Time-Fractional Telegrapher's Equation (P1) Approximation for the Transport Equation

In this technical note we develop a new approximation from the solution of the time-dependent Boltzmann equation, which includes a fractional constitutive equation of the neutron current density, for a general medium. The fractional constitutive equation in combination with the conservation law that governs the particle collision and reaction processes (P-1) approximation for the transport equation gives a time-fractional telegrapher's equation (TFTE). The wave velocity found with this approximation is 3(-gamma/2) for gamma < 1. The numerical results are compared with the exact solution and Heizler's approximation. We found that the TFTE gives the best estimate for a purely absorbing medium, where most approximations fail. The asymptotic diffusion coefficient was applied for a heterogeneous medium, and the results show that the behavior of the TFTE improves.