On fractional parabolic systems of vector order

TL;DR

This paper investigates the existence of classical solutions for a system of fractional parabolic PDEs with vector orders of differentiation, where the orders differ across equations and are not necessarily rational, extending prior work with uniform fractional orders.

Contribution

It introduces conditions for the existence of classical solutions to fractional parabolic systems with non-uniform, possibly irrational, fractional orders, a novel extension over previous studies with uniform orders.

Findings

Established sufficient (and sometimes necessary) conditions for solutions.

Extended the theory to systems with non-uniform fractional orders.

Addressed systems with elliptic diagonal operators and triangular structure.

Abstract

The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of differential operators $\mathbb{A}(D)$ is triangular (elements above or below the diagonal are zero). Operators located on the diagonal are elliptic. The main distinctive feature of this system is that the vector-order $\mathcal{B}$ has different components $\beta_j\in (0,1]$, and $\beta_j$ are not necessarily rational. Sufficient conditions (in some cases they are necessary) on the initial function and the right-hand side of the equation are found to ensure the existence of a classical solution. Note that the existence of a classical solution to systems of fractional differential equations was studied by various authors, but in all these works the fractional order had the same components for each equation: $\beta_j=\beta$, $j=1,...,m$.