Abstract
We study a time-fractional diffusion--wave equation governed by a Sturm--Liouville operator with singular potential, singular initial data, and singular forcing, for fractional orders between zero and two. For bounded nonnegative potentials, we derive spectral solution representations in terms of Mittag–Leffler functions and prove well-posedness with estimates uniform in the potential. We distinguish the subdiffusive and diffusion–wave regimes through their appropriate initial data and establish additional time regularity in the latter case. We then treat distributional potentials and possibly singular data and forcing by positivity-preserving regularisation. Within the framework of very weak solutions, we prove existence and uniqueness modulo negligible nets. Finally, we establish consistency: when the coefficients and data are regular, the regularised solutions converge in the natural continuous-in-time square-integrable topology to the classical solution.