Stochastic solution to a time-fractional attenuated wave equation

Mark M Meerschaert, Peter Straka, Yuzhen Zhou, Robert J McGough

Abstract

The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.