Numerical evaluation of time-domain Green's functions for space-fractional wave equations

Abstract

Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency. Several different models for the power law attenuation of medical ultrasound have been developed using fractional calculus, where each contains one or more time- or space-fractional derivatives. For certain time-fractional models, exact and approximate time-domain Green's functions have been derived and evaluated numerically, but a similar analysis has not yet been performed on the space-fractional models. To address this deficiency, time-domain Green's functions are numerically calculated here for two space-fractional models, namely the Chen-Holm and Treeby-Cox space-fractional wave equations. Numerical results are computed for both of these in breast and liver with power law exponents of 1.5 and 1.139, respectively. The results show that these two space-fractional wave equations are causal everywhere. Away from the origin, the time-domain Green's function for the dispersive Treeby-Cox space-fractional wave equation is very similar to the time-domain Green's functions calculated for the corresponding time-fractional wave equations, but the time-domain Green's function for the nondispersive Chen-Holm space-fractional wave equation is quite different. To highlight the similarities and differences between these, time-domain Green's functions are compared and evaluated at different distances for both breast and liver.