New formulae for higher order derivatives and applications

Authors:
Richard M. Slevinsky;Hassan Safouhi
Affiliations:
Mathematical Section, Campus Saint-Jean, University of Alberta, 8406, 91 Street, Edmonton (AB), Canada T6C 4G9;Mathematical Section, Campus Saint-Jean, University of Alberta, 8406, 91 Street, Edmonton (AB), Canada T6C 4G9
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

We present new formulae (the Slevinsky-Safouhi formulae I and II) for the analytical development of higher order derivatives. These formulae, which are analytic and exact, represent the kth derivative as a discrete sum of only k+1 terms. Involved in the expression for the kth derivative are coefficients of the terms in the summation. These coefficients can be computed recursively and they are not subject to any computational instability. As examples of applications, we develop higher order derivatives of Legendre functions, Chebyshev polynomials of the first kind, Hermite functions and Bessel functions. We also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations. We also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical Bessel integral functions.