An H1 convergence of the spectral method for the time-fractional non-linear diffusion equations

The generalized discrete Gronwall inequality is applied to analyze the optimal H 1 error estimate of the time-stepping spectral method for the time-fractional diffusion equations, where the time-fractional derivative is discretized by the second-order fractional backward difference formula or the second-order generalized Newton-Gregory formula. The methodology is extended to analyze the fractional Crank–Nicolson spectral method and the time-stepping spectral method for the multi-term time-fractional differential equations. Numerical simulations are provided to support the theoretical analysis.