The higher-order heat-type equations via signed Lévy stable and generalized Airy functions

We study the higher-order heat-type equation with first time and Mth spatial partial derivatives, M = 2, 3, …. We demonstrate that its exact solutions for M even can be constructed with the help of signed Lévy stable functions. For M odd the same role is played by a special generalization of the 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 spatial and temporary evolution of particular solutions for simple initial conditions. (paper)