Taylor and Maclaurin Series Representations of the Nield-Kuznetsov Function of the First Kind

Abstract

Taylor and Maclaurin series and polynomial approximations of the Standard Nield-Kuznetsov function of the first kind are obtained in this work. Convergence and error criteria are developed. The obtained series represent alternatives to the existing asymptotic and ascending series approximations of this integral function, and are expected to provide an efficient method of computation that is valid for all values of the argument.

A Note on Taylor Series Representation of the Nield-Kuznetsov Function of the Second Kind

“…In a recent article, Alderson and Hamdan, [1], investigated Taylor and Maclaurin series and polynomial representations of the standard Nield-Kuznetsov function of the first kind, 𝑁𝑖(𝑥). This function arises in the solution to Airy's inhomogeneous ordinary differential equation, [2], of the form…”

“…where 𝑅 ∈ ℜ. Hamdan and Kamel, [3], showed that the general solution to (1) is expressible in the form:…”

“…When the forcing function is a differentiable function, 𝑓(𝑥), that replaces 𝑅 in (1), general solution to (1) is expressible in the form, [3]: 𝑦 = 𝑐 1 𝐴𝑖(𝑥) + 𝑐 2 𝐵𝑖(𝑥) + 𝜋{𝐾𝑖(𝑥) − 𝑓(𝑥)𝑁𝑖(𝑥)} (5) where the function 𝐾𝑖(𝑥) is the Standard Nield-Kuznetsov Function of the second kind. This function takes the following equivalent forms, [3]:…”

“…Taylor and Maclaurin series representations for 𝑁𝑖(𝑥), [1], provided a convenient, time-saving method of computations pf 𝑁𝑖(𝑥), for values of 𝑥 that are small enough for ascending series evaluations. Accurate results can be obtained using a Maclaurin polynomial of degree less than 15.…”

“…Accurate results can be obtained using a Maclaurin polynomial of degree less than 15. Motivated by the findings of Alderson and Hamdan, [1], the current work aims at obtaining Taylor and Maclaurin series expansions and polynomial representations of 𝐾𝑖(𝑥). To this end, higher derivatives of 𝐾𝑖(𝑥), together with Airy's polynomials, are needed in the analysis that follows.…”