SOME RESULTS ON THE COEFFICIENTS OF INTEGRATED EXPANSIONS OF ULTRASPHERICAL POLYNOMIALS AND THEIR INTEGRALS

摘要

The formula of expressing the coefficients of an expansion ofultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.CLC Number：O17 Document ID：AAuthor Resume：E. H. Doha，e-mail: eiddoha@frcu, eun. eg References：[1]Canuto,C. ,Spectral Methods in Fluid Dynamics,Springer,Belrin,1988.[2]Doha,E.H.,An Accurate Solution of Parabolic Equations by Expansion in Ultraspherical Polynomials,Comput. Math. Appl. ,19(1990),75-88.[3]Doha,E. H. ,The Coefficients of Differentiated Expansions and Derivatives of Ultraspherical Polynomials,Comput. Math. Appl.,21(1991),115-122.[4]Doha,E.H. ,The Chebyshev Coefficients of General order Derivatives of an Infinitely Differen-tiable Function in Two or Three Variables,Ann. Univ. Sci. Budapest. Sect. Comput. ,13(1992),83-91.[5]Doha,E. H.,On the Cefficients of Differentiable Expansions of Double and Triple Legendre Polynomials,Ann Univ. Sci. Budapest. Sect. Comput. ,15(1995),25-35.[6]Doha,E.H. ,The Ultraspherical Coefficients of the Moments of a General-Order Derivatives of an Infinitely Differentiable Function,J. Comput. Math. ,89(1998),53-72.[7]Doha,E.H. ,The Coefficients of Differentiated Expansions of Double and Triple Ultraspherical Polynomials,Annales Univ. Sci. Budapest.,Sect. Comp.,19(200),57-73.[8]Doha,E.H. and Al-Kholi,F. M. R. ,An Efficient Double Legerdre Spectral Method for Parabolic and Elliptic Partial Differential Equations,Intern. J. Computer. Math. (toAppear).[9]Fox,L. and Parker,I.B. ,Chebyshev Polynomials in Numerical Analysis,Clarendon Press,Oxford,1972.[10]Doha,E.H. and Helal,M. A. ,An Accurate Double Chebyshev Spectral Approximation for Parabolic Partial Differential Equations,J. Egypt. Math. Soc.,5 (1997),No. 1,83-101.[11]Gottlieb,D. and Orszag,S.A. ,Numerical Analysis of Spectral Methods: Theory and Applications,CBMS-NSF Regional Conf. Series in Applied Mathematics,Vol.[2]6,Society for Industrial and Applied Mathamatics,Philadelphia,PA,1977.[12]Karageorghis,A. ,Chebyshev Spectral Methods for Solving Two-Point Boundary Value Problems Arising in Heat Transfer,Comput. Methods Appl. Mech. Eng. ,70(1988),103-121.[13]Karageorghis,A. ,A Note on the Chebyshev Coefficients of the General-Order Derivative of an Infinitely Differentiable Function,J. Comput. Appl. Math.,21(1988),129-132.[14]Karageorghis,A. ,A Note on the Chebyshev Coefficients of the Moments of the General Order Derivative of an Infinitely Differentiable Function,J. Comput. Appl. Math. ,21(1988),383-386.[15]Karageorghis,A. and Phillips,T.N. ,On the Coefficients of Differentiated Expansions of UItraspherical Polynomials,ICASE Report No. 89- 65,NASA Langley Research Center,Hampton,VA,1989 and Appl. Num. Math.,9(1992),133-141.[16]Luke,Y. ,The Special Functions and Their Approximations,Vol. 1,Academic Press,New York,1969.[17]Phillips,T.N. ,On the Legendre Coefficients of a General Order Derivative of an Inifintely Differentiable Function,IMA J. Numer. Anal. ,8(1988),455-459.[18]Phillps,T.N. and Karageorghis,A. ,On the Coefficients of Integrated Exapansions of Ultraspherical Polynomials,SIAM J. Numer. Anal. ,27(1990),823-830.Manuscript Received：2000年4月27日Manuscript Revised：2001年5月15日Published：2001年9月1日