$\alpha$-robust error analysis of two nonuniform schemes for Caputo-Hadamard fractional reaction sub-diffusion problems

In this paper, we focused on the Caputo-Hadamard fractional reaction sub-diffusion equations. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in the temporal domain, we formulated two efficient numerical schemes, where the second order difference method was used in the spatial dimension. Furthermore, we derived the stability and convergence of these proposed schemes. Remarkably, both derived numerical methods exhibited $\alpha$-robustness, that is, it remained valid when $\alpha \to 1^{-}$. Numerical experiments were given to demonstrate the theoretical statements.