Linear and nonlinear wave equation models with power law attenuation

Abstract

Motivated by the need to model high intensity focused ultrasound in lossy media we study linear and nonlinear wave equations that contain non local time fractional derivatives, whose inclusion in our models incorporates the effects of acoustic attenuation. This is characterized by a frequency dependent power law parametrized by a non integer γ ∈ (0, 2), leading to the need for fractional derivative damping. Issues with such integro-differential equations arise in the continuous and discrete analysis due to singularities that occur at t = 0 and as a result of their non local nature. To address these issues we present results that carefully show how to treat such equations for smooth and non smooth solutions, and we derive fast and efficient numerical schemes that pay particular attention to the handling of these non local operators. As well as acoustic attenuation the models we consider will need to account for nonlinear propagation effects that result from the focusing of the ultrasound waves and be modelled on an unbounded domain. To combat this issue, we use a perfectly matched layer. That is, we truncate the unbounded domain to a finite computational domain and impose an absorbing, non reflecting boundary layer around it.