Inhomogeneous Airy’s and Generalized Airy’s Equations with Initial and Bounday Conditions

Abstract

Inhomogeneous Airy’s and Generalized Airy’s equations with initial and boundary date are considered in this work. Solutions are obtained for constant and variable forcing functions, and general solutions are expressed in terms of Standard and Generalized Nield-Kuznetsov functions of the first- and second-kinds. Series representations of these functions and their efficient computation methodologies are presented with examples.

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

“…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!…”

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

“…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.…”