In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials

Rababah, Abedallah

English

Abedallah Rababah

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Conformable Chebyshev differential equation of first kind

Institute of Advanced Engineering and Science

International Journal of Electrical and Computer Engineering (IJECE)Vol. 11, No. 1, February 2021, pp. 628∼635ISSN: 2088-8708, DOI: 10.11591/ijece.v11i1.pp628-635 r 628Conformable Chebyshev differential equation of first kindAbedallah RababahDepartment of Mathematical Sciences, United Arab Emirates University, United Arab EmiratesDepartment of Mathematics, Jordan University of Science and Technology, JordanArticle InfoArticle history:Received Oct 31, 2019Revised Jun 20, 2020Accepted Jul 1, 2020Keywords:Chebyshev-I differentialequationChebyshev-I polynomialsConformable differentiationRodrigues’ formulaABSTRACTIn this paper, the Chebyshev-I conformable differential equation is considered.A proper power series is examined; there are two solutions, the even solution andthe odd solution. The Rodrigues’ type formula is also allocated for the conformableChebyshev-I polynomials.This is an open access article under the CC BY-SA license.Corresponding Author:Abedallah Rababah,Department of Mathematical Sciences,United Arab Emirates University, 15551 Al Ain, United Arab Emirates.E-mail: rababah@uaeu.ac.ae1. INTRODUCTIONIt is well known that the year 1695 is considered as the birthday of the so called field of fractionalcalculus. L’Hospital wrote a letter on September 30, 1695 to Leibniz wondering about the notation used in thepublications for the derivative of the functionf(x) = x,dnfdxn.L’Hospital asked Leibniz the following question: what would the result be if n = 12 . Leibniz wroteback to L’Hospital: an apparent paradox, from which one day useful consequences will be drawn. Since then,many Mathematicians tried their definitions for a fractional order derivative; most of these definitions are basedon integration formulas . The commonly used definitions are those given by Riemann-Liouville, Caputo, andGrünwald-Letnikov of non-integer derivatives. These definitions are summarized in the following lines [1–5].Let n be a positive integer, α ∈ (n− 1, n], and x ≥ a.− Riemann-Liouville (left-sided) α fractional derivative of f is defined by:Dαa [f(x)] =1Γ(n− α)dndxn∫ xa(x− ξ)n−1−αf(ξ) dξ,where Γ(x) is the well known Gamma function.− Grünwald-Letnikov (left-sided) α fractional derivative of f is defined by:Dαa [f(x)] = limh→01hα[n]∑k=0(−1)k Γ(α+ 1)f(x− kh)Γ(k + 1)Γ(α− k + 1).Journal homepage: http://ijece.iaescore.comInt J Elec & Comp Eng ISSN: 2088-8708 r 629− Caputo (left-sided) derivative α fractional derivative of f is defined by:Dαa [f(x)] =1Γ(n− α)∫ xa(x− ξ)n−1−α dndξn[f(ξ)] dξ., x ≥ a.For a complete list of fractional derivatives, see the survey in [2].An acceptable definition for fractional derivatives should agree with the ordinary derivatives.As noticed in [3], the only property inherited by all definitions of fractional derivatives is the linearity property;they counted the following drawbacks of one definition or another: the Riemann-Liouville derivative does notreproduce the derivative of a constant to be 0. All definitions do not satisfy the product, quotient, and chainrules. There are other drawbacks for the existing definitions of fractional derivatives.The well-known Chebyshev differential equation of first kind is given in the form [6–13]:(1− x2) y′′ − x y′ + n2 y = 0, n ∈ N.Expanding the solution using a power series around the ordinary point x = 0 yields the well-known Chebyshevpolynomials of first kind, Tn(x). For Multivariate case, see [14–16]. The Chebyshev-I polynomials fulfill thefollowing orthogonality conditions [6, 7]:∫ 1−1Tn(x) Tm(x)√1− x2dx = 0, whenever n 6= m. (1)2. RESEARCH METHODIn this section, the conformable Chebyshev equation of first kind is defined and solved to give theconformable Chebyshev functions of first kind. For α ∈ (0, 1], the α conformable derivative of y is denoted byDαy; the second α conformable derivative is denoted by DαDαy. Recently, in [3] the conformable derivativeis given in the following definition, see also [4].Definition Let f : [0,∞) 7→ R, then the conformable derivative of f of order α is defined by:Dαf(x) = limε→0f(x+ εx1−α)− f(x)ε, x > 0, α ∈ (0, 1). (2)If f is α differentiable in (0, a), a > 0, andlimx→0+Dαf(x)exists, then we setDαf(0) = limx→0+Dαf(x).Let a, b, c, p ∈ R and f and g be α differentiable at a point x > 0, then the conformable derivativesatisfies the following properties:Dα(c) = 0,Dα(xp) = pxp−α,Dα(af + bg) = aDαf + bDαg. (3)It is satisfactory to consider the α conformable derivative for α ∈ (0, 1]. If α ∈ (n, n + 1], n ∈ N ,then the conformable derivative, if it exist, is defined byDαf(x) = Dn(Dα−nf(x)).A fractional power series of α is defined as a series of the form∞∑k=0akxkα, α ∈ (0, 1]. (4)Conformable Chebyshev differential equation of first kind (Abedallah Rababah)630 r ISSN: 2088-8708Throughout this paper, let Pnbe the set of all polynomials of degree ≤ n.For α ∈ (0, 1], we define the conformable Chebyshev differential equation of the first kind by:(1− x2α) DαDαy − α xα Dαy + α2 n2 y = 0, n ∈ N. (5)For few applications using the Chebyshev polynomials, see [17–26]. A solution of (5) in a properpower series of α is considered. Let this solution be given by a conformable power series of α asy =∞∑k=0ak xkα.3. RESULTS AND ANALYSISUsing the rules of conformable derivatives (3), then the α first and second conformable derivatives aregiven by:Dαy =∞∑k=1α k ak x(k−1)α,DαDαy =∞∑k=2α2 k(k − 1) ak x(k−2)α. (6)Substituting the proper power series and its conformable derivatives in (5) yields the following equa-tion:(1− x2α)∞∑k=2α2k(k − 1)akx(k−2)α − αxα∞∑k=1α k akx(k−1)α + α2 n2∞∑k=0akxkα = 0.This is further simplified to∞∑k=2α2k(k − 1)akx(k−2)α −∞∑k=2α2k(k − 1)akxkα − αxα∞∑k=1αkakx(k−1)α+α2n2∞∑k=0akxkα = 0.In the last equation, substitute k + 2 for k in the first term to get:∞∑k=0α2(k + 2)(k + 1)ak+2xkα −∞∑k=2α2k(k − 1)akxkα −∞∑k=1α2kakxkα+∞∑k=0α2n2akxkα = 0.Rewriting the last equation so that the summations start counting with k = 2 yields:(2α2a2 + α2n2a0) +[6α2a3 − α2a1 + α2n2a1]xα+∞∑k=2[α2(k + 2)(k + 1)ak+2 − α2k(k − 1)ak − α2kak + α2n2ak]xkα = 0.To compute the values of the parameters of the fractional power series, the coefficients on both sidesare compared. The constant coefficient yields2α2a2 + α2n2a0 = 0.Int J Elec & Comp Eng, Vol. 11, No. 1, February 2021 : 628 – 635Int J Elec & Comp Eng ISSN: 2088-8708 r 631Solving for a2 givesa2 =−n22a0. (7)The coefficient of xα yields6α2a3 − α2a1 + α2n2a1 = 0.Solving for a3 givesa3 = −n2 − 16a1. (8)The general term for the coefficient of xkα yieldsα2(k + 2)(k + 1)ak+2 − α2k(k − 1)ak − α2kak + α2n2ak = 0.This is solved for ak+2 to yield:ak+2 =k2 − n2(k + 2)(k + 1)ak.This can be rewritten in the formak+2 = −(n− k)(n+ k)(k + 2)(k + 1)ak. (9)The fractional power series contains either even or odd terms. Consequently, there are two independent so-lutions. The first solution is the even-terms solution, and the second solution is the odd-terms solution. Bothsolutions diverge at x = ±1. The only interesting solutions are polynomial solutions. For the even terms,substituting k = 0 in the last formula gives (7) for a2. Substituting k = 2, 4, 6, . . . in the last formula givesa4, a6, a8, . . . as follows:a4 =(n− 2)(n+ 2)4(3)n22a0,a6 = −(n− 4)(n+ 4)6(5)(n− 2)(n+ 2)4(3)n22a0,a8 =(n− 6)(n+ 6)8(7)(n− 4)(n+ 4)6(5)(n− 2)(n+ 2)4(3)n22a0.The general term looks as follows:a2k = (−1)k(n− 2(k − 1))(n+ 2(k − 1))2k(2k − 1)· · · (n− 4)(n+ 4)6(5)(n− 2)(n+ 2)4(3)n(n)2(1)a0.This can be written as follows:a2k =(−1)k(2k)!k−1∏i=0(n− 2i)(n+ 2i) a0, (10)where∏is the product symbol. If we seta0 = (−1)[n2 ]gives the following conformable Chebyshev polynomials:T0(x) = 1,T2(x) = 2x2α − 1,Conformable Chebyshev differential equation of first kind (Abedallah Rababah)632 r ISSN: 2088-8708T4(x) = 8x4α − 8x2α + 1,T6(x) = 32x6α − 48x4α + 18x2α − 1,T8(x) = 128x8α − 256x6α + 160x4α − 32x2α + 1.Similarly, when the fractional power series contains only odd terms, then the odd terms are given in thefollowing compact form:a2k+1 =(−1)k(2k + 1)!k−1∏i=0(n− 1− 2i)(n+ 1 + 2i) a1. (11)If we seta1 = (−1)[n2 ]ngives the following conformable Chebyshev polynomials:T1(x) = xα,T3(x) = 4x3α − 3xα,T5(x) = 16x5α − 20x3α + 5xα,T7(x) = 64x7α − 112x5α + 56x3α − 7xα.To derive a Rodrigues’ type formula for the Chebyshev-I polynomials, Tn(x), a polynomial tn(x) is to benecessitated that fulfills the Chebyshev orthogonality:∫ 1−1tn(x)pn−1(x)√1− x2dx = 0, ∀pn−1 ∈ Pn−1. (12)Rewrite the integrand in (12) in the formtn(x)√1− x2pn−1(x) =(dndxnφn(x))pn−1(x)and integrate the orthogonality conditions by parts n times to get0 =∫ 1−1(dndxnφn(x))pn−1(x) dx =n−1∑i=0(−1)iφ(n−1−i)n (x) p(i)n−1(x)]1x=−1. (13)The last formula is valid for any polynomial of degree ≤ n − 1, thus we acquire the following conditions onφn(x) and its derivatives at x = −1, 1 as followsφn(−1) = φ′n(−1) = · · · = φ(n−1)n (−1) = 0,φn(1) = φ′n(1) = · · · = φ(n−1)n (1) = 0. (14)The polynomial tn(x) is of degree n and thus0 =dn+1tn(x)dxn+1=dn+1dxn+1(√1− x2 dnφn(x)dxn). (15)The (15) with the conditions in (14) form a differential equation with 2n boundary conditions and has thesolutionφn(x) = bn(1− x2)n− 12 , for some constant bn. (16)Consequently, the polynomial tn(x) has the formInt J Elec & Comp Eng, Vol. 11, No. 1, February 2021 : 628 – 635Int J Elec & Comp Eng ISSN: 2088-8708 r 633tn(x) = bn√1− x2 dndxn[(1− x2)n− 12 ] . (17)Normalizingtn(1) = Tn(1)yields(−1)n(2n− 1)(2n− 3) · · · (5)(3)(1) bn = 1.And thusbn =(−1)n(2n− 1)!!, (18)where (2n− 1)!! = (2n− 1)(2n− 3)(· · · (3)(1) denotes the double factorial. Since(2n− 1)!! = (2n)!n!2nthusbn =(−2)n n!(2n)!. (19)Thus, the Chebyshev-I polynomials are given by the following Rodrigues’ type formula:Tn(x) =(−2)nn!(2n)!(1− x2) 12 dndxn[(1− x2)n− 12]. (20)The conformable Chebyshev-I polynomials can be expressed by the following Rodrigues’ type for-mula:Tn(x) =(−2)nn!(2n)!αn(1− x2α) 12 Dnα[(1− x2α)n− 12], n ∈ N, α ∈ (0, 1]. (21)It is also interesting to discuss and find a similar generalization to orthogonal polynomials on triangular do-mains [14].4. CONCLUSIONIn this paper, the conformable Chebyshev differential equation of the first kind is introduced in formula(5), using the conformable derivative defined in formula (2). Then the explicit form of the conformable Cheby-shev functions are derived in the form of a power series and a Ridrigue’s type formula is also derived in (20).We consider the conformable derivative given in formula (2) of a function f(x).This means that the ordinary derivative f ′(x) exists and f ′(x) = D1f(x). Moreover, Dαf(x) = x1−αf ′(x)andDαDαf(x) = (1−α)x1−2αf ′(x)+x2−2αf ′′(x) for α ∈ (0, 1]. From these formulas, it is clear that, if theordinary Chebyshev polynomials y(x) = Tn(x) are solutions of the ordinary Chebyshev differential equation,then the functions u(x) = y(xα) = Tn(xα) are solutions of the conformable differential (5).ACKNOWLEDGEMENTThe author owes thanks for the reviewers for invaluable suggestions to improve an earlier version ofthis paper. This research was funded by Jordan University of Science and Technology.REFERENCES[1] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J., ”Fractional Calculus: Models And NumericalMethods,” Fractional Calculus: Models And Numerical Methods, 2016.Conformable Chebyshev differential equation of first kind (Abedallah Rababah)634 r ISSN: 2088-8708[2] Baleanu, D. and , Mustafa, O, ”Asymptotic Integration and Stability: For Ordinary, Functional and Dis-crete Differential Equations of Fractional Order,” Asymptotic Integration and Stability: For Ordinary,Functional and Discrete Differential Equations of Fractional Order, 2015.[3] Khalil R., Al Horani M., Yousef A., and Sababheh M, ”A new definition of fractional derivative,” Journalof Computational Applied Mathematics, vol. 264, pp. 65-70, 2014.[4] Abdeljawad T., ”On conformable fractional calculus,” Journal of Computational Applied Mathematics,vol. 279, pp. 57-66, 2015.[5] SET, E., CHOI, J. and DEMIRBAS, S.̧ ”Some New Chebyshev Type Inequalities Forfractional IntegralOperator Containing A Furtherextension Of Mittag-Leffler Function In The Kernel,” 2019.[6] Mason J. and Handscomb, D. ”Chebyshev Polynomials,” Chebyshev Polynomials,2003.[7] Szegö., ”G. Orthogonal Polynomials,” Orthogonal Polynomials, 1975.[8] Rababah., et al., ”Computing Derivatives of Jacobi Polynomials using Bernstein Transformation and Dif-ferentiation Matrix,” Numerical Functional Analysis and Optimization, vol. 29, no. 5-6, pp. 660-673,2008.[9] Rababah, A. , ”Transformation of Chebyshev-Bernstein polynomial basis,” Comput. Meth. Appl. Math.,vol. 3, no. 4, pp. 608-622, 2003.[10] Rababah, A., ”Jacobi-Bernstein basis transfoprmation,” Comput. Meth. Appl. Math., vol. 4, no. 2, pp.206-214, 2004.[11] Rababah, A., ”Integration of Jacobi and Weighted Bernstein Polynomials Using Bases Transformations,”Comput. Meth. Appl. Math., vol. 7, no. 3, pp. 221-226, 2007.[12] Y. Simsek and G. Ozdemir, ”Some Identities Of The Humbert And Generalized Chebyshev Polynomials”,International Conference on Analysis and Its Applications (ICAA-2016), pp. 124-124, 2016.[13] Y. Simsek and M. Gunay, ”On Bernstein type polynomials and their Applications,” Advances in DifferenceEquations, vol. 79, 2015.[14] Rababah A. and Alqudah M., ”Jacobi-weighted orthogonal polynomials on triangular domains,” Journalof Applied Mathematics, pp. 205-217, 2005.[15] A. Rababah, ”L-2 degree reduction of triangular Bèzier surfaces with common tangent planes at vertices,”International Journal of Computational Geometry and Applications, vol. 15, no. 5, pp. 477-490, 2005.[16] A. Rababah, ”Distances with rational triangular Bèzier surfaces,” Applied Mathematics and Computation,vol. 160, pp. 379-386, 2015.[17] A. Rababah., et al., ”Multiple Degree Reduction and Elevation of Bézier Curves Using Jacobi–BernsteinBasis Transformations,” Numerical Functional Analysis and Optimization, vol. 28, no. 9-10, pp. 1179-1196, 2007.[18] A. Rababah, ”Approximation von Kurven mit Polynomen und Splines,” Ph. Dissertation, Stuttgart Uni-versität, West Germany, 1992.[19] A. Rababah, ”Taylor theorem for planar curves,” Proceedings of the American Mathematical Society,vol. 119, no. 3, pp. 803-810, 1993.[20] A. Rababah, ”The best uniform quadratic approximation of circular arcs with high accuracy,” Open Math-ematics, vol. 14, no. 1, pp. 118-127, 2016.[21] A. Rababah, ”The best quintic Chebyshev approximation of circular arcs of order ten,” InternationalJournal of Electrical and Computer Engineering (IJECE), vol. 9, no. 5, pp. 3779-3785, 2019.[22] Ashwak ALabaichi, Maisa’a Abid Ali K. Al-Dabbas, and Adnan Salih, ”Image steganography using leastsignificant bit and secret map techniques,” International Journal of Electrical and Computer Engineering(IJECE), vol. 10, no. 1, pp. 935-946, 2020.[23] Shima Ramesh Maniyath and Thanikaiselvan V., ”Novel efficient multiple encryption algorithmfor realtime images,” International Journal of Electrical and Computer Engineering (IJECE), vol. 10, no. 2, pp.1327-1336, 2020.[24] R. Rizal Isnanto, Munawar Agus Riyadi, and Muhammad Fahmi Awaj, ”Herb Leaves Recognition usingGray Level Co-occurrence Matrix and Five Distance-based Similarity Measures,” International Journalof Electrical and Computer Engineering (IJECE), vol. 8, no. 3, pp. 1920-1932, 2018.[25] A. Rababah and M. Jaradat, ”Approximating offset curves using B ézier curves with high accuracy,”International Journal of Electrical and Computer Engineering, vol. 10, no. 2, pp. 1648-1654, 2020.[26] A. Rababah and E. Rababah, The best sextic approximation of hyperbola with order twelve,” InternationalJournal of Electrical and Computer Engineering (IJECE), vol. 10, no. 2, pp. 2192-2199, 2020.Int J Elec & Comp Eng, Vol. 11, No. 1, February 2021 : 628 – 635Int J Elec & Comp Eng ISSN: 2088-8708 r 635BIOGRAPHIES OF AUTHORSAbedallah Rababah is a professor of mathematics at United Arab Emirates University and is onleave from Jordan University of Science and Technology. He is working in the field of ComputerAided Geometric Design, abbreviated CAGD. In particular, his research is on degree raising and re-duction of Bézier curves and surfaces with geometric boundary conditions, Bernstein polynomials,and their duality. He is known for his research in describing approximation methods that significantlyimprove the standard rates obtained by classical (local Taylor, Hermite) methods. He proved the fol-lowing conjecture for a particular set of curves of nonzero measure: Conjecture: A smooth regularplanar curve can, in general, be approximated by a polynomial curve of degree n with order 2n.The method exploited the freedom in the choice of the parametrization and achieved the order 4n/3,rather than n + 1. Generalizations were also proved for space curves. Professor Rababah is also do-ing research in the fields of classical approximation theory, orthogonal polynomials, Jacobi-weightedorthogonal polynomials on triangular domains, and best uniform approximations. Since 1992,He has been teaching at German, Jordanian, American, Canadian, and Emirates’ universities. Heis active in the editorial boards of many journals in mathematics and computer science. Further infocan be found on his homepage at ResearchGate:https://www.researchgate.net/profile/Abedallah−Rababahor at http://www.just.edu.jo/eportfolio/Pages/Default.aspx?email=rababahConformable Chebyshev differential equation of first kind (Abedallah Rababah)

Conformable chebyshev differential Equation of first kind

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