Approximate analytical time-domain Green's functions for the Caputo fractional wave equation

Area of Science:

• Geophysics
• Wave Propagation
• Mathematical Physics

Background:

• The Caputo fractional wave equation models viscoelastic and ultrasound wave propagation, accounting for power-law attenuation and dispersion.
• This model is derived from a causal, fractional constitutive equation.

Purpose of the Study:

• Derive an approximate analytical time-domain Green's function for the 3D Caputo fractional wave equation.
• Investigate the Green's function for power-law exponents greater than one.
• Analyze the behavior of Green's functions in 1D, 2D, and 3D.

Main Methods:

• Developed an approximate analytical time-domain Green's function for the Caputo equation.
• Utilized shifted and scaled maximally skewed stable distributions.
• Computed approximate 1D and 2D Green's functions in terms of stable distributions.

Main Results:

• The 3D Green's function is a scaled stable distribution multiplied by a spherical spreading factor.
• Approximate 1D and 2D Green's functions were computed.
• Decomposition of the Green's function into loss and diffraction components was achieved.

Conclusions:

• The Caputo wave equation can be approximated by a coupled lossless wave equation and a fractional diffusion equation.
• The derived Green's function provides insights into wave phenomena with power-law attenuation and dispersion.