Fixed-grid schemes for fractional moving-boundary problems

Abstract

The Landau transformation xi = x/s(t) is the usual way to hold a moving boundary in place. It fails for fractional time derivatives because they have no chain rule: the transformed operator gains a nonlocal integral with a closed form only for self-similar fronts. We develop a sharp-interface finite-difference scheme for the conservation-consistent memory-flux fractional Stefan problem on a fixed grid, storing the memory limit h(x) = s^-1(x) as the time each node melts. The front is advanced using an integrated energy balance, with the singular source timed to the sub-step crossing instant. Against a known self-similar solution, the front constant is accurate to 0.02 to 0.11%, whereas competing formulations differ by 17 to 27% at alpha = 0.6. A remaining positivity violation follows from dt/t0 ~ s0^(1 - 2/alpha) and disappears on a front-locked time grid. We also derive two exact solutions with non-power-law fronts: a source-dominated solution and a Mittag-Leffler traveling wave, and verify the scheme against both. Neural-network solvers for moving boundaries are often trained in normalized coordinates; for alpha < 1, this silently changes the problem, while the change vanishes at alpha = 1, where checks are commonly performed. In the classical limit, our scheme is more accurate in 12 ms than a reimplemented moving-boundary network in 177 s.