Time-fractional derivatives

The definitions of the fractional integral, Grünwald-Letnikov fractional derivative, Riemann-Liouville fractional derivative, and Caputo fractional derivative are presented. The numerical approximations for the Riemann-Liouville fractional derivative based on the shifted Grünwald-Letnikov formula are provided. The L1 interpolation approximation and L2-1σ interpolation approximation for the Caputo fractional derivative are given. The finite difference methods based on Grünwald-Letnikov formula, L1 formula and L2-1σ formula for the fractional ordinary equation are derived. Four finite difference schemes based on the first-order Grünwald-Letnikov formula, the second-order shifted Grünwald-Letnikov formula, the L1 formula and the L2-1σ formula are constructed for the time-fractional subdiffusion equations. Two difference schemes by using the L1 formula and the L2-1σ formula are developed for the time-fractional diffusion-wave equations. For each scheme, the convergence result is given.