On the zeros of the Pearcey integral and a Rayleigh-type equation

TL;DR

This paper investigates the zeros of the Pearcey integral, introduces a related Rayleigh-type ODE, and develops a method to solve certain heat equation boundary problems using generalized Airy functions.

Contribution

It identifies functions where the Pearcey integral vanishes and links them to a nonlinear ODE, providing a new approach to boundary problems in heat equations.

Findings

Identified functions where the Pearcey integral is zero.

Connected these functions to a Rayleigh-type ODE.

Developed a methodology for solving heat equation boundary problems.

Abstract

In this work we find a sequence of functions at which the Pearcey function is identically zero. The sequence of functions can be expressed in terms of a second order non-linear ODE, which happens to be the Rayleigh-type. As a byproduct of these facts, we develop a methodology to find a class of functions which solve the moving boundary problem of the heat equation. To this end, we make use of generalized Airy functions, which in some particular cases fall within the category of functions with infinitely many real zeros, studied by P\'olya.