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1 Joural of omputatoal Physcs 5 (013) otets lsts avalable at ScVerse SceceDrect Joural of omputatoal Physcs Fractoal Sturm Louvlle ege-problems: Theory ad umercal approxmato Mohse Zayerour, George Em Karadas Dvso of Appled Mathematcs, Brow Uversty, 18 George, Provdece, RI 091, USA artcle fo abstract Artcle hstory: Receved 11 March 013 Receved revsed form 5 Jue 013 Accepted 7 Jue 013 Avalable ole 4 July 013 Keywords: Regular/sgular fractoal Sturm Louvlle operators Poly-fractoomal egefuctos Fractal expaso set Expoetal covergece We frst cosder a regular fractoal Sturm Louvlle problem of two ds RFSLP-I ad RFSLP-II of order ν (0, ). The correspodg fractoal dfferetal operators these problems are both of Rema Louvlle ad aputo type, of the same fractoal order μ ν/ (0, 1). We obta the aalytcal egesolutos to RFSLP-I & -II as opolyomal fuctos, whch we defe as Jacob poly-fractoomals. These egefuctos are orthogoal wth respect to the weght fucto assocated wth RFSLP-I & -II. Subsequetly, we exted the fractoal operators to a ew famly of sgular fractoal Sturm Louvlle problems of two ds, SFSLP-I ad SFSLP-II. We show that the prmary regular boudary-value problems RFSLP-I & -II are deed asymptotc cases for the sgular couterparts SFSLP-I & -II. Furthermore, we prove that the egevalues of the sgular problems are real-valued ad the correspodg egefuctos are orthogoal. I addto, we obta the ege-solutos to SFSLP-I & -II aalytcally, also as opolyomal fuctos, hece completg the whole famly of the Jacob poly-fractoomals. I umercal examples, we employ the ew poly-fractoomal bases to demostrate the expoetal covergece of the approxmato agreemet wth the theoretcal results. 013 Elsever Ic. All rghts reserved. 1. Itroducto The Sturm Louvlle theory has bee the eystoe for the developmet of spectral methods ad the theory of self-adot operators [1]. For may applcatos, the Sturm Louvlle Problems (SLPs) are studed as boudary-value problems []. However, to date mostly teger-order dfferetal operators SLPs have bee used, ad such operators do ot clude ay fractoal dfferetal operators. Fractoal calculus s a theory whch ufes ad geeralzes the otos of teger-order dfferetato ad tegrato to ay real- or complex-order [3 5]. Over the last decade, t has bee demostrated that may systems scece ad egeerg ca be modeled more accurately by employg fractoal-order rather tha teger-order dervatves [6 8]. I most of the fractoal Sturm Louvlle formulatos preseted recetly, the ordary dervatves a tradtoal Sturm Louvlle problem are replaced wth fractoal dervatves, ad the resultg problems are solved usg some umercal schemes such as Adoma decomposto method [9], or fractoal dfferetal trasform method [10], or alteratvely usg the method of Haar wavelet operatoal matrx [11]. However, such umercal studes, roud-off errors ad the pseudo-spectra troduced approxmatg the fte-dmesoal boudary-value problem as a fte-dmesoal egevalue problem prohbt computg more tha a hadful of egevalues ad egefuctos wth the desred precso. Furthermore, these papers do ot exame the commo propertes of Fractoal Sturm Louvlle Problems (FSLPs) such as orthogoalty of the egefuctos of the fractoal operator addto to the realty or complexty of the egesolutos. * orrespodg author. E-mal address: g@dam.brow.edu (G.E. Karadas) /$ see frot matter 013 Elsever Ic. All rghts reserved.

2 496 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Establshg the aforemetoed fudametal propertes for FSLPs s very mportat establshg proper umercal methods, e.g. the egesolutos of the RFSLP may be complex [1]. To ths ed, some results have bee recetly provded [13,14], where the fractoal character of the problem has bee cosdered through defg a classcal Sturm Louvlle operator, exteded by the term that cludes a sum of the left- ad rght-sded fractoal dervatves. More recetly, a Regular Fractoal Sturm Louvlle Problem (RFSLP) of two types has bee defed [15], where t has bee show that the egevalues of the problem are real, ad the egefuctos correspodg to dstct egevalues are orthogoal. However, the dscreteess ad smplcty of the egevalues have ot bee addressed. I addto, the spectral propertes of a regular FSLP for dffuso operator have bee studed [16] demostratg that the fractoal dffuso operator s self-adot. The recet progress FSLPs s promsg for developg ew spectral methods for fractoal PDEs, however, the egesolutos have ot bee obtaed explctly ad o umercal approxmato results have bee publshed so far. The ma cotrbuto of ths paper s to develop a spectral theory for the Regular ad Sgular Fractoal Sturm Louvlle Problems (RFSLP & SFSLP) ad demostrate ts utlty by costructg explctly proper bases for umercal approxmatos of fractoal fuctos. To ths ed, we frst cosder a regular problem of two ds,.e., RFSLP-I & -II. The, we obta the aalytcal egesolutos to these problems explctly for the frst tme. We show that the explct egevalues to RFSLP-I & -II are real, dscrete ad smple. I addto, we demostrate that the correspodg egefuctos are of o-polyomal form, called Jacob poly-fractoomals. We also show that these egefuctos are orthogoal ad dese L w [, 1], formga complete bass the Hlbert space. We subsequetly exted the regular problem to a sgular fractoal Sturm Louvlle problem aga of two ds SFSLP-I & -II, ad prove that the egevalues of these sgular problems are real ad the egefuctos correspodg to dstct egevalues are orthogoal; these too are computed aalytcally. We show that the egesolutos to such sgular problems share may fudametal propertes wth ther regular couterparts, wth the explct egefuctos of SFSLP-I & -II completg the famly of the Jacob poly-fractoomals. Fally, we complete the spectral theory for the regular ad sgular FSLPs by aalyzg the approxmato propertes of the egefuctos of RFSLPs ad SFSLPs, whch are employed as bass fuctos approxmato theory. Our umercal tests verfy the theoretcal expoetal covergece approxmatg o-polyomal fuctos L w [, 1]. We compare wth the stadard polyomal bass fuctos (such as Legedre polyomals) demostratg the fast expoetal covergece of the poly-fractoomal bases. I the followg, we frst preset some prelmary of fractoal calculus Secto, ad we proceed wth the theory o RFSLP ad SFSLP Sectos 3 ad 4. I Secto 5 we preset umercal approxmatos of selected fuctos ad we summarze our results Secto 6.. Deftos Before defg the boudary-value problem, we start wth some prelmary deftos of fractoal calculus [4]. The left-sded ad rght-sded Rema Louvlle tegrals of order μ, whe 0 < μ < 1, are defed, respectvely, as ad ( I μ ) x f (x) 1 Γ(μ) x ( x I μ ) x R f (x) 1 x R Γ(μ) x f (s) ds (x s) 1 μ, x >, (1) f (s) ds (s x) 1 μ, x < x R, () where Γ represets the Euler gamma fucto. The correspodg verse operators,.e., the left-sded ad rght-sded fractoal dervatves of order μ, are the defed based o (1) ad (), as ad ( D μ ) x f (x) d ( x d I 1 μ ) x f (x) 1 d Γ(1 μ) dx ( x D μ ) x R f (x) d ( x I 1 μ ) ( 1 d x dx R f (x) Γ(1 μ) dx x f (s) ds (x s) μ, x >, (3) ) x R x f (s) ds (s x) μ, x < x R. (4) Furthermore, the correspodg left- ad rght-sded aputo dervatves of order μ (0, 1) are obtaed as ( xl D μ ) ( ) x f (x) I 1 μ df 1 x (x) dx Γ(1 μ) x f (s) ds (x s) μ, x >, (5)

3 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) ad ( x D μ ) ( x R f (x) x I 1 μ df x R dx ) (x) 1 Γ(1 μ) x f (s) ds (x s) μ, x < x R. (6) The two deftos of the left- ad rght-sded fractoal dervatves of both Rema Louvlle ad aputo type are led by the followg relatoshp, whch ca be derved by a drect calculato ad ( D μ ) f ( ) x f (x) Γ(1 μ)(x ) μ + ( xl D μ ) x f (x), (7) ( x D μ x R f ) (x) f (x R ) Γ(1 μ)(x R x) μ + ( x D μ x R f ) (x). (8) Moreover, the fractoal tegrato-by-parts for the aforemetoed fractoal dervatves s obtaed as ad x R x R f (x) μ x Dx R g(x) dx f (x) D μ x g(x) dx x R x R g(x) D μ x f (x) dx f (x) x Iμ x R g(x) x R x, (9) g(x) x D μ x R f (x) dx + f (x) I μ x g(x) x R. (10) x Fally, we recall a useful property of the Rema Louvlle fractoal dervatves. Assume that 0 < p < 1 ad 0 < q < 1ad f ( ) 0 x >,the D p+q x f (x) ( D p )( x D q ) ( x f (x) D q )( x D p ) x f (x). (11) 3. Part I: Regular fractoal Sturm Louvlle problems of d I & II We cosder a regular fractoal Sturm Louvlle problem (RFSLP) of order ν (0, ) [15], where the dfferetal part cotas both left- ad rght-sded fractoal dervatves, each of order μ ν/ (0, 1) as D μ[ p (x) D μ Φ () λ (x)] + q (x)φ () λ (x) + λw (x)φ () λ (x) 0, x [, x R ], (1) where {1, }, wth 1 deotg the RFSLP of frst d, where D μ x Dμ x R (.e., rght-sded Rema Louvlle fractoal dervatve of order μ) ad D μ x L D μ x (.e., left-sded aputo fractoal dervatve of order μ), ad correspodgtotherfslpofsecoddwhch D μ D μ x ad D μ x Dμ x R (.e., respectvely, left-sded Rema Louvlle ad rght-sded aputo fractoal dervatve of order μ). I such settg, μ (0, 1), p (x) 0, w (x) s a o-egatve weght fucto, ad q (x) s a potetal fucto. Also, p, q ad w are real-valued cotuous fuctos the terval [, x R ]. The boudary-value problem (1) s subect to the boudary codtos a 1 Φ () λ () + a I 1 μ[ p (x) D μ Φ () λ (x)] xxl 0, (13) b 1 Φ () λ (x R) + b I 1 μ[ p (x) D μ Φ () λ (x)] xxr 0, (14) where a 1 + a 0, b 1 + b 0. I ths otato, I 1 μ x I1 μ x R (.e., rght-sded Rema Louvlle fractoal tegrato of order 1 μ) whe 1 for RFSLP of frst d, whle, I 1 μ I 1 μ x (.e., left-sded Rema Louvlle fractoal tegrato of order 1 μ) whe for RFSLP of secod d. The problem of fdg the egevalues λ such that the boudary-value problems (1) (14) have o-trval solutos yelds the egefucto of the regular fractoal Sturm Louvlle egevalue problem. The followg theorem characterzes the egesolutos we obta: Theorem 3.1. (See [15].) The egevalues of (1) are real, ad the egefuctos, correspodg to dstct egevalues each problem, are orthogoal wth respect to the weght fuctos w (x).

4 498 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Regular boudary-value problem defto I ths study, we shall solve two partcular forms of RFSLP (1) (14) deoted by RFSLP-I whe 1 ad RFSLP-II whe oforderν μ (0, ), where the potetal fuctos q (x) 0, both problems. To ths ed, the followg o-local dfferetal operator s defed L μ : D μ[ K D μ ( ) ], (15) where K s costat, ad by the otato we troduced, L μ 1 : x Dμ x R [Kx L D μ x ( )] RFSLP-I (.e., frst we tae the left-sded μ-th order aputo dervatve of the fucto multpled by a costat, ad the we tae the rght-sded Rema Louvlle dervatve of order μ), ad for the case of RFSLP-II we reverse the order of the rght-sded ad left-sded dervatve for the : D μ x [K x Dμ x R ( )], where μ (0, 1). I fact, we have set er ad outer fractoal dervatves the operator,.e., L μ p (x) K, a cotuous o-zero costat fucto x [, 1]. We referred to K as stffess costat, whch yelds the regularty character to the boudary-value problem. That beg defed, we cosder the RFSLP (-I & -II) as L μ Φ () λ (x) + λ(1 x) μ (1 + x) μ Φ () λ (x) 0, {1, }, x [, 1]. (16) We shall solve (16) subect to a homogeeous Drchlet ad a homogeeous fractoal tegro-dfferetal boudary codto to the problems RFSLP-I ad RFSLP-II, respectvely, as ad Φ (1) λ () 0, x I 1 μ [ 1 K Dμ x Φ (1) λ (x)] x+1 0, (+1) 0, [ I1 μ x K x D μ 1 Φ() λ (x)] x 0. Φ () λ The boudary codtos (17) ad (18) are atural o-local calculus ad fractoal dfferetal equatos, ad they are motvated by the fractoal tegrato-by-parts (9) ad (10). I fact, the fudametal propertes of egesolutos the theory of classcal Sturm Louvlle problems are coected wth the tegrato-by-parts formula ad the choce of the boudary codtos. For stace, the cotuty or dscreteess of the ege-spectrum boudary-value problems s hghly depedet o the type of boudary codtos eforced. I the settg chose here, we shall show that the ege-spectra of RFSLP-I ad RFSLP-II are smple ad fully dscrete. 3.. Aalytcal egesolutos to RFSLP-I & -II Here, we obta the aalytcal soluto Φ () λ (x) to RFSLP-I & -II, (16), subect to the homogeeous Drchlet ad tegrodfferetal boudary codtos (17) ad (18). Before that, we recall the followg lemmas for the stadard Jacob polyomals P α,β : Lemma 3.. (See [17].) For μ > 0, α >, β>,ad x [, 1] (1 + x) (x) () Γ(β+ μ + 1) Γ(β+ 1)Γ (μ)p α,β () α μ,β+μ β+μ P P α μ,β+μ x (17) (18) (1 + s) β P α,β (s) (x s) 1 μ ds. (19) By the left-sded Rema Louvlle tegral (1) ad evaluatg the specal ed-values P α μ,β+μ () ad P α,β (), we ca re-wrte (19) as { Iμ x (1 + x) β α,β P (x) } Γ( + β + 1) Γ( + β + μ + 1) (1 + x)β+μ P α μ,β+μ (x). (0) Lemma 3. cabereducedtothecasewheα +μ ad β μ as { Iμ x (1 + x) μ μ, μ P (x) } Γ( μ + 1) P (x), (1) Γ( + 1) where P (x) P 0,0 (x) represets the Legedre polyomal of degree. O the other had, we ca set α β 0(0) ad tae the fractoal dervatve Dμ x o both sdes of (0) to obta Dμ x { (1 + x) μ P μ,μ } Γ( + μ + 1) Γ( + 1) P (x). ()

5 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Lemma 3.3. (See [17].) For μ > 0, α >, β>,ad x [, 1] (1 x) (x) (+1) Γ(α + μ + 1) Γ(α + 1)Γ (μ)p α,β (+1) α+μ,β μ α+μ P P α+μ,β μ x (1 s) α P α,β (s) (s x) 1 μ ds. (3) By the rght-sded Rema Louvlle tegral () ad evaluatg the specal ed-values P α μ,β+μ (+1) ad P α,β (+1), we ca re-wrte (3) as x I μ 1 { (1 x) α α,β P (x) } Γ( + α + 1) Γ( + α + μ + 1) (1 x)α+μ P α+μ,β μ (x). (4) Smlarly, Lemma 3.3 for the case α μ ad β +μ leads to x I μ 1 { (1 x) μ P μ,+μ (x) } Γ( μ + 1) P (x). (5) Γ( + 1) O the other had, oe ca set α β 0(4) ad tae the fractoal dervatve x Dμ 1 f x Dμ 1 { (1 x) μ P μ, μ } Γ( + μ + 1) Γ( + 1) Relatos (1), (), (5) ad (6) are the ey to provg the followg theorem. o both sdes of (4) to obta P (x). (6) Theorem 3.4. The exact egefuctos to (16), whe 1,.e., RFSLP-I, subect to (17) are gve as Φ (1) (x) (1 + x) μ P μ,μ (x), 1, ad the correspodg dstct egevalues are λ (1) KΓ( + μ), 1. Γ( μ) Moreover, the exact egefuctos to (16), whe,.e., RFSLP-II subect to (18), aregveas (7) (8) Φ () (x) (1 x) μ P μ, μ (x), 1 (9) where the correspodg dstct egevalues are gve as λ () λ (1) KΓ( + μ), 1. Γ( μ) Proof. We splt the proof to three parts. Part a: Frst,weprove(7) ad (8). From(7), t s clear that Φ (1) sce Φ (1) () 0, by property (7), we could replace Dμ x { x I 1 μ [ +1 K { x I 1 μ [ K +1 { x I 1 μ +1 Dμ x Φ (1) (x) ]} x+1 Dμ x Φ (1) [ K Dμ x (x) ]} x+1 ( (1 + x) μ μ,μ P (x))]} x+1 by Dμ x,hece, (ad by carryg out the fractoal dervatve the bracet usg ()) { [ ]} x I 1 μ Γ( + μ) Γ( + μ) { +1 K P (x) K x I 1 μ [ +1 Pf (x) ]} Γ() x+1 Γ(). x+1 By worg out the fractoal tegrato usg (4) (whe α β 0), we obta K { (1 x) μ P μ, μ } 0. x+1 (30) () 0. To chec the other boudary codto, Now, we eed to show that (7) deed satsfes (16), whe 1, wth the egevalues (8). Frst, we tae the fractoal tegrato of order μ o both sdes of (16) tag 1, K Dμ x Φ (1) (x) λ x Iμ +1 { (1 x) μ (1 + x) μ Φ (1) (x) }.

6 500 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Substtutg (7) ad replacg the aputo dervatve by the Rema Louvlle oe, thas to (7), weobta [ K Dμ x (1 + x) μ μ,μ P (x)] { λ x Iμ +1 (1 x) μ μ,μ P (x)}. Fally, the fractoal dervatve o the left-had sde ad the fractoal tegrato o the rght-had sde are wored out usg () ad (5), respectvely, as Γ( + μ) Γ( μ) K P (x) λ P (x). Γ() Γ() Sce, P α+1, β (x) s o-zero almost everywhere [, 1], we ca cacel ths term out from both sdes ad get λ λ (1) KΓ( + μ), 1, Γ( μ) whch shows that the egevalues of RFSLP-I are real-valued ad dscrete. I fact, ths result agrees wth Theorems 3.1. Moreover, the orthogoalty of the egefuctos (7) wth respect to w 1 (x) (1 x) μ (1 + x) μ s show as w 1 (x)φ (1) (x)φ (1) (x) dx w 1 (x) [ (1 + x) μ] μ,μ P (1 x) μ (1 + x) μ P μ,μ μ, μ (x)p (x) dx μ,μ (x)p (x) dx μ,μ δ, where μ,μ deotes the orthogoalty costat of the famly of Jacob polyomals wth parameters μ,μ. Part b: The proof of the ege-soluto to RFSLP-II, (9) ad (30), follows the same steps as Part a. It s clear that Φ () (1) 0. To chec the other boudary codto (18), sceφ () (1) 0, by (8), we ca replace x Dμ 1 by x Dμ 1 ;hece, by substtutg (9), ad worg out the mddle fractoal dervatve usg (6), { x I 1 μ +1 [ K x D μ 1 Φ(1) (x) ]} x K Γ() Γ( + μ) { I1 μ x [ P (x) ]} x, ad by worg out the fractoal tegrato usg (4) (whe α β 0), we obta K { (1 + x) μ P μ,+μ } 0. x To chec f (9) satsfes (16), whe, we ca substtute (9) to (16) ad carry out the fractoal tegrato of order μ o both sdes usg (0). The, by worg out the fractoal dervatve o the left-had sde usg (5) we verfy that (9) satsfes the boudary-value problem, provded that (30) are the real-values dstct egevalues of RFSLP-II. Fally, the orthogoalty of the egefuctos (9) wth respect to w (x) (1 x) μ (1 + x) μ s show as w (x)φ () (x)φ () (x) dx (1 x) μ (1 + x) μ P μ, μ μ, μ (x)p (x) dx μ, μ δ, where μ, μ represets the orthogoalty costat of the famly of Jacob polyomals wth parameters μ, μ. Part c: It s left to prove that the set{φ () (x): 1,,...} forms a bass for the fte-dmesoal Hlbert space L w [, 1], ad λ(), the correspodg egevalue for each, s smple. Let f (x) L w [, 1] ad the clearly g(x) (1 ± x) μ f (x) L w [, 1], as well whe μ (0, 1). Hece N a Φ () (x) f (x) 1 L w [,1] N a (1 ± x) μ P μ,±μ (x) f (x) 1 L w [,1] ( N (1 ± x)μ a P μ,±μ (x) (1 ± x) μ f (x)) 1 L w [,1]

7 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) ( N (1 ± x)μ a P μ,±μ (x) g(x)) 1 (1 ± x) μ N L w [,1] a P μ,±μ (x) g(x) 1 N c a P μ,±μ (x) g(x), 1 L w [,1] L w [,1] L w [,1] (by auchy Schwartz) where the upper sgs are correspodg to RFSLP-I, 1, ad the lower sgs are correspodg to the case,.e., RFSLP-II. Hece, N lm a N Φ () (x) f (x) lm c N a N P μ,±μ (x) g(x) 0, (31) 1 L w [,1] 1 L w [,1] by Weerstrass theorem. Therefore, N 1 a Φ () (x) L w f (x), mplyg that {Φ () (x): 1,,...} s dese the Hlbert space ad t forms a bass for L w [, 1]. To show the smplcty of the egevalues, assume that correspodg to the egevalue λ (), there exts aother ege- fucto Φ () (x) L w (x),. By the desty of the egefuctos set,.e., (31), we ca represet Φ () Φ () Φ () (x) 1 [, 1] addto to Φ() (x), whch s by Theorem 3.1 orthogoal to the rest of the egefuctos (x) as a Φ () (x). Now, by multplyg both sdes by Φ () (x), 1,,... ad, ad tegratg wth respect to the weght fucto w(x) we obta w(x)φ () (x)φ () (x) dx 1 (3) a w(x)φ () (x)φ () (x) dx a 0, (33) whch cotradcts to Theorem 3.1. Therefore, the egevalues λ () are smple, ad ths completes the proof. The growth of the magtude of the egevalues of RFSLP-I & -II, λ, {1, }, splottedfg. 1, correspodg to three values of μ 0.35, μ 0.5, ad μ We observe that there are two growth modes, depedg o ether μ (0, 1/), s observed, or, μ (1/, 1), where a superlear subquadratc growth mode s Φ (1) where a sublear growth λ 1 λ otced. The case μ 1/ leads to a exactly lear growth mode. I order to vsually get more sese of how the egesolutos loo le, Fg. we plot the egefuctos of RFSLP-I, (x) of dfferet orders ad correspodg to dfferet values of μ used Fg. 1. I each plot we compare the egesolutos wth the correspodg stadard Jacob polyomals P μ,μ (x). I a smlar fasho, we plot the egefuctos of RFSLP-II, Φ () (x), of dfferet orders compared to P +μ, μ (x) Fg. 3. So far, the egefuctos have bee defed the terval [, 1]. The followg lemma provdes a useful shfted defto of the Φ (), whch s ot oly more coveet to wor wth but also helps explot some terestg propertes. Fg. 1. Magtude of the egevalues of RFSLP-I ad RFSLP-II, λ (1) λ (), versus, correspodg to μ 0.35, left: sublear growth, μ 0.5; mddle: lear growth, ad μ 0.99; rght: superlear subquadratc growth. The blue le deotes the lear growth.

8 50 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg.. Egefuctos of RFSLP-I, Φ (1),versusx, for 1 (frst row), (secod row), 5 (thrd row), ad 10 (last row), correspodg to the fractoal order μ ν/ 0.35 (left colum), μ ν/ 0.5 (mddle colum), ad μ ν/ 0.99 (rght colum). Lemma 3.5. The shfted egesolutos to RFSLP-I & -II, deoted by Φ () (t), {1, },aregveby ) Φ () (t) μ ( () )( 1 + () +1 μ 1 where t [0, 1] the mapped doma, case of RFSLP-I, ad t [, 0] RFSLP-II. t +μ, (34)

9 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg. 3. Egefuctos of RFSLP-II, Φ (),versusx, for 1 (frst row), (secod row), 5 (thrd row), ad 10 (last row), correspodg to the fractoal order μ ν/ 0.35 (left colum), μ ν/ 0.5 (mddle colum), ad μ ν/ 0.99 (rght colum). Proof. We frst obta the shfted RFSLP-I by performg a affe mappg from terval [, 1] to [0, 1]. Todoso,we recall the power expaso of the Jacob polyomal P α,β (x) as ( )( )( ) P α,β + α + β + + α x 1 (x), x [, 1] (35) 0

10 504 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) ad from the propertes of the Jacob polyomals we have P α,β ( x) () P β,α (x). We obta the shfted egesoluto Φ (1) (t) utlzg (35) ad (36) (7) ad performg the chage of varable x t 1 as Φ (1) (t) μ ( () (36) )( ) 1 + μ t +μ. (37) 1 I order to obta the shfted Φ () (t), we follow the same steps, except that ths tme we do the chage of varable x t + 1, whch maps [, 1] to [, 0]. Defto 3.6. A fractoomal s defed as a fucto f : of o-teger power, deoted as t +μ, where Z + ad μ (0, 1), whch the power ca be represeted as a sum of a teger ad o-teger umber. Moreover, deoted by F +μ e s the fractal expaso set, whch s defed as the set of all fractoomals of order less tha or equal + μ as F +μ e spa { t +μ : μ (0, 1) ad 0, 1,..., }. (38) Remar 3.7. All fractoomals are zero-valued at t 0. Moreover, asymptotcally, whe μ 0, a fractoomal of order +μ approaches the moomal t. Defto 3.8. A poly-fractoomal of order + μ <, {0, 1,,..., N < }, ad μ (0, 1), s defed as a lear combato of the elemets the fractal expaso set F +μ e,as F +μ (t) a 0 t μ + a 1 t 1+μ + +a t +μ, where a, {0, 1,...,} are costats. Moreover, deoted by F +μ s the space of all poly-fractoomals up to order + μ. ByRemar 3.7, all elemets F +μ asymptotcally approach the correspodg stadard polyomal of order wth coeffcets a. Remar 3.9. It s observed that F +μ L w sce μ (0, 1), ad hece, all poly-fractoomals F+μ ca be represeted as a fte sum terms of the shfted egefuctos of RFSLP-I & -II. It s true by the desty of the egefucto L w, show Part c of the proof Theorem 3.4. Lemma Ay fractoal aputo dervatve of order μ (0, 1) of all polyomals up to degree N les the space of polyfractoomals F + μ,where N 1,ad μ 1 μ (0, 1). Proof. Let f (t) N 0 a t be a arbtrary polyomal of order N,.e., a N 0. From [4] ad for μ (0, 1), wehave { 0, < μ, 0 Dμ t t Hece, by (39), Γ(+1) Γ(+1 μ) t μ, 0 < μ. (39) where b proof. 0 Dμ t f (t) N 0 a 0 Dμ t t N 1 a Γ( + 1) Γ( + 1 μ) t μ Γ(+1) Γ(+1 μ) a.tag N 1 ad μ 1 μ (0, 1), ad the fact that b N N b t μ, (40) 1 Γ(+1) Γ(+1 μ) a N 0 completes the Theorem The shfted egesolutos to (16), Φ () (t), N ad <, form a complete herarchcal bass for the ftedmesoal space of poly-fractoomals F +μ,whereμ (0, 1). Proof. From (34), t s clear that dm F +μ dm { Φ (), {1,,...,}}. (41) Moreover, we ca re-wrte (34) as T t Φ (), (4)

11 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) where t t μ t 1+μ.. t +μ ad Φ (1) 1 (t) (t),. (t) Φ () Φ () Φ () ad fally, T {T } s a matrx obtaed as,1 T {T },1 ()+ ( 1 + )( ) 1 + () +1 μ, 1 whch s a lower-tragular matrx. Thas to the orthogoalty of the Φ (), the egefuctos are learly depedet, therefore, the matrx T s vertble. Let T T, whch s also lower tragular. Hece, t T Φ (). (43) I other words, each elemet the poly-fractoomal space F +μ,sayt m+μ,0 m 1, ca be uquely represeted through the followg expaso t m+μ 1 c Φ () (t) m {T m }Φ () (t) {T m }Φ () (t), (44) 1 1 where the last equalty holds sce T s a lower-tragular matrx. As see (38), the fractal expaso set F +μ e F +1+μ e, whch dcates that the shfted ege-solutos Φ () form a herarchcal expaso bass set Propertes of the egesoluto to RFSLP-I & -II Next, we lst a umber of propertes of the solutos to RFSLP-I & -II (16): No-polyomal ature: From Φ () (x) show (7) whe 1, ad (9) correspodg to, t s uderstood that the egefuctos exhbt a o-polyomal behavor, thas to the multpler (1 ± x) μ of fractoal power. Hece, to dstgush them from the stadard Jacob polyomals, we refer to Φ () (x) as Jacob poly-fractoomal of order + μ. Asymptotc egevalues λ () : The growth the magtude of egevalues RFSLP wth s depedet o the fractoal dervatve order μ, as show (30). Sceμ (0, 1), there are two modes of growth the magtude of λ (),thesublear mode correspodg to 0 < μ < 1/, ad superlear subquadratc mode whch correspods to 1/ < μ < 1. Partcularly, whe μ 1/, the egevalues grow learly wth. Hece, the asymptotc values are summarzed as λ () K, μ 1, K, μ 1/, (45) K, μ 0. Recurrece relatos: Thas to the herarchcal property of the egefuctos Φ (), we obta the followg recurrece relatos: Φ () 1 (x) (1 ± x)μ, Φ () (x) (1 ± x)μ (x μ),. a Φ () +1 (x) b xφ () (x) c Φ () (x), a 4 ( 1), b ( 1)( ), c 4( 1 μ)( 1 ± μ), where the upper sgs correspod to 1, soluto to RFSLP-I, ad the lower sgs correspod to RFSLP-II whe. (46)

12 506 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Orthogoalty: (1 x) μ (1 + x) μ Φ () (x)φ() m (x) dx J α,β 1 (1 x) α (1 + x) β P α,β (x)p α,β m (x) dx J α,β δ, (47) Γ( + α )Γ ( + β ), (48) ( 1)!Γ() where (α 1,β 1 ) ( μ, μ) ad (α,β ) (μ, μ). Fractoal dervatves: Dμ x Φ (1) Dμ x Φ (1) x Dμ 1 Φ() x Dμ 1 Φ() where P (x) deotes that stadard Legedre polyomal of order 1. Orthogoalty of the fractoal dervatves: Γ( + μ) P (x), (49) Γ() ( ) D μ Φ () Dμ Φ () Γ( + μ) dx Γ() 1 δ, (50) where D μ ca be ether Dμ x Frst dervatves: dφ (1) (x) dx dφ () (x) dx Specal values: Φ (1) () 0, Φ (1) or Dμ x, whe 1, ad ca be ether x Dμ 1 or x Dμ 1 whe. μ(1 + x) μ P μ,μ (x) + (1 + x)μ P 1 μ,1+μ (x), (51) μ(1 x) μ P μ, μ (x) + (1 x)μ P 1+μ,1 μ (x). (5) ( (+1) μ 1 μ 1 Φ () (+1) 0, ), Φ () () () Φ (1) (+1). 4. Part II: Sgular fractoal Sturm Louvlle problems of d I & II I the secod part of the paper, we beg wth our defto of the sgular fractoal Sturm Louvlle of frst d I (SFSLP-I) ad secod d II (SFSLP-II) of order ν μ (0, ), wth parameters < α < μ, ad <β<μ 1 SFSLP-I ( 1), ad < α < μ 1, ad <β< μ SFSLP-II ( ), for x [, 1] as D μ{ (1 x) α+1 (1 + x) β+1 D μ P () (x) } + Λ () (1 x) α+1 μ (1 + x) β+1 μ P () (x) 0, (57) where the fractoal order μ (0, 1) ad {1, }, where 1 deotes the SFSLP-I whch D μ x Dμ +1 ad D μ Dμ x ;also correspods to the RFSLP-II where D μ Dμ x ad D μ x Dμ +1. The sgular boudary-value problem s subect to the followg boudary codtos P ()( () ) 0, (58) { I 1 μ[ p(x) D μ P () (x) ]} 0, x() +1 (59) (53) (54) (55) (56) where I 1 μ x I1 μ +1 whe 1 SFSLP-I, ad I 1 μ I1 μ x case of SFSLP-II; p(x) (1 x) α+1 (1 + x) β+1, used the fractoal dfferetal operator (57), vashes at the boudary eds x ±1. We also ote that the weght fucto w(x) (1 x) α+1 μ (1 + x) β+1 μ (57) s a o-egatve fucto. Theorem 4.1. The egevalues of SFSLP-I & -II (57) (59) are real-valued, moreover, the egefuctos correspodg to dstct egevalues of SFSLP-I & -II (57) (59) are orthogoal wth respect to the weght fucto w(x) (1 x) α+1 μ (1 + x) β+1 μ.

13 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Proof. Part a: LetL α,β;μ be the fractoal dfferetal operator of order μ as L α,β;μ : D μ{ (1 x) α+1 (1 + x) β+1 D μ ( ) }, (60) ad assume that Λ () s the egevalue of (57) (59) correspodg the egefucto η () (x), where {1, }. The the followg set of equatos are vald for η () (x) L α,β;μ η () (x) + Λ () w(x)η () (x) 0 (61) subect to the boudary codtos η ()( () ) 0, { I 1 μ[ p(x) D μ η () (x) ]} x() (+1) 0, ad ts complex cougate η () (x) L α,β;μ I η () (x) + Λ () w(x) η () (x) 0, correspodg to the followg boudary codtos (6) η ()( () ) 0, { I 1 μ[ p(x) D μ η () (x) ]} x() Now, we multply (61) by η () (x), ad (6) by η () (x) ad subtract them as ( Λ () Λ ()) w(x)η () (x) η () (x) η () (x)l α,β;μ η () (x) η () (x)l α,β;μ η () (x). (63) Itegratg over the terval [, 1] ad utlzg the fractoal tegrato-by-parts (9) ad (10), we obta +1 ( Λ () Λ ()) w(x) η () (x) dx η ( () +1){ I 1 μ[ p(x) D μ η () (x) ]} x() +1 + η ()( () +1){ I 1 μ[ p(x) D μ η () (x) ]} x() +1 + η ( () ) η ( () ), (64) where we re-terate that I 1 μ x I1 μ +1 ad D μ Dμ x whe 1 SFSLP-I, also I 1 μ I1 μ x x D μ +1 case of,.e., SFSLP-II. Now, by applyg the boudary codtos for η() (x) ad η () (x) we obta +1 ( Λ () Λ ()) w(x) η () (x) dx 0. ad D μ (65) Therefore, Λ () Λ () because η () (x) s a o-trval soluto of the problem, ad w(x) s o-egatve terval [, 1]. Part b: Now, we prove the secod statemet o the orthogoalty of the egefuctos wth respect to the weght fucto w(x). Assume that η () 1 (x) ad η() (x) are two egefuctos correspodg to two dstct egevalues Λ() 1 ad Λ (), respectvely. The they both satsfy (57) (59) as L α,β;μ η () 1 (x) + Λ() 1 w(x)η() 1 (x) 0 (66) subect to η () ( ) 1 () 0, { I 1 μ[ p(x) D μ η () 1 (x)]} 0, x() +1 ad L α,β;μ η () (x) + Λ() w(x)η() (x) 0, correspodg to the followg boudary codtos η ( () ) 0, { I 1 μ[ p(x) D μ η () (x)]} x() (+1) 0. (67)

14 508 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) It ca be show that ( () Λ 1 Λ() ) w(x)η () 1 (x)η() (x) η() Itegratg over the terval [, 1] yelds ( Λ () 1 Λ() ) +1 1 (x)lα,β;μ η () (x) η() (x)lα,β;μ w(x)η () (x)η () (x) dx η 1(+1) { I 1 μ[ p(x) D μ η () 1 (x)]} x+1 η () 1 (x). (68) + η () (+1){ I 1 μ[ p(x) D μ η () (x)]} x+1 + η () 1 () η() (), (69) ad usg fractoal tegrato-by-parts (9) ad (10), alsosceλ () 1 Λ() 0, we obta +1 w(x)η () 1 (x)η() (x) dx 0, whch completes the proof. Theorem 4.. The exact egefuctos of SFSLP-I (57) (59), whe 1,aregveas P (1) (x) (1) P α,β,μ ad the correspodg dstct egevalues are Λ (1) (x) (1 + x) β+μ P α μ+1, β+μ (x), (71) (1) Λ α,β,μ Γ( β + μ 1)Γ ( + α + 1) Γ( β 1)Γ ( + α μ + 1), (7) ad furthermore, the exact egefuctos to SFSLP-II (57) (59), caseof,aregveas P () (x) () P α,β,μ ad the correspodg dstct egevalues are (x) (1 x) α+μ P α+μ,β μ+1 (x), (73) Λ () () Λ α,β,μ Γ( α + μ 1)Γ ( + β + 1) Γ( α + μ 1)Γ ( + β μ + 1). (74) Proof. The proof follows smlar steps as show Theorem 3.4. Hece, weolyprove(71) ad (7) detal. From (71), t s clear that (1) P α,β,μ () 0. So, we eed to mae sure that the other boudary codto s satsfed. Sce (1) P α,β,μ () 0, the property (7) helps replacg Dμ x by Dμ x.osequetly, { x I 1 μ [ +1 p(x) Dμ (1) x P α,β,μ (x) ]} x+1 { x I 1 μ +1 { x I 1 μ +1 [ p(x) Dμ (1) x P α,β,μ [ p(x) Dμ x (x) ]} x+1 ( (1 + x) β+μ P α μ+1, β+μ (x) )]} x+1 ad by carryg out the fractoal dervatve usg Lemma 3. { [ ]} x I 1 μ Γ( 1 β + μ) +1 p(x) (1 + x) β P 1+α, β Γ( 1 β) x+1 Γ( 1 β + μ) { x I 1 μ [ +1 (1 x) 1+α 1+α, β]} P Γ( 1 β), x+1 ad by worg out the fractoal tegrato usg Lemma 3.3 we obta Γ( 1 β + μ) Γ( 1 β) Γ( + α + 1) Γ( + α μ + 1) { (1 x) +α μ 1+α, β} P 0. x+1 The ext step s to show that (71) satsfes (57) wth egevalues (7). Frst, we tae a fractoal tegrato of order μ o both sdes of (57) ad substtute (71). The, aga by replacg the aputo dervatve by the Rema Louvlle oe, thas to (7), weobta (70)

15 (1 x) α+1 β+1 (1 + x) Λ (1) x Iμ +1 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) [ (1 + x) β+μ P α μ+1, β+μ (x) ] Dμ x { (1 x) α+1 μ α μ+1, β+μ P (x) }. Fally, the fractoal dervatve o the left-had sde ad the fractoal tegrato o the rght-had sde s wored out usg () ad (4) as Γ( 1 β + μ) (1 x) α+1 P α+1, β (x) Λ Γ( 1 β) (1) Γ( + α + μ + 1) Γ( + α + 1) (1 x) α+1 P α+1, β (x). By a smlar argumet o the (1 x) α+1 P α+1, β (x) beg o-zero almost everywhere, we ca cacel ths term out o both sdes ad obta Λ (1) (1) Λ α,β,μ Γ( β + μ 1)Γ ( + α + 1) Γ( β 1)Γ ( + α μ + 1). Now, we eed to chec Theorem 4.1, to see f(7) verfes that the egevalues are deed real-valued ad dstct, ad the orthogoalty of the egefuctos wth respect to w(x) (1 x) 1+α μ (1 + x) 1+β μ s vald: w(x) (1) P α,β,μ (x) (1) P α,β,μ (x) dx w(x) [ (1 + x) β+μ] P α μ+1, β+μ (1 x) 1+α μ (1 + x) β+μ P α μ+1, β+μ (x)p α μ+1, β+μ (x) dx (x)p α μ+1, β+μ (1 x) α (1 + x) β P α,β (x)p α,β (x) dx α,β ( 1)δ, where α α μ + 1, β β + μ 1, ad deoted by α,β () s the orthogoalty costat of the famly of Jacob polyomals. The smplcty of the egevalues ca be also show a smlar fasho as Part c the proof of Theorem 3.4, ad ths completes the proof. Lemma 4.3. The shfted egefuctos to SFSLP-I & -II, deoted by () P α,β,μ (t), aregveas () P α,β,μ (t) μ() 0 (x) dx ( )( () () +1 μ (1) ) 1 t + μ(), (75) 1 where case of the SFLSP-I ( 1), t [0, 1], μ (1) β + μ 1 ad 0 < μ (1) < μ, adforsfslp-ii( ), t [, 0], μ () α + μ 1 also 0 < μ () < μ. Proof. The proof follows the oe Lemma 3.5. Theorem 4.4. The shfted egesolutos to (57), () P α,β,μ (t), form a complete herarchcal bass for the fte-dmesoal space of poly-fractoomals F + μ (),where μ (1) β + μ 1 ad μ () α + μ 1,where0 < μ (1) < μ,also0 < μ () < μ. Proof. The proof follows the oe Theorem The growth of the magtude the egevalues of SFSLP-I, Λ (1), exhbts a smlar behavor as oe observed RFSLP-I & -II. However, there are aother two degrees of freedom the choce of parameters α ad β, whch affect the magtude of the egevalues. It turs out that case of SFSLP-I ( 1), the optmal hghest magtude s acheved whe α μ ad β, μ (0, 1). The growth of the Λ (1) correspodg to three values of μ 0.35, μ 0.5, ad μ 0.99 s show Fg. 4. Aga, we observe about the two growth modes of Λ (1), depedg o ether μ (0, 1/), where a sublear growth Λ 1 s observed, or, μ (1/, 1), where a superlear subquadratc growth mode s vald; the case μ 1/ leads to a exactly lear growth mode. orrespodg to the aforemetoed fractoal-orders μ, Fg. 5, we plot the egefuctos of SFSLP-I, P (1) (x), of dfferet orders ad correspodg to dfferet values of μ used Fg. 4. Ia smlar fasho, we compare the egesolutos wth the correspodg stadard Jacob polyomals P α μ+1, β+μ (x) each plot.

16 510 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg. 4. Magtude of the egevalues of SFSLP-I, Λ (1), versus, correspodg to α 0 ad β 0.7, correspodg to dfferet fractoal-order μ 0.35, left: sublear growth, μ 0.5; mddle: lear growth, ad μ 0.99; rght: superlear subquadratc growth. Here we compare the growth of the egevalues to the optmal case whe α μ ad β. I Fg. 6, the growth of the magtude Λ (), correspodg to three values of μ 0.35, μ 0.5, ad μ 0.99 s plotted. I SFSLP-II ( ), the optmal hghest magtude the egevalues s acheved whe α ad β μ. Moreover, Fg. 7, we plot the egefuctos of SFSLP-II, P () (x), of dfferet fractoal-orders ad correspodg to dfferet μ used Fg. 6. Ths tme, we compare the egesolutos wth the correspodg stadard Jacob polyomals P α+μ,β μ+1 (x) each plot Propertes of the ege-solutos to SFSLP-I & -II We lst a umber of propertes of the egesolutos to SFSLP-I & -II as follows. No-polyomal ature: From (71) ad (73), the egefuctos exhbt a o-polyomal (fractal) behavor, thas to the fractoomal multplers (1 + x) β+μ SFSLP-I ad (1 x) α+μ SFSLP-II. Ideed, these poly-fractoomals are the geeralzato of those troduced RFSLP (7) ad (9). We realze that whe α ad β smultaeously, the ege-solutos to the sgular problems SFSLP-I & -II, oly asymptotcally, approach to that of the regular couterparts. However, specal atteto should be tae due to the fact that whe α ad β, the goverg equatos (57) the become o-sgular ad equvalet to the regular problems RFSLP-I & -II (16) at the frst place. Here, we refer to () P α,β,μ (x) as the geeralzato of the whole famly of the Jacob poly-fractoomal correspodg to the trple α,β,μ, where < α < μ, ad <β<μ SFSLP-I ( 1), ad < α < μ 1, ad <β< μ SFSLP-II ( ). Asymptotc egevalues Λ () : The growth the magtude of egevalues SFSLP wth s depedet o three parameters: the fractoal dervatve order μ, α ad β. From(7) ad (74), t s easy to show that α ad β oly affect the magtude ad ot the behavor (.e., order) of the growth. As show (30), sceμ (0, 1), there are two modes of growth the magtude of Λ () referred to as sublear mode correspodg to 0 < μ < 1/, ad superlear subquadratc mode whch correspods to 1/ < μ < 1. Partcularly, whe μ 1/, the egevalues grow learly wth. The optmal hghest magtude of Λ (1) acheved whe α μ ad β SFSLP-I, ad case of the SFSLP-II whe α ad β μ the optmal egevalues are obtaed. The asymptotc cases are summarzed as Λ (), μ 1,, μ 1/, 1, μ 0. Recurrece relatos: A recurrece relatos s obtaed for the Jacob poly-fractoomals () P α,β,μ (1) P α,β,μ 1 (x) (1 + x) β+μ, (1) P α,β,μ (x) 1 (1 + x) β+μ[ α + β μ + + (α β + ] )x,. a (1) P α,β,μ +1 (x) (b + c x) (1) P α,β,μ (x) d (1) P α,β,μ (x), a ( + α β)( + α β ), (x) as (76)

17 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg. 5. Egefuctos of SFSLP-I, P (1),versusx, for 1 (frst row), (secod row), 5 (thrd row), ad 10 (last row), correspodg to the fractoal-order μ ν/ 0.35 (left colum), μ ν/ 0.5 (mddle colum), ad μ ν/ 0.99 (rght colum). Here, we tae the same values α 0 ad β 0.7, as show Fg. 4. b ( α + β 1)(α β)(α + β μ + ), c ( α + β)( α + β 1)( α + β ), d ( α + μ )( + β μ)( α + β), (77)

18 51 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg. 6. Magtude of the egevalues of SFSLP-II, Λ (), versus, correspodg to α 0.7 ad β 0, correspodg to dfferet fractoal-order μ 0.35, left: sublear growth, μ 0.5; mddle: lear growth, ad μ 0.99; rght: superlear subquadratc growth. Here we compare the growth of the egevalues to the optmal case whe α adβ μ. ad () P α,β,μ 1 (x) (1 x) α+μ, () P α,β,μ (x) 1 (1 x) α+μ[ α β + μ + ( α + β + ] )x,. a () P α,β,μ +1 (x) ( b + c x) () P α,β,μ (x) d () P α,β,μ (x), ( α + β)( α + β ), a b c d Orthogoalty: where ad ( α + β 1)(α β)(α + β μ + ), ( + α + β)( + α + β 1)( + α + β ), ( + α μ)( β + μ )( + α β). (78) (1 x) α+1 μ (1 + x) β+1 μ() P α,β,μ (1) α,β () α,β α β+1 + α β 1 α+β+1 α + β 1 Fractoal dervatves: ( (1) D β+μ+1 x P α,β,μ ) D β+μ+1 x ad x D α+μ 1 where P α β,0 Frst dervatves: d dx ( () P α,β,μ (x) () P α,β,μ (x) dx () α,β δ, (79) Γ( + α μ + 1)Γ ( β + μ 1), ( 1)!Γ( + α β) Γ( α + μ 1)Γ ( + β μ + 1). ( 1)!Γ( α + β) ) x D α+μ 1 ( (1) P α,β,μ ( () P α,β,μ (x) ad P 0,β α (x) deote the stadard Jacob polyomals. ( (1) P α,β,μ (x) ) ( β + μ 1)(1 + x) β+μ P α μ+1, β+μ (x) + 1 ( + α β)(1 + x) β+μ P α μ+, β+μ (x), ) Γ( + μ) P α β,0 (x), (80) Γ() ) Γ( + μ) P 0,β α (x), (81) Γ()

19 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg. 7. Egefuctos of SFSLP-II, P (),versusx, for 1 (frst row), (secod row), 5 (thrd row), ad 10 (last row), correspodg to the fractoal-order μ ν/ 0.35 (left colum), μ ν/ 0.5 (mddle colum), ad μ ν/ 0.99 (rght colum). Here, we tae the same values α 0.7 ad β 0, as show Fg. 6. ad d dx ( () P α,β,μ (x) ) (+α μ + 1)(1 x) α+μ P α+μ,β μ+1 (x) + 1 ( α + β)(1 x) α+μ P α+μ,β μ+ (x).

20 514 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Specal values: ad (1) P α,β,μ () 0, (1) P α,β,μ (+1) β+μ ( + α μ 1 () P α,β,μ (+1) 0, () P α,β,μ 5. Numercal approxmato () α+μ ( + β μ 1 ), (8) ). (83) As dscussed Secto 4.1, tagα β SFLP-I & -II essetally elmates the sgularty the defto of SFSLP-I & -II (57). Accordgly, we are ot allowed to tae such values for α ad β, uless asymptotcally, the SFSLP-I & -II. However, the Jacob poly-fractoomals () P α,β,μ (x), {1, }, regardless of where they are comg from, are the geeralzato of the poly-fractoomals Φ () (x) whch are ow as the egefuctos of FSLP-I & -II. Therefore, we ca represet the whole famly of the Jacob poly-fractoomals () P α,β,μ (x) as { () P α,β,μ egefuctos of RFSLPs (16), α β, (x) egefuctos of SFSLPs (57), otherwse, (84) where {1, }. By Theorems 3.11 ad 4.4, we ca employ such bass fuctos for umercal approxmato. I such settg, we ca study the approxmato propertes of the famly of Jacob poly-fractoomals () P α,β,μ (x) a ufed fasho. To ths ed, we represet a fucto f (x) L w [, 1] as f (x) f N (x) N 1 ˆf () P α,β,μ (x), x [, 1] (85) where f (x) satsfed the same boudary codtos as () P α,β,μ (x) (85). Now, the ma questo s how fast the expaso coeffcets ˆf decay. By multplyg (85) by L α,β;μ ( () P α,β,μ (x)), 1,,...,N, ad tegratg the terval [, 1], we obta f (x)l α,β;μ ( () P α,β,μ (x) ) dx ( N 1 ) ˆf () P α,β,μ (x) L α,β;μ ( () P α,β,μ (x) ) dx, where L α,β;μ ( () P α,β,μ (x)) o the rght-had sde ca be substtuted by the rght-had sde of (57),.e., Λ () w(x) () P α,β,μ (x) as f (x)l α,β;μ ( () P α,β,μ (x) ) dx N 1 ˆf Λ () ad thas to the orthogoalty property (79) we get ˆf () α,β Λ () or equvaletly by (60), ˆf () α,β Λ () f (x)l α,β;μ ( () P α,β,μ (x) ) dx, (1 x) α+1 μ (1 + x) β+1 μ() P α,β,μ (x) () P α,β,μ (x), f (x) D μ{ (1 x) α+1 (1 + x) β+1 D μ( () P α,β,μ (x) )} dx. (86)

21 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) We recall that 1 correspods to D μ x Dμ +1 ad D μ Dμ x, also whe wehave D μ x D μ +1. Now, by carryg out the fractoal tegrato-by-parts (9) ad (10), weget Dμ x ad D μ ˆf () α,β Λ () whch s equvalet to ˆf () α,β Λ () (1 x) α+1 (1 + x) β+1( D μ f (x) )( D μ() P α,β,μ (x) ) dx, (87) (1 x) α+1 (1 + x) β+1( D μ() P α,β,μ (x) )( D μ f (x) ) dx () P α,β,μ (x) x Iμ x R f (x) +1. x (88) We realze that the last term (88) s detcally zero. Aga, by the fractoal tegrato-by-parts (9) ad (10), we obta ˆf or equvaletly ˆf () α,β () α,β Λ () Λ () () P α,β,μ (x) D μ{ (1 x) α+1 (1 + x) β+1 D μ f (x) } dx, () P α,β,μ (x)l α,β;μ [ ] f (x) dx, f deoted by f (1) (x) L α,β;μ [ f (x)] L w [, 1]. By carryg out the fractoal tegrato-by-parts aother (m 1) tmes, ad settg f (m) (x) L α,β;μ [ f (m) (x)] L w [, 1], weobta ˆf f(m) Λ () (x) L, 1,,...,N. (89) m w osequetly, f the fucto f (x) [, 1], we recover the expoetal decay of the expaso coeffcets ˆf. Remar 5.1. Although whe 0 < μ < 1/ the magtude of the egevalues grows sublearly, such decay behavor does ot affect fudametally the expoetal character of the decay the coeffcets f f (x) possesses the requred regularty Numercal tests I the followg examples, we test the covergece rate approxmatg some poly-fractoomals addto to some other type of fuctos volvg fractoal character. By Theorems 3.11 ad 4.4, we ca exactly represet ay polyfractoomal F N+μ of order N + μ terms of the frst N regular Jacob fractal bass fuctos (16), or alteratvely, usg the frst N sgular Jacob fractal bass fuctos (75). However, ths s ot the case whe other types of bass fuctos, such as the stadard (shfted) Legedre polyomals P (x), areemployed. We frst approxmate the smplest fractal fucto f (t) t usg our regular ad sgular Jacob poly-fractoomals, where we see that oly oe term s eeded to exactly represet the fractoomal,.e., f (t) f 1 (t). To mae a comparso, we also plot the L -orm error terms of N, the umber of expaso terms (85) Fg. 8 (left), whe the stadard Legedre polyomals are employed as the bass fuctos. Moreover, we represet the poly-fractoomal f (t) t 1/3 + t 4+1/3 + t 7+1/3 by our regular ad sgular Jacob poly-fractoomals to compare the effcecy of such expaso fuctos to other stadard polyomal bases. The fast (super) spectral covergece of the our fractal bass fuctos show Fg. 8 (rght), compared to that of the Legedre expaso, hghlghts the effcecy of Jacob poly-fractoomal bass fuctos approxmatg o-polyomal fuctos. Next, we approxmate aother two fuctos whch are ot poly-fractoomals. I Fg. 9, we show the L -orm error (85), where the covergece to f (t) t 1/3 s(t) s show o the left ad the error the approxmato of f (t) s(3 t) s plotted o the rght. Oce aga, we observe spectral (expoetal) covergece of (85) whe the regular ad the sgular egefuctos are employed as the bass fuctos, compared to the case whe the stadard Legedre polyomals are employed. Fally, we also test how well smooth fuctos are approxmated usg a o-polyomal bass Fg. 10. As expected, we see that the Legedre polyomal bass s outperformg the poly-fractoomal bass but oly slghtly ad we stll observe expoetal covergece of the latter. Here we employed μ 1/ for the both RFSLP ad SFSLP bases but other choces are also possble to optmze the covergece rate.

22 516 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) Fg. 8. L -orm error f (t) f N (t) L versus N, the umber of expaso terms (85) whe Legedre polyomals are used as the bass fuctos. Here, f (t) s a poly-fractoomal; left: f (t) t, where oly oe term,.e., () P α,β,μ 1 s eeded to exactly capture t,adrght: f (t) t 1/3 + t 4+1/3 + t 7+1/3 ; here α β 0. Fg. 9. L -orm error f (t) f N (t) L versus N, the umber of expaso terms (85), where f (t) s ot a poly-fractoomal; left: f (t) t 1/3 s(t), ad rght: f (t) s(3 t); hereα β 0. Fg. 10. L -orm error f (t) f N (t) L versus N, the umber of expaso terms (85), where f (t) s a polyomal; left: f (t) t 6 + t 11 + t 15,adrght: f (t) t 5 exp t/4 1; here α β Summary We have cosdered a regular fractoal Sturm Louvlle problem of two ds RFSLP-I ad RFSLP-II of order ν (0, ), [15] wth the fractoal dfferetal operators both of Rema Louvlle ad aputo type, of the same fractoal-order μ ν/ (0, 1). Ths choce, tur, motvated a proper fractoal tegrato-by-parts. I the frst part of the paper, we obtaed the aalytcal egesolutos to RFSLP-I & -II as o-polyomal fuctos, whch we defed as Jacob polyfractoomals. These egefuctos were show to be orthogoal wth respect to the weght fucto, assocated wth the

23 M. Zayerour, G.E. Karadas / Joural of omputatoal Physcs 5 (013) RFSLP-I & -II. I addto, these egefuctos were show to be herarchcal, ad a useful recursve relato was obtaed for each type of the egefuctos. Moreover, a detaled lst of other mportat propertes of such poly-fractoomals was preseted at the ed of the frst part of the paper. We exteded the fractoal operators to a ew famly of sgular fractoal Sturm Louvlle problems of two ds, SFSLP-I ad SFSLP-II, the secod part of the paper. We showed that the regular boudary-value problems RFSLP-I & -II are deed asymptotc cases for the sgular couterparts SFSLP-I & -II. We also proved that the egevalues of the sgular problems are real-valued ad the egefuctos correspodg to dstct egevalues are orthogoal. Subsequetly, we obtaed the ege-solutos to SFSLP-I & -II aalytcally, also as o-polyomal fuctos, whch completed the whole famly of the Jacob poly-fractoomals. I a smlar fasho, a umber of useful propertes of such egesolutos was troduced. Fally, we aalyzed the umercal approxmato propertes of the egesolutos to RFSLP-I & -II ad SFSLP-I & -II a ufed fasho. The expoetal covergece approxmatg fractal fuctos such as poly-fractoomals addto to some other fractal fuctos such as fractoal trgoometrc fuctos hghlghted the effcecy of the ew fractal bass fuctos compared to Legedre polyomals. Acowledgemets Ths wor was supported by the ollaboratory o Mathematcs for Mesoscopc Modelg of Materals (M4) at PNNL fuded by the Departmet of Eergy, by a AFOSR MURI ad by NSF/DMS. Refereces [1] W.O. Amre, A.M. Hz, D.B. Pearso, Sturm Louvlle Theory: Past ad Preset, Brhäuser, Basel, 005. [] A. Zettl, Sturm Louvlle Theory, vol. 11, Amerca Mathematcal Socety, 010. [3] K.S. Mller, B. Ross, A Itroducto to the Fractoal alculus ad Fractoal Dfferetal Equatos, Joh Wley ad Sos, Ic., New Yor, NY, [4] I. Podluby, Fractoal Dfferetal Equatos, Academc Press, Sa Dego, A, USA, [5] A.A. Klbass, H.M. Srvastava, J.J. Trullo, Theory ad Applcatos of Fractoal Dfferetal Equatos, Elsever, Amsterdam, Netherlads, 006. [6] A. arpter, F. Maard, Fractals ad Fractoal alculus otuum Mechacs, Sprger-Verlag Telos, [7] B.J. West, M. Bologa, P. Grgol, Physcs of Fractal Operators, Sprger-Verlag, New Yor, NY, 003. [8] R.L. Mag, Fractoal alculus Boegeerg, Begell House Ic., Reddg, T, 006. [9] Q.M. Al-Mdallal, A effcet method for solvg fractoal Sturm Louvlle problems, haos, Soltos & Fractals 40 (1) (009) [10] V.S. Ertür, omputg egeelemets of Sturm Louvlle problems of fractoal order va fractoal dfferetal trasform method, Mathematcal ad omputatoal Applcatos 16 (3) (011) 71. [11] A. Neamaty, R. Darz, S. Zaree, B. Mohammadzadeh, Haar wavelet operatoal matrx of fractoal order tegrato ad ts applcato for egevalues of fractoal Sturm Louvlle problem, World Appled Sceces Joural 16 (1) (01) [1] B. J, R. Wllam, A verse Sturm Louvlle problem wth a fractoal dervatve, Joural of omputatoal Physcs 31 (01) [13] J. Q, S. he, Egevalue problems of the model from olocal cotuum mechacs, Joural of Mathematcal Physcs 5 (011) [14] T.M. Ataacovc, B. Staovc, Geeralzed wave equato olocal elastcty, Acta Mechaca 08 (1) (009) [15] M. Klme, O.P. Agrawal, O a regular fractoal Sturm Louvlle problem wth dervatves of order (0, 1), : Proceedgs of 13th Iteratoal arpatha otrol oferece, I, July, 01, [16] E. Bas, F. Met, Spectral propertes of fractoal Sturm Louvlle problem for dffuso operator, preprt, arxv: , 01, pp [17] R. Asey, J. Ftch, Itegral represetatos for Jacob polyomals ad some applcatos, Joural of Mathematcal Aalyss ad Applcatos 6 (1969)