On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

An analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by Phillips and Karageorghis (27 (1990) 823). A new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given. The tensor product of ultraspherical polynomials is used to approximate a function of more than one variable. Formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described.

[1]  L. Fox,et al.  Chebyshev polynomials in numerical analysis , 1970 .

[2]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[3]  Eid H. Doha,et al.  An accurate solution of parabolic equations by expansion in ultraspherical polynomials , 1990 .

[4]  Eid H. Doha,et al.  The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function , 1998 .

[5]  Timothy Nigel Phillips,et al.  On the coefficients of differentiated expansions of ultraspherical polynomials , 1992 .

[6]  T. Phillips,et al.  Preconditioners for the spectral multigrid method , 1986 .

[7]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[8]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[9]  Chebyshev spectral methods for solving two-point boundary value problems arising in heat transfer , 1988 .

[10]  Eid H. Doha,et al.  Efficient Spectral-Galerkin Algorithms for Direct Solution of Second-Order Equations Using Ultraspherical Polynomials , 2002, SIAM J. Sci. Comput..

[11]  Timothy Nigel Phillips,et al.  On the Legendre Coefficients of a General-Order Derivative of an Infinitely Differentiable Function , 1988 .

[12]  Eid H. Doha,et al.  The coefficients of differentiated expansions and derivatives of ultraspherical polynomials , 1991 .

[13]  Eid H. Doha,et al.  An Efficient Double Legendre Spectral Method for Parabolic and Elliptic Partial Differential Equations , 2001, Int. J. Comput. Math..

[14]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[15]  Andreas Karageorghis,et al.  A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function , 1988 .

[16]  Timothy Nigel Phillips,et al.  On the coefficients of integrated expansions of ultraspherical polynomials , 1990 .

[17]  Guo Ben-Yu,et al.  Gegenbauer Approximation and Its Applications to Differential Equations on the Whole Line , 1998 .

[18]  Thomas Hagstrom,et al.  An efficient spectral method for ordinary differential equations with rational function coefficients , 1996, Math. Comput..