The higher-order heat-type equations via signed Lévy stable and generalized Airy functions (Q2857962)

This article presents a very interesting study of higher-order heat-type equations (HOHTE) with first time and \(M\)th spatial partial derivatives, where \(M=2,3,\dots\). This is related to the idea of generalizations of Brownian motion. In this approach it is assumed that for the normal heat equation one can compute the \(M\)th-order derivative having different models with different coefficients that depend on the original heat equation. It is possible to compute some general solutions for this approach and in the literature one can find some interesting examples. Especially, in this paper, we have a purpose for a new type of integral transform which will furnish the long-time behavior for an integer \(M>2\).NEWLINENEWLINEThis paper is divided into six sections. After the introduction, in Section 2, there is a presentation of the operational methods for generalizing the Gauss-Weierstrass transform. Having this, the authors show in Section 3 and 4 exact and explicit forms of the integral kernels. In Section 3, one can also find the study of Lévy signed functions associated with the even values of \(M\). For odd \(M\) and generalized Airy function one is asked to look into Section 4, respectively. In Section 5, there is a presentation of solutions of specific examples of HOHTE. The conclusions can be found in Section 6.