Fractional-Hyperbolic Systems

TL;DR

This paper introduces a class of fractional hyperbolic systems involving Caputo derivatives, constructs their fundamental solutions, and analyzes decay properties outside a fractional light cone, extending classical hyperbolic PDE theory.

Contribution

It defines fractional hyperbolic systems with Caputo derivatives and constructs fundamental solutions with decay estimates, advancing the understanding of fractional PDEs.

Findings

Fundamental solutions exhibit exponential decay outside fractional light cone.

The systems generalize classical hyperbolic PDEs to fractional derivatives.

Provides a framework for analyzing fractional evolution systems.

Abstract

We describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order $\alpha \in (0,1)$ in the time variable $t$ and the first order derivatives in spatial variables $x=(x_1,...,x_n)$, which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone $\{(t,x):\ |t^{-\alpha}x|\le 1\}$.