A note on fractional derivative modeling of broadband frequency-dependent absorption: Model III

Keywords: fractal geometry, fractional derivative, fractional Fourier transform, fractional power of a matrix, self similarity, complex partial differential equation, broadband ultrasound, frequency-dependent attenuation, time domain. 1. Backgrounds The rational behind this model is schematically illustrated below: Fractal geometry (irregular soft tissues) → Fractional Fourier transform (frequency-dependent attenuation: , y∈[0,2] is real valued) → Fractional derivative (Fourier transform () y ω α ω α 0 = () () (ω ω P j FT s +) t t p s s − =         ∂ ∂ , s is real valued) → Fractional power of the positive discretization matrix of Laplacian (the modified mode superposition model, see Model I) → Macro damping effect (negative real part of frequency domain solution: absorption)