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Numerical computation of time-domain Green's functions for the Treeby-Cox space-fractional wave equation

The attenuation of ultrasound in soft tissue follows a frequency-dependent power law. As an alternative to the time-fractional partial differential equations that describe power law attenuation and dispersion, Treeby and Cox derived a space-fractional model that accounts for both power law attenuation and dispersion. The Treeby-Cox space-fractional wave equation is convenient for simulations with the pseudo-spectral approach; however, numerical calculations of the time-domain Green's function for the Treeby-Cox wave equation are more challenging because numerical evaluation of a highly oscillatory improper integral is required. When applied to this problem, most standard numerical integration techniques perform poorly, so an alternative approach is required.