Higher Derivatives and Polynomials of the Standard Nield-Kuznetsov Function of the First Kind

Abstract

In this fundamental work, higher derivatives of the standard Nield-Kuznetsov function of the first kind, and the polynomials arising from this function and Airy’s functions, are derived and discussed. This work provides background theoretical material and computational procedures for the arising polynomials and the higher derivatives of the recently introduced Nield-Kuznetsov function, which has filled a gap that existed in the literature since the nineteenth century. The ease by which the inhomogeneous Airy’s equation can now be solved is an advantage offered by the Nield-Kuznetsov functions. The current analysis might prove to be invaluable in the study of inhomogeneous Schrodinger, Tricomi, and Spark ordinary differential equations.

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

“…Using the value 𝐿𝑖 2 (−1) = − 𝜋 2 12 in (21), and subsequent use in (15), the general solution to (14) can be written as:…”

“…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.…”