A new analytical technique of the [formula omitted]-type difference schemes for time fractional mixed sub-diffusion and diffusion-wave equations

Abstract

A time fractional mixed sub-diffusion and diffusion-wave equation involving at least two Caputo time derivatives of order γ∈(1,2) and α∈(0,1) is considered. A new analytical technique is introduced to analyze the standard finite difference method based on L1 scheme. Both stability and convergence of the scheme are proved rigorously. The final convergence result shows clearly that the temporal accuracy arrives at the order of O(τmin{2−α,3−γ}) in a discrete H1-norm. This novel analytical technique can provide new insights in analyzing other multi-term mixed time fractional differential equations.