Fractional Generalization of Higher-Order Diffusion

TL;DR

This paper introduces a fractional extension of higher-order diffusion theory, deriving fundamental solutions by properly handling the fractional Laplacian, thus broadening the mathematical framework for diffusion models.

Contribution

It presents a novel fractional generalization of higher-order diffusion theory, including the derivation of fundamental solutions and a proper treatment of the fractional Laplacian.

Findings

Derived fundamental solutions for the fractional higher-order diffusion equation

Extended the theory from integer to fractional cases using fractional Laplacian

Established a mathematical framework analogous to fractional gradient elasticity

Abstract

A fractional generalization of the second author's higher-order diffusion theory is given and fundamental solutions are obtained. The extension from the integer to the fractional case involves a proper treatment of the fractional Laplacian of Riesz type. This is done with analogy to an earlier treatment in extending the second author's gradient elasticity model from the integer to the fractional case.