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 . DOI: 10.1016/S0252-9602(15)60004-2

[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