The Higher-Order Heat-Type Equations via signed L\'{e}vy stable and generalized Airy functions

TL;DR

This paper develops explicit solutions for higher-order heat-type equations using signed Lévy stable functions for even orders and a generalized Airy function for odd orders, enabling detailed analysis of their evolution.

Contribution

It introduces a novel approach to constructing exact heat kernels for higher-order equations using specialized functions, expanding analytical tools in this area.

Findings

Exact solutions for even-order equations via signed Lévy stable functions.

Introduction of a generalized Airy function for odd-order equations.

Graphical analysis of solution evolution over space and time.

Abstract

We study the higher-order heat-type equation with first time and M-th spatial partial derivatives, M = 2, 3, ... . We demonstrate that its exact solutions for M even can be constructed with the help of signed Levy stable functions. For M odd the same role is played by a special generalization of Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spacial and temporary evolution of particular solutions for simple initial conditions.