Modelling power-law ultrasound absorption using a time-fractional, static memory, Fourier pseudo-spectral method

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.