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

We describe and implement a numerical method for modelling the frequency-dependent power-law absorption of ultrasound in tissue, as governed by 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. The resulting numerical method requires a fixed (static) memory cost, irrespective of the number of time steps. The spatial domain is treated by the Fourier spectral method. 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 operator is the local treatment of the power-law, allowing for a spatially varying frequency power-law.