Interpretation of the failure of the time-independent diffusion equation near a point source

Abstract

The time-independent diffusion equation (DE) yields inaccurate solutions near sources. To investigate this issue we solve a forward problem of light propagation from a time-independent isotropic point source in an infinite medium of constant refractive index, constant absorption coefficient and constant reduced scattering coefficient using the standard radiative transfer equation (RTE) and three recently reported RTEs. Only one of those equations yields solutions consistent with the results of Monte Carlo simulations in near field, due to its ability to model ray divergence. From that equation we derive a condition for neglecting the divergence of rays emitted by a point source in forward problems that is the same as the reported condition for the accuracy of the time-independent DE. These results strongly suggest that the inaccuracy of the time-independent DE near sources is due to a missing term describing ray divergence, a drawback inherited from the RTE. One of the studied RTEs and the derived equations are free of that drawback.