Causality analysis of the fractional state and wave equations based on fractional Laplacians

Abstract

This paper is concerned with the investigation of the state and wave equations based on fractional Laplacians which have some popularity in Photoacoustic tomography (PAT). It is shown that such state equations are nonlocal, more precisely, a local density variation causes an instant global pressure variation and a local pressure variation can only be caused by an instant global density variation. This is in contrast to all frequency dependent dissipative state equations known to the author. Moreover, it is shown that the Green function G of the respective wave equation does not have a finite wave front speed. To obtain a fractional wave equation with a finite wave front speed, we propose a local fractional state equation that is similar to the original fractional state equation. We note that our approach is readily applied to many state and wave equations based on fractional space derivatives. Finally, we present numerical simulations that clearly visualize the noncausal behavior of the underlying space fractional state equations.