  We summarise and implement a numerical method for evaluating a numerical
method for modelling the frequency dependent power-law absorption within
ultrasound using the first order linear wave equations with a loss taking the
form of a fractional time derivative. The (Caputo) fractional time derivative
requires the full problem history which is contained within an iterative
procedure with the resulting numerical method requiring a static memory at
across all time steps without loss of accuracy. The Spatial domain is treated
by the Fourier k-space method, with derivatives on a uniform grid. Numerically
comparisons are made against a model for the same power-law absorption with
loss described by the fractional- Laplacian operator. One advantage of the
fractional time derivative over the Fractional Laplacian is the local treatment
of the power-law, allowing for a spatially varying frequency power-law.
