Approximate analytical time-domain Green’s functions for space-fractional wave equations

Abstract

Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the power law exponent and attenuation constant for breast and for liver. The results for both breast and liver converge very close to the source, and similar performance is observed in time-domain Green’s functions computed for linear with frequency attenuation. Despite minor differences in the arguments, the approximate analytical time-domain Green’s functions derived for dispersive time-fractional and space-fractional wave equations are also quite similar. [Work supported in part by NIH Grants EB023051 and EB012079.] Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the po...