On the Asymptotic Forms of the Solutions of Ordinary Linear Differential Equations of the Third Order in a Region Containing a Turning Point

On the Ni(x) integral function and its application to the Airy’s non-homogeneous equation

“…In particular, Ni(x) satisfies a third-order ODE (Eq. (A.4.3) of Appendix A) that is a special case of Langer's comparison ODE, [4,5,12]. We have only considered the definition of Ni(x) for x > 0 and no attempt has been made to consider its extensions of this function over the entire real line.…”

“…Many problems in mathematical physics that are governed by ordinary differential equations (ODE) are apt to reduction to Airy's ODE by an appropriate transformation or a change of variable [4,5,8,11,12]. A well-known form of Airy's ODE in the unknown function y(x) is given by Eq.…”

“…The aim of the current work is to present formulation, and outline the method of solution to some initial value problems involving ODE (1), using the Nield-Kuznetsov function, and to discuss some of its elementary properties and the differential equations it satisfies (such as Langer's comparison ODE, [4,5,12]). No claim is made in the current work that the properties discussed and developed are exhaustive; indeed, many additional would-be interesting properties of the Nield-Kuznetsov function are yet to be discovered.…”

“…ODE (A.4.3)is a special case of the so-called comparison ODE that received interest in the literature (see[12], page 108, and Langer,[4,5]), namely:BiðtÞdt ¼ p Bi 0 ðxÞGiðxÞ À Gi 0 ðxÞBiðxÞ…”

Fundamental solutions of a linear ordinary differential equation of the third order in the neighborhood of a simple second order turning point

“…The method of associated regions [3] is employed for this purpose. This is followed by important lemmas on the boundedness of certain types of functions and integrals which occur in deducing solutions of the given equation from those of the rela-ted equation. It has been shown (c.f.…”