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Singular source terms in sub-diffusion equations may lead to theunboundedness of solutions, which will bring a severe reduction of convergenceorder of existing time-stepping schemes. In this work, we propose two efficienttime-stepping schemes for solving sub-diffusion equations with a class ofsource terms mildly singular in time. One discretization is based on theGr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimatewith respect to time is rigorously established for singular source terms andnonsmooth initial data. The other scheme derived from the second-order backwarddifferentiation formula (BDF) is proved to possess second-order accuracy intime. Further, piecewise linear finite element and lumped mass finite elementdiscretizations in space are applied and analyzed rigorously. Numericalinvestigations confirm our theoretical results.

Two time-stepping schemes for sub-diffusion equations with singular source terms