ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

doi:10.1016/S0252-9602(15)60004-2

Abstract

Abstract:

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

Key words: Chebyshev polynomials of the third and fourth kinds, expansion coefficients, generalized hypergeometric functions, boundary value problems

CLC Number:

42C10

Eid H. DOHA, Waleed M. ABD-ELHAMEED, Mahmoud A. BASSUONY. ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS[J].Acta mathematica scientia,Series B, 2015, 35(2): 326-338.

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References

[1] Bialecki B, Fairweather G, Karageorghis A. Matrix decomposition algorithms for elliptic boundary value problems: a survey. Numer Algor, 2011, 56: 253-295

[2] Doha E H, Abd-Elhameed W M. Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. SIAM J Sci Comput, 2002, 24: 548-571

[3] Doha E H, Abd-Elhameed W M. Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for the direct solution of (2n+1)th-order linear differential equations. Math Comput Simulat, 2009, 79: 3221-3242

[4] Doha E H, Bhrawy A H. A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer Meth Part D E, 2009, 25: 712-739

[5] Doha E H, Abd-Elhameed W M, Youssri Y H. Efficient spectral-Petrov-Galerkin methods for third-and fifth-order differential equations using general parameters generalized Jacobi polynomials. Quaest Math, 2013, 36(1): 15-38

[6] Gheorghiu C I. Spectral methods for differential problems. Cluj-Napoca: Casa Cartii de Stiinta Publishing House, 2007

[7] Lewanowicz S. Recurrence relations for the coefficients in Jacobi series solutions of linear differential equations. SIAM J Math Anal, 1986, 17: 1037-1052

[8] Lewanowicz S. A new approach to the problem of constructing recurrence relations for the Jacobi coeffi- cients. Appl Math, 1991, 21: 303-326

[9] Lewanowicz S. Quick construction of recurrence relations for the Jacobi coefficients. J Comput Appl Math, 1992, 43: 355-372

[10] Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral methods in fluid dynamics. New York: Springer-Verlag, 1988

[11] Karageorghis A. A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function. J Comput Appl Math, 1988, 21: 129-132

[12] Phillips T N. On the Legendre coefficients of a general-order derivative of an infinitely differentiable function. IMA J Numer Anal, 1988, 8: 455-459

[13] Karageorghis A, Phillips T N. On the coefficients of differentiated expansions of ultraspherical polynomials. Appl Num Math, 1992, 9: 133-141

[14] Doha E H. The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. Comput Math Appl, 1991, 21: 115-122

[15] Doha E H. On the coefficients of differentiated expansions and derivatives of Jacobi polynomials. J Phys A: Math Gen, 2002, 35: 3467-3478

[16] Doha E H. On the connection coefficients and recurrence relations arising from expansions in series of Hermite polynomials. Integr Transf Spec F, 2004, 15: 13-29

[17] Doha E H. On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials. J Phys A: Math Gen, 2003, 36: 5449-5462

[18] Doha E H, Ahmed H M. Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomilas. J Phys A: Math Gen, 2004, 37: 8045-8063

[19] Chandrasekhar S. Hydrodynamic and hydromagnetic stability. Oxford: Clarenden Press, 1961

[20] Wazwaz A M. Approximate solutions to boundary value problems of higher order by the modified decom- position method. Comput Math Appl, 2000, 40(6): 679-691

[21] Siddiqi S S, Twizell A H. Spline solutions of linear twelfth-order boundary-value problems. J Comput Appl Math, 1997, 78: 371-390

[22] Siddiqi S S, Akram G. Solutions of twelfth order boundary value problems using thirteen degree spline. Appl Math Comput, 2006, 182: 1443-1453

[23] Islam S-U, Haq S, Ali J . Numerical solution of special 12th-order boundary value problems using differential transform method. Commun Nonlinear Sci Numer Simulat, 2009, 14: 1132-1138

[24] Agarwal R P. Boundary value problems for high ordinary differential equations. Singapore: World Scientific, 1986

[25] Boyd J. Chebyshev and Fourier spectral methods. Mineola: Courier Dover Publications, 2001

[26] Doha E H. The first and second kind Chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function. Int J Comput Math, 1994, 51: 21-35

[27] Fox L, Parker I B. Chebyshev polynomials in numerical analysis. Oxford: OUP, 1972

[28] Mason J C. Chebyshev polynomial approximations for the L-membrane eigenvalue problem. SIAM J Appl Math, 1967, 15: 171-186

[29] Aghigh K, Masjed-Jamei M, Dehghan M. A survey on third and fourth kind of Chebyshev polynomials and their applications. Appl Math Comput, 2008, 199: 2-12

[30] Eslahchi M R, Dehghan M, Amani S. The third and fourth kinds of Chebyshev polynomials and best uniform approximation. Math Comput Model, 2012, 55: 1746-1762

[31] Gautschi W. On mean convergence of extended Lagrange interpolation. J Comput Appl Math, 1992, 43: 19-35

[32] Mason J C. Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. J Comput Appl Math, 1993, 49: 169-178

[33] Notaris S E. Interpolatory quadrature formulae with Chebyshev abscissae of the third or fourth kind. J Comput Appl Math, 1997, 81: 83-99

[34] Doha E H, Abd-Elhameed W M. On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds. Bull Malays Math Sci Soc (2), 2014, 37(2): 383-398

[35] Mason J C, Handscomb D C. Chebyshev polynomials. London: Chapman&Hall, 2003

[36] Shen J, Tang T, Wang L L. Spectral methods: algorithms, analysis and applications. Springer, 2011