Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

In 1986, Dixon and McKee (Z Angew Math Mech 66:535–544, 1986) developed a discrete fractional Gronwall inequality, which can be seen as generalization of the classical inequality. However, this generalized inequality its variant (Al-Maskari Karaa in SIAM J Numer Anal 57:1524–1544, 2019) have not been widely applied numerical analysis time-stepping methods for time-fractional evolution equations. The main purpose paper is to show how apply prove convergence class nonlinear subdiffusion equations, including popular backward difference type order one two, Crank-Nicolson methods. We obtain optimal $$L^2$$ error estimate space discretization multi-dimensional problems. fast also proved simple manner. present work unifies several existing schemes. Numerical examples are provided verify effectiveness method.