Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions

TL;DR

This paper investigates Nield-Kuznetsov functions related to the parabolic Weber equation, deriving connection formulas, asymptotic expansions, and explicit Laplace transform representations, with applications in fluid flow analysis.

Contribution

It introduces new connection formulas, a novel complementary Nield-Kuznetsov function, and explicit Laplace transform representations for parabolic cylinder functions.

Findings

Derived asymptotic expansions valid for large parameters and arguments.

Constructed connection formulas between solutions and the new function.

Explicitly represented Laplace transforms of parabolic cylinder functions.

Abstract

Nield-Kuznetsov functions of the first kind are studied, which are solutions of an inhomogeneous parabolic Weber equation, and have applications in fluid flow problems. Connection formulas are constructed between them, numerically satisfactory solutions of the homogeneous version of the differential equation, and a new complementary Nield-Kuznetsov function. Asymptotic expansions are then derived that are uniformly valid for large values of the parameter and unbounded real and complex values of the argument. Laplace transforms of the parabolic cylinder functions $W(a,x)$ and $U(a,x)$ are subsequently shown to be explicitly represented in terms of the complementary Nield-Kuznetsov function and closely related functions, and from these uniform asymptotic expansions are derived for the integrals.