Nield-Kuznetsov Functions: Current Advances and New Results

Abstract

In this article, we discuss a class of functions known as the Nield-Kuznetsov functions, introduced over the past decade. These functions arise in the solutions to inhomogeneous Airy’s and Weber’s equations. Derivations of these functions are provided, together with their methods of computations

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

“…Values of 𝑁𝑖 (𝑚) (𝑥 0 ), for 𝑚 = 2,3, … , 𝑁, can be evaluated using the following derivative formula, (cf. [6][7][8]):…”

“…Using (27), the following values of 𝑁 (𝑚) (0) are obtained, for 𝑚 = 0,1,2, … ,10: 𝑁𝑖(0) = 𝑁𝑖 ′ (0) = 𝑁𝑖 ′′′ (0) = 𝑁𝑖 (4) (0) = 𝑁𝑖 (6) (0) = 𝑁𝑖 (7) (0) = 𝑁𝑖 (9) (0) = 𝑁𝑖 (10) (0) = 0, 𝑁𝑖 ′′ (0) = − 1 𝜋 ; 𝑁𝑖 (5) = − 3 𝜋 ; 𝑁𝑖 (8) = − 18 𝜋 . Using the values of 𝑟 𝑚 (0) and 𝑞 𝑚 (0) of Table 3 and the above input in (28) results in: 3 3!…”

“…These methods have been developed in the literature, and received considerable validation (cf. [6][7][8] and the references therein).…”

The Sigmoid Neural Network Activation Function and its Connections to Airy’s and the Nield-Kuznetsov Functions

“…Airy's ODE, [8][9][10], has received considerable attention in the literature, and general approaches to solutions of Airy's inhomogeenoeus ODE have been introduced, (cf. [11][12][13][14][15][16][17][18][19] and the references therein). In addition to its importance in mathematical physics, solutions to the inhomogeneous Airy's ODE when its forcing function is a general function of the independent variable, give rise to new functions that are important in the advancement of our mathematical library of functions.…”