Chebyshev spectral methods for solving two-point boundary value problems arising in heat transfer
Computer Methods in Applied Mechanics and Engineering
Journal of Computational and Applied Mathematics
On the coefficients of integrated expansions of ultraspherical polynomials
SIAM Journal on Numerical Analysis
On the coefficients of differentiated expansions of ultraspherical polynomials
Applied Numerical Mathematics
An efficient spectral method for ordinary differential equations with rational function coefficients
Mathematics of Computation
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Journal of Computational and Applied Mathematics
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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.