The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different fractional partial differential equations. These models for the power law attenuation of medical ultrasound have been developed using fractional calculus, where each contains one or more time-fractional or space-fractional derivatives. To demonstrate the similarities and differences in the solutions to causal and noncausal fractional partial differential equations, time-domain Green's functions are calculated numerically for the fractional wave equations. For three time-fractional wave equations, namely the power law wave equation, the Szabo wave equation, and the Caputo wave equation, these Green's functions are evaluated for water with a power law exponent of y=2, liver with a power law exponent of y=1.139, and breast with a power law exponent of y=1.5. Simulation results show that the noncausal features of the numerically calculated time-domain response are only evident in the extreme nearfield region and that the causal and the noncausal Green's functions converge to the same time-domain waveform in the farfield. When noncausal time-domain Green's functions are convolved with finite-bandwidth signals, the noncausal behavior in the time-domain is eliminated, which suggests that noncausal time-domain behavior only appears in a very limited set of circumstances and that these time-fractional models are equally effective for most numerical calculations.For the calculation of space-fractional wave equations, time-domain Green's functions are numerically calculated for two space-fractional models, namely the Chen-Holm and Treeby-Cox wave equations. Numerical results are computed for these in breast and liver. 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 breast and liver parameters. When time-domain Green's functions are convolved with finite-bandwidth signals, the phase velocity difference in these two space-fractional wave equations is responsible for a time delay that is especially evident in the farfield.The power law wave equation is also utilized to implement a perfectly matched layer (PML) for numerical calculations with the Khokhlov - Zabolotskaya - Kuznetsov (KZK) equation. KZK simulations previously required a computational grid with a large radial distance relative to the aperture radius to delay the reflections from the boundary. To decrease the size of the computational grid, an absorbing boundary layer derived from the power law wave equation. Simulations of linear pressure fields generated by a spherically focused transducer are evaluated for a short pulse. Numerical results for linear KZK simulations with and without the absorbing boundary layer are compared to the numerical results with a sufficiently large radial distance. Simulation results with and without the PML are also evaluated, where these show that the absorbing layer effectively attenuates the wavefronts that reach the boundary of the computational grid.
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