Some properties of Boubaker polynomials and applications

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1 Some properties of Boubaker polyomials ad applicatios Gradimir V. Milovaović ad Duša Joksimović Citatio: AIP Cof. Proc. 179, 1050 (2012); doi: / View olie: View Table of Cotets: Published by the America Istitute of Physics. Additioal iformatio o AIP Cof. Proc. Joural Homepage: Joural Iformatio: Top dowloads: Iformatio for Authors:

2 Some Properties of Boubaker Polyomials ad Applicatios Gradimir V. Milovaović ad Duša Joksimović Mathematical Istitute of the Serbia Academy of Scieces ad Arts, Kez Mihailova 36, P.O. Box 367, Beograd, Serbia Graduate School of Busiess Studies, Megatred Uiversity, Goce Delčeva, Belgrade, Serbia Abstract. Some ew properties of Boubaker polyomials, as well as a applicatio of these polyomials for obtaiig approximate aalytical solutio of Love s itegral equatio are preseted. Keywords: Chebyshev polyomials; Boubaker polyomials; zeros; recurrece relatio; Fredholm itegral equatio; Love s itegral equatio. PACS: Gp, Mv, Rz, Gf, Nm, J INTRODUCTION There are several papers o the so-called Boubaker polyomials ad their applicatios i differet problems i physics ad other computatioal ad applied scieces (cf. [1, 2, 3, 9] ad refereces therei). Such polyomials are defied i a similar way as Chebyshev polyomials of the first ad secod kid T (x) ad U (x), which are orthogoal o ( 1,1) with respect to the weights fuctios 1/ 1 x 2 ad 1 x 2, respectively. The moic Boubaker polyomials are defied as [/2] B (x)= ( 1) k ( k k ad B 0 (x)=1. Alteratively, they ca be expressed by a three-term recurrece relatio where B 0 (x)=1, B 1 (x)=x, B 2 (x)=x The ext seve members of this polyomial sequece are: ) k k x 2k, 1, (1) B m+1 (x)=xb m (x) B m 1 (x), m = 2,3,..., (2) B 3 (x)=x 3 + x, B (x)=x 2, B 5 (x)=x 5 x 3 3x, B 6 (x)=x 6 2x 3x 2 + 2, B 7 (x)=x 7 3x 5 2x 3 + 5x, B (x)=x x 6 + x 2 2, B 9 (x)=x 9 5x 7 + 3x x 3 7x. Otherwise, the polyomials (1) ca be expressed i terms of Chebyshev polyomials of the first ad secod kid, T (x) ad U (x). Namely, we ca prove (e.g., by the mathematical iductio) the followig result: Theorem 1. For m 1 the followig formula B m (x)=2t m (x/2)+u m 2 (x/2) holds, where U 1 (x) 0. THREE-TERM RECURRENCE RELATION AND ZEROS As we ca see, the relatio (2) is ot true for m = 1. I order to provide a relatio for each m N, we ca defie a sequece β m } m N by β 1 = 2 ad β m = 1 for m 2, ad the we have the three-term recurrece relatio i the form B m+1 (x)=xb m (x) β m B m 1 (x), m = 1,2,..., with B 0 (x)=1, B 1 (x)=0. (3) Usig this relatio for m = 0,1,..., 1, ad defiig -dimesioal vectors b (x)=[b 0 (x) B 1 (x)... B 1 (x)] T ad e =[ ] T (the last coordiate vector), we obtai the equatio (xi M )b (x)=b (x)e, () Numerical Aalysis ad Applied Mathematics ICNAAM 2012 AIP Cof. Proc. 179, (2012); doi: / America Istitute of Physics /$

3 where I is the idetity matrix of order ad M is a tridiagoal matrix of order, 0 1 O 0 1 O β M =. β = O β 1 0 O 1 0. Accordig to () we coclude that the zeros of the polyomial B (x) are also eigevalues of the matrix M. Also, usig Gerschgori s theorem, it is easy to see that these eigevalues are i the uit circle z < 2 (see also [9]). It is well-kow that for orthogoal polyomials o a symmetric iterval ( a, a), which satisfy a three-term recurrece relatio of the form (3), it ca be defied two ew polyomial systems which are orthogoal o (0,a 2 ) (cf. [7, pp ]). I a similar way, we ca itroduce here also two ew (oorthogoal) systems of (moic) polyomials P = p m (t)} ad Q = q m (t)} via Boubaker polyomials B m (x), so that B 2m (x)=p m (x 2 ) ad B 2m+1 (x)=xq m (x 2 ). Theorem 2. Let β m,m 1, be recursive coefficiets i the recurrece relatio (3). The, p m+1 (t)=(t a m )p m (t) b m p m 1 (t) ad q m+1 (t)=(t c m )q m (t) d m q m 1 (t), with p 0 (t)=q 0 (t)=1, p 1 (t)=q 1 (t)=, where the recursive coefficiets are give by 2, m = 0, a m = β 2m + β 2m+1 = 2, m 1, 2, m = 1, b m = β 2m β 2m 1 = 1, m 2, ad 1, m = 0, c m = β 2m+1 + β 2m+2 = 2, m 1, d m = β 2m β 2m+1 = 1, m 1. Thus, this theorem gives two systems of polyomials: P = 1, t +2, t 2 2, t 3 2t 2 3t +2, t t 3 +t 2,... } ad Q = 1, t + 1, t 2 t 3, t 3 3t 2 2t + 5, t 5t 3 + 3t t 7,... }. I order to ivestigate zeros of the polyomials B (z) o the imagiary axis we put z = iy ad cosider B (iy)/i m, 2, i.e., the sequece of polyomials y 2 2, y(y 2 1), y 2, y(y + y 2 3), y 6 + 2y 3y 2 2,... For t > 0 we itroduce two sequeces of polyomials e m (t) ad o m (t), m = 1,2,...,by e m (t)=( 1) m B 2m (i t) ad o m (t)=( 1) m B 2m+1(i t) i. t Accordig to (1) ad Theorem 2, it is clear that e m (t)=( 1) m m ( ) 2m k 2m k p m ( t)= k 2m k tm k, o m (t)=( 1) m m ( ) 2m k 2m k + 1 q m ( t)= k 2m 2k + 1 tm k. (5) Theorem 3. For ay m N the polyomials e m (t) ad o m (t) have oly oe positive zero. Proof. I the proof we use the umber of sig variatios (differeces) betwee cosecutive ozero coefficiets of a polyomial ordered by descedig variable expoet. We ote that the coefficiets i (5) are positive for k < m/2 ad egative for k > m/2, so that we have oly oe sig variatio. Accordig to Descartes Rule the umber of positive zeros is either equal to the umber of sig differeces betwee cosecutive ozero coefficiets, or less tha it by a multiple of 2. Sice e m (0)= 2 < 0 ad e m (T ) > 0 for each sufficietly large positive T, we coclude that e m (t) has oly oe positive zero. A similar proof ca be doe for polyomials o m (t). Usig this theorem oe ca prove the followig result o the zero distributio: 1051

4 Theorem. Every polyomial B (x), 2, has two complex cojugate zeros ±i γ, γ > 0, ad other zeros are real ad symmetrically distributed i ( 2,2), where lim γ = /3. + Thus, B 2m (x)=(x 2 m + γ 2m ) (x 2 τ 2m,ν ), B 2m+1 (x)=x(x 2 m + γ 2m+1 ) (x 2 τ 2m+1,ν ), ν=1 where > τ,1 >...>τ,m > 0 ad = 2m or = 2m + 1. ν=1 APPLICATIONS The polyomials B m (x)} plays importat role i applicatios. Solutios to several applied physics problems based o the so-called Boubaker Polyomials Expasio Scheme (BPES) (cf. [9] ad refereces therei). It is easy to prove that these polyomials satisfy the relatio (cf. [3]) B (m+1) (x)=(x x 2 + 2)B m (x) β m B (m 1) (x), m 1, where β m is defied before. Recetly, for example, Kumar [] has preseted a method for obtaiig a aalytical solutio of Love s itegral equatio (see [5, 6]) 1 f (x) μ 1 r r 2 f (y)dy = 1, 1 < x < 1, (6) +(x y) 2 for a particular physical (electrostatical) system, based o the Boubaker polyomials expasio scheme (BPES). A approximatio to the solutio of (6), i the case r = 1 ad μ = 1/π, was give by Love [6], f (x) f L (x)= x x x x. (7) As a approximate solutio solutio i the set of polyomials of degree at most (i otatio P ), Kumar [] used the expasio f (1) (x)= c m B m (x), but i his approach was a error. The corrected versio of the method leads to the equatio 1 ( r 1 ) rb m (y) c m B m (x) μ 1 r 2 +(x y) 2 c m B m (y)dy = B m (x) μ 1 r 2 +(x y) 2 dy c m = 1. Takig collocatio poits as the positive zeros of T 2 (x) we get a system of liear equatios for determiig the coefficiets c m, m = 1,...,. I the same case r = 1 ad μ = 1/π, the correspodig solutios for = 1 ad = 2, are f (1) (x)= B (x) ad f (1) (x)= B (x) b (x), or i the expadig form f (1) (x)= x ad f (1) (x)= x x x x. However, we ca get better solutios takig the costat term (B 0 (x) =1) i the correspodig expasio of the approximate polyomial solutio, i.e., (x) = c m B m (x). I that case, usig the positive zeros of T 2+2 (x) as m=0 collocatio poits, we obtai the followig approximative solutios ad (x)= b 0 (x) B (x)= x (x) = B 0 (x) B (x) B (x) = x x x x. Moreover, i the previous set of polyomials we ca get much better results if we take the complete basis of (eve) polyomials. Thus, i order to fid a approximate solutio i the set P 2, we put f 2 (x)= c m B 2m (x). For example, i this case we fid f (x)=2.6399b 0 (x) B 2 (x) b (x) ad m=0 f (x) = B 0 (x) B 2 (x) b (x) b 6 (x) B (x) = x x x x. 1052

5 TABLE 1. Maximal relative errors of the approximate solutios Approximate Maximal relative errors solutio = 1 = 2 = 3 = f L (x) 1.69( 3) f (1) (x) 3.2( 1) 1.27( 1) 3.57( 2) 1.22( 2) (x) 2.3( 2) 1.16( 3) 1.52( ) 1.07( 5) f (x) 2.3( ) 1.37( 6) 9.65( 9) 2.1( 10) Maximal relative errors of the previous approximate solutios, icludig Love s solutio (7), are displayed i Table 1, where we used as the exact solutio oe obtaied by a efficiet method for solvig Fredholm itegral equatios of the secod kid []. Numbers i paretheses idicate decimal expoets. The solutios f (x) for μ = 1/π ad r = 0.1, r = 1, ad r = 10 are preseted i Figure 1. FIGURE 1. The solutios f (x) of Love s equatio (6) for r = 1/10 (dotted lie), r = 1 (dashed lie) ad r = 10 (solid lie) ACKNOWLEDGMENTS The authors were supported i part by the Serbia Miistry of Educatio ad Sciece. REFERENCES 1. K. Boubaker, Boubaker polyomials expasio scheme (BPES) solutio to Boltzma diffusio equatio i the case of strogly aisotropic eutral particles forward backward scatterig, Aals of Nuclear Eergy 3, (2011). 2. K. Boubaker, L. Zhag, Fermat-liked relatios for the Boubaker polyomial sequeces via Riorda matrices aalysis, Joural of the Associatio of Arab Uiversities for Basic ad Applied Scieces (2012) (to appear). 3. B. Karem Be Mahmoud, Temperature 3D profilig i cryogeic cylidrical devices usig Boubaker polyomials expasio scheme (BPES), Cryogeics 9, (2009).. A.S. Kumar, A aalytical solutio to applied mathematics-related Love s equatio usig the Boubaker polyomials expasio scheme, J. Frakli Ist. 37, (2010). 5. E.R. Love, The electrostatic field of two equal circular co-axial coductig disks, Quart. J. Mech. Appl. Math. 2, 2 51 (199). 6. E.R. Love, The potetial due to a circular parallel plate codeser, Mathematika 37, (1990). 7. G. Mastroiai, G.V. Milovaović, Iterpolatio Processes Basic Theory ad Applicatios, Spriger Moographs i Mathematics, Berli Heidelberg: Spriger Verlag, G. Mastroiai, G.V. Milovaović, Well-coditioed matrices for umerical treatmet of Fredholm itegral equatios of the secod kid, Numer. Liear Algebra Appl. 16, (2009). 9. T.G. Zhao, L. Naig, W.X. Yue, Some ew features of the Boubaker polyomials expasio scheme BPES, Math. Notes V7, No. 2, (2010). 1053