  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.
