2 3)2 5) 2 L 2x) ) 2 L x), 3.4) where L x) is the Legedre polyoial of -th degree. The GJP/Fs i 3.) ad 3.4) were used i [32] as basis fuctios to approxiate the solutios of secod- ad forth-order equatios with hoogeeous Dirichlet boudary coditios, while the GJP/Fs i 3.2) ad 3.3) were used as basis fuctios for the test ad trial spaces i the dual- Petrov Galeri ethod for third-order differetial equatios i [33]. A ai advatage of usig the GJP/Fs as basis fuctios is that the GJP/Fs satisfy all give boudary coditios of the uderlyig proble. Hece, there is o eed to costruct special quadratures ivolvig derivatives at ed-poits as i a collocatio approach [7,27,29]. for third-order equatios ad i [7] for fourth-order equatios. The spectral approxiatios usig GJP/Fs lead to well-coditioed, sparse for probles with costat or polyoial coefficiets cf. [32,33]), systes that ca be efficietly ipleeted. Moreover, usig the GJP/Fs siplifies theoretical aalysis, ad leads to ore precise error estiates as deostrated below. 3.. Spectral-Galeri ethods for high order equatios We cosider the followig 2-th order liear equatio: 2 L 2 u := ) b 0 u 2) + b 2 u ) = f, i I,, =0 u ) ±) = 0, 0, where {b j } 0 j 2 ad f are give, ad we assue b 0 > 0. We itroduce the biliear for associated with 3.5): a u, v) = b 0 x u, x v) + ) b x u, x v) + ) b 2 x u, x v ) + +b 2 u, v), u, v H I). 3.6) As usual, we assue that the biliear for is cotiuous ad elliptic i H0 I), i.e., 3.5)

10 020 B.-Y. Guo et al. / Applied uerical Matheatics ) a u, v) C 0 u v, u, v H0 I), 3.7a) a u, u) C u 2, u H 0 I) 3.7b) where C 0 ad C are two positive costats depedig oly o b j, 0 j 2. The variatioal forulatio for 3.5) is: Give f H I), fid u H0 I) such that a u, v) = f, v), v H0 I), 3.8) ad the correspodig spectral-galeri approxiatio is: Give f C I), fid u V := P H0 I) such that a u,v ) = f, v ), v V, 3.9) where, ) is the ier product associated to the Legedre Gauss Lobatto quadrature. The well-posedess of 3.8) ad 3.9) is esured by 3.7a) 3.7b) Error estiates Let us deote π = π,. We ote iediately that x π u u ), x v ) = ) π u u, 2 x v ) = ) π u u, ω, x 2 v ) ω, = 0, v V, 3.0) which is a cosequece of 2.28) ad the fact ω, x 2v V. I other words, π is siultaeously orthogoal projectors associated with, ) ω, ad x, x ). For siplicity, we assue that {b j } are costats, ad let u ad u be respectively the solutios of 3.8) ad 3.9). The, we have the followig result: Theore 3.. Assuig u H0 I) Br ω, I) ad f x 2 ) B ρ I),, r, ρ with r, ρ, ω the for 0 μ, we have 0,0 u u μ μ r r x u ω r,r + ρ ρ x f x 2 ) ) ω ρ,ρ. 3.) Proof. We deote ê = π u u ad e = u u = u π u) +ê. We first prove 3.) for μ =. We derive fro 3.8) ad 3.9) that ) a ê,v ) = a π u u, v + f, v ) f, v ), v V. 3.2) By usig the Hardy iequality cf., for exaple, Sectio A.4 i [3]), it is easy to show that v 2 x 2 ) 2 dx x v) 2 x 2 ) 2+2 dx x v ) 2 dx, v H 0 I). 3.3) I I For v V,letf = f x 2 ) ad ṽ = v x 2 ), the by usig the properties of the Legedre Gauss Lobatto quadrature cf. [3]) ad 3.3), f, v ) f, v ) = f,ṽ ) f,ṽ ) = f π 0 f,ṽ ) I f π 0 f,ṽ ) f π 0 f + I f π 0 f ) ṽ f π 0 f + f I f ) x v C f π 0 f 2 + f I f 2) + C 2 v ) We recall fro Theore 4.0 i [26] ad Theore 2.2 with α, β) = 0, 0) that f π 0 f + f I f ρ x ρ f ω ρ,ρ. 3.5) Thas to 3.0), the first ter ivolvig the derivative of the highest order vaishes i the expressio of a π u u, v ). Moreover, we have fro 2.39) with coditio iii) that for certai suitable sall ε>0, 0, l +,+ l 2 ad,l, I

11 B.-Y. Guo et al. / Applied uerical Matheatics ) x π u u ), x l v ) x π u u ) ω +, + x l v ω, r r x u ω r,r v l ε v 2 l + c 4ε 2 2r x r u 2 ω r,r. 3.6) Taig v =ê i 3.2) ad 3.6), we derive fro 3.6), 3.7b), 3.4) ad 3.5 that C 2 ê 2 ε ê 2 + ) 2 2r r x u 2 ω r,r + 2ρ ρ x f 2 ω ρ,ρ =0 ε ê r 2 x r u 2 ω r,r + 2ρ x ρ f ω ρ,ρ. 3.7) Thus, ê r r x u ω r,r + ρ ρ x f ω ρ,ρ. 3.8) O the other had, we have fro 2.39) with coditio iii) that π u u π u u B r ω, x u ω r,r. So 3.) follows fro the triagle iequality, 3.8) ad the above estiate. We ow tur to the case μ = 0. For give g L 2 I), we cosider the auxiliary proble: Fid w H0 I) such that a z, w) = g, z), z H0 I). 3.9) We ow fro 3.7a) ad 3.7b) that 3.9) has a uique solutio with the regularity w 2 g. 3.20) Fro 3.8) ad 3.9), a u u,v ) = f, v ) f, v ), v V. 3.2) Taig z = u u i 3.9), we fid fro Theore 2.2, 3.7a), 3.), 3.20) 3.2) ad 3.4) that u u,g)= a u u,w)= a u u,w π w) + f,π w) f,π w) u u π w w + f π 0 f + I f π 0 f ) x π w r x r u ω r,r x 2 w ω, + ρ x ρ f ω ρ,ρ w r r x u ω r,r + ρ ρ x f ω ρ,ρ) g. Cosequetly, u u = sup g L 2 I) g 0 u u,g) g r u ω r,r + ρ x ρ f ω ρ,ρ. This iplies the result with μ = 0. For 0 <μ<,letθ = μ. Clearly 0 <θ<. Sice H I) is cotiuously ebedded ad dese i L 2 I),we ca defie the iterpolatio space [H I), L 2 I)] θ as i [5]. Ideed, as is show i Theore.6 of [9] see also [22]), [H I), L 2 I)] θ = H θ) I) = H μ I). Therefore, by the Gagliardo ireberg iequality ad the previous results, u u μ u u θ u u θ μ r u r,ω, + ρ x ρ f ω ρ,ρ. This eds the proof. Rear 3.. Usig the GJP/F approxiatio ot oly greatly siplifies the error aalysis, but also leads to ore precise error estiates. For istace, if we use the H0 -orthogoal projectio results i [8] ad [22], the the best error estiate will be u u r u r + ρ f ρ, 0 r. 3.22)

12 022 B.-Y. Guo et al. / Applied uerical Matheatics ) Therefore, the result 3.) is uch sharper tha 3.22) for probles with sigularities at the edpoits. As a exaple, let ux) = x) γ vx), v C I), γ >, x I, 3.23) 2γ + ε be a solutio of 3.5). It ca be easily checed that u H γ +/2 ε I) Bω, I) ε >0) ad f H γ 2+/2 ε I), f x 2 ) 2γ )+ ε B I) ε>0). Hece, Theore 3. with μ = iplies that ω 0,0 u u 2γ +2 +ε, 3.24) while the usual aalysis cf. Berardi ad Maday [7]) oly leads to u u 3 γ +2+/2+ε. 3.25) 3.3. Matrix for of 3.9) I view of the hoogeeous boudary coditios satisfied by j,,wehave V = spa { j, 2,j, 2+,...,j, }. Usig the facts that ω, x 2j, l V l ad j, is orthogoal to V l if >l, we fid that x j,, x j, ) l = ) j,, x 2 j, ) l = j,,ω, x 2 j, ) l ω, = ) By syetry, the sae is true if <l. Hece, lettig φ x) = c, j, with a suitable c,, we ca have x φ, x φ ) l = δl. Hece, by settig f = f, φ ), f = f 2,f 2+,...,f ) T, u = û l φ l, u = û 2, û 2+,...,û ) T, l=2 a l = a φ l,φ ), A = a l ) 2,l, the atrix syste associated with 3.9) becoes Au = f. 3.27) Thas to 3.7a) 3.7b), we have C 0 u 2 l 2 = C 0 u 2 a u,u ) = Au, u) l 2 C u 2 = C u 2 l 2, 3.28) which iplies that coda) C /C 0 ad is idepedet of. It ca be easily show that A is a sparse atrix with badwidth 2 +. The sae arguet as above shows that 3.28) is still valid for probles with variable coefficiets as log as 3.7a) 3.7b) are satisfied. Therefore, eve though A becoes full for probles with variable coefficiets but the product of A with a vector x ca be coputed efficietly without the explicit owledge of the etries of A so the associated liear syste ca still be solved efficietly with a suitable iterative ethod such as the Cojugate Gradiet ethod. The geeralized Jacobi polyoials/fuctios were also successfully used for uerical solutios of partial differetial equatios of odd orders cf. [25,34]).

13 B.-Y. Guo et al. / Applied uerical Matheatics ) Fig.. The axiu poitwise error arer ) ad the L 2 -error arer ) agaist various i sei-log scale for Exaple. 4. uerical results We preset soe uerical exaples to illustrate the perforace of the proposed spectral ethods usig geeralized Jacobi polyoials as basis fuctios. As a exaple, we cosider the sixth-order equatios, which are ow to arise i astrophysics [,]. I the coputatios, we use the spectral-galeri schee 3.9) with = 3 ad the } as the basis fuctios. We first cosider a exaple discussed i [7], where the uerical solutios are obtaied by a Sic-Galeri ethod. geeralized Jacobi polyoials {J 3, 3 Exaple. Cosider u 6) x) ux) = fx), x, ), 4.) with boudary coditios for u±), u ±), u ±) ad fx)such that the exact solutio is ux) = x)e x. I Fig., we plot the axiu poitwise error ad the L 2 -error agaist various. It is clear that the errors decay expoetially fast, cosistet with the results i Theore 3. sice both the solutio u ad the fuctio f are aalytic. ote that with the sae coputatioal cost, say, = 6, the Sic-ethod i [7] oly achieves a accuracy O0 4 ), see Table 4.3 i [7], while our ethod is uch ore accurate. Exaple 2. We cosider 4.) with boudary coditios for u±), u ±), u ±) ad fx) such that the exact solutio is ux) = + x) γ e x, x, ). Whe γ is ot a iteger, the solutio has a fiite regularity ad it ca be easily checed that cf. Rear 3.) 2γ 2 ε u B I), f x 2 ) 3 2γ 5 ε B I) ε>0). Hece, Theore 3. with = 3 ad μ = 3 iplies that ω 3, 3 ε 2γ +5 u u 3 ε>0). ω 0,0 We plot i Fig. 2 the H 3 -error agaist various with γ = 3., 3.5, 3.8, 4.2. ote that for these values of γ, f is ot eve i L 2 I). The approxiate slopes of these lies are respectively.9, 2.0, 2.65 ad These covergece rates are very close to the predicted covergece rate of 2γ 5 i 4.2). 4.2)

14 024 B.-Y. Guo et al. / Applied uerical Matheatics ) Fig. 2. H 3 -errors agaist various i log log scale for Exaple 2 with several γ. 5. Cocludig rears We itroduced i this paper a faily of geeralized Jacobi polyoials/fuctios with idexes α, β R based o the priciple that they are utually orthogoal with respect to the correspodig Jacobi weights ad that they iherit selected iportat properties of the classical Jacobi polyoials. We established two sets of approxiatio results by usig the Stur Liouville operator ad the derivative recurrece relatios. A iportat applicatio of GJP/Fs is to serve as basis fuctios for spectral approxiatios of differetial equatios with suitable boudary coditios which are autoatically satisfied by correspodig GJP/Fs. This is especially coveiet for high-order differetial equatios. Ulie i a collocatio ethod for which special quadratures ivolvig derivatives at the ed poits eed to be developed, the ipleetatios usig GJP/Fs are siple ad straightforward. Moreover, the use of geeralized Jacobi polyoials/fuctios leads to uch siplified aalysis, ore precise error estiates ad well coditioed algoriths. Appedix A. Properties of the classical Jacobi polyoials The classical Jacobi polyoials are the eigefuctios of the Stur Liouville proble: x x) α+ + x) β+ x J x) ) + μ x)α + x) β J x) = 0, 0, A.) with the correspodig eigevalues μ = + α + β + ). A alterative for of A.) is see [35]) x 2 ) x 2 Y + [ α β + α + β 2)x ] x Y + + ) + α + β)y = 0 A.2) where Y x) = ω x)j x) ad ω x) = x) α + x) β. The classical Jacobi polyoials with idexes α, β > satisfy the followig recurrece relatios see Szegö [35], Asey [2] ad Raiville [3]): J J x) = ) J β,α x); x) = J J x) = + α + β x)j α+,β x) = A.3) x) J α,β x), α, β > 0, ; A.4) [ + β)j x) + + α)j α,β x) ], >0; A.5) α + β + 2 [ + α + )J x) + )J + x)] ; A.6)

15 + x)j + x) = B.-Y. Guo et al. / Applied uerical Matheatics ) α + β + 2 [ + β + )J x) + + )J + x)] ; A.7) x J x) = α+,β+ + α + β + )J 2 x), ; A.8) ω x)j xj x) = a J x) = ) )! 2! x) + b J where a,b,c are costats see [35] for their expressios). Appedix B. Soe proofs B.. The proof of 2.0) d { ω α+,β+ dx x)j α+,β+ x) }, 0; A.9) x) + c J + x), A.0) We first cosider the case α, β) ℵ. Taig Y x) = ω α, β x)j α, β x 2 ) 2 x j x) + [ β α) α + β + 2)x ] x j x) + λ j x) = 0. x) = j x) i A.2), we fid that Multiplyig ω x) o both sides of the above equatio, we ca rewrite the resultig equatio as 2.0) with λ = + ) α β). that ext, let α, β) ℵ 2. By the defiitio 2.7), we have J x x) α+ + x) β+ x x) α j which ca be siplified to x 2 ) 2 x j x) = x) α j x). We plug it ito A.) to get x) )) + μ + x) β j x) = 0, x) + [ β α) α + β + 2)x ] x j x) + μ αβ + ) ) j x) = 0. Multiplyig ω x) o both sides of the above equatio, we ca get the resultig equatio 2.0) with α, β) ℵ 2. We ca prove the case α, β) ℵ 3 siilarly. Fially 2.0) with α, β) ℵ 4 is a direct cosequece of A.) ad 2.7). B.2. The proof of Lea 2. We first prove 2.6). For α, β 2, let = [ α] [ β] 0, ad by A.9) ad 2.7), j α+,β+ x) 2.7) = x) α + x) β J α, β + x) A.9) = 2 + ) x x) α + x) β J α, β x) ) 2.7) = 2 [ α] [ β]+) xj x). B.) This leads to 2.6) for the case i) of the coditio 2.5). I fact, B.) also holds for α = ad β 2, which, alog with 2.7) the cases,β) ℵ ad 0,β + ) ℵ 3 ), leads to j 0,β+ x) = + x) β J 0, β + x) = 2 + ) x ω, β x)j, β x) ) = 2 + ) xj,β x), = [ β] 0. B.2) Hece, 2.6) holds for the case ii) of the coditio 2.5). Siilarly, we ca prove the case: α 2 ad β =, while taig α = β = i B.) gives 2.6) for the case iv) of the coditio 2.5). We ow tur to the proof of 2.8). If α 2 ad β>, the, usig A.6), A.8) ad 2.7) with = [ α], yields that for 0,

16 026 B.-Y. Guo et al. / Applied uerical Matheatics ) x j x) 2.7) = x x) α J x) ) = x) α αj x) + x) x J x) ) A.8) = x) α αj x) + 2 α + β + ) x)j α+,β+ x) A.6) = x) α αj x) + α + β + α)j + 2 α + β + x) J + x) )). Usig A.4) gives J + x) = J x) J α,β+ x), ad pluggig it ito the above forula leads to { x j x) = x) α αj x) + α + β + α)j 2 α + β + x) α)j α,β+ x) J + x) )} = x) α { Thas to A.5), we have α + β + )J + Cosequetly, 2 α + β + α) α + β + ) 2 α + β + + β + )J x) α + β + )J + x) ) } J α,β+ x) x) = + β + )J x j x) = x) α α) 2 α + β + J α,β+ = α) x) α J α,β+ x) 2.7) = [ α] α ) j α+,β+ x).. x) + α)j α,β+ x). x) α) α + β + ) 2 α + β + ) ) J α,β+ x) Hece, 2.8) holds for the case i) of coditio 2.7). ote that the above procedure is also valid for α = ad β>, aely, x j,β x) = J 0,β+ x) = j 0,β+ x),. Here, we used the defiitio 2.7) with 0,β+ ) ℵ 4 ) to derive the last idetity. This iplies 2.8) for the case ii) of coditio 2.7). Fially, 2.20) ca be verified by usig the property 2.3) ad 2.8). B.3. The proof of Lea 2.2 We first prove 2.2). If, the we ow fro the coditio 2.7) that the derivative relatio 2.8) is valid for α = ad β>, ad usig it iductively leads to the desired result 2.2) i case of. ext, thas to A.8), we derive that for a,b >, x p Jq a,b Ɣq + p + a + b + ) x) = 2 p Ɣq + a + b + ) J a+p,b+p q p x), q p, p,q. B.3) Thus, for >, we deduce fro 2.8) with = ad the above forula that

17 B.-Y. Guo et al. / Applied uerical Matheatics ) x j,β x) = x x j,β x) 2.2) = ) i) 2.7) i=0 ) ) = ) i) B.3) i=0 ) = ) i) 2.7) = D,β i=0, j +,β+ x). x x j 0,β+ x) J 0,β+ x) Ɣ + + β + ) 2 Ɣ + β + ) J,β+ x) We used the defiitio 2.7) with +, β + ) ℵ 4 to derive the last idetity. The result 2.23) follows fro 2.3) ad 2.2) 2.22). We ow tur to the proof of 2.24). For the first case: l, we ca derive the result by usig 2.6) iductively. For the secod case: l<, we use the above result with = l,l < ad 2.2) with l l, to deduce that ) x j, l l x) = x l x l j, l x) = 2) l l + i) i= = 2) l l i= l + i) = E,l, j +, l+ x). ) l ) l i=0 l x j +l,0 l x) ) l i) j +, l x) We ca prove the result with l< i the sae aer. Fially, the result 2.26) follows fro 2.25) ad 2.3). B.4. Derivatio of 3.) 3.4) Let α, β < ad = [ α] [ β]. By the defiitio 2.7) ad A.6), A.7), j α,β x) = j x) = 2 2 α β 2 2 α β Hece, taig α = β = 0 leads to [ α)j x) j x) ], [ α)j x) + j x) ]. B.4) j,0 x) = L x) L x), j 0, x) = L x) + L x). B.5) ext, we verify fro A.6), A.7) that for a,b >, where x 2 )J a+,b+ A a,b = x) = A a,b J a,b x) + Ba,b J a,b 4 + a) + b) 2 + a + b)2 + a + b + ), Ba,b = x) + C a,b J a,b + x), B.6) 4a b) 2 + a + b)2 + a + b + 2), C a,b 4 + ) = 2 + a + b + )2 + a + b + 2). B.7) Taig a = α, b = β ad =, we derive fro B.6), B.7) ad the defiitio 2.7) that j α,β + x) = A α, β j x) + B α, β j x) + C α, β j + x). B.8)

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