The Higher-Order Heat-Type Equations via signed Lévy stable and generalized Airy functions

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.

11 pages, 6 figures; several typos corrected

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• Theory of relativistic heat polynomials and one-sided Lévy distributions
• Green's functions and the Cauchy problem of the Burgers hierarchy and forced Burgers equation
• On Fundamental Solutions of Higher-Order Space-Fractional Dirac equations
• On the zeros of the Pearcey integral and a Rayleigh-type equation
• Higher-order heat equation and the Gelfand-Dickey hierarchy