Limitations on nonlinear fractional calculus models used in biomedical acoustics

Abstract

Fractional calculus models used for biomedical ultrasound are associated with attenuation proportional to ω y, where y is typically in the range 1 < y < 2. To determine whether the attenuation and accompanying dispersion are sufficient to stabilize shock formation, the models are formulated as a Burgers equation with the traditional loss term replaced by a fractional derivative of order y. For y < 1 the resulting equation predicts unphysical solutions beyond the shock-formation distance. The second example pertains to nonlinear Lucassen interface waves, a model equation for which has been proposed to describe mechanical perturbations that accompany the transmission of nerve impulses. Linear Lucassen waves are defined by a second-order space derivative and a fractional time derivative of order 3/2, which falls between order 2 in the wave equation and order 1 in the diffusion equation. The resulting attenuation is proportional to ω3/4, and the corresponding nonlinear “fractional diffusive waves,” while strongly attenuated on the scale of a wavelength, may be lacking essential physics beyond the predicted shock-formation distance. Calculations are presented that determine wave amplitudes and propagation distances for which these two fractional calculus models may be of questionable physical significance due to nonlinearity. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]