Nonuniform time-stepping methods are promising for Caputo reaction sub-diffusion problems because they would be simple and effectiveness in resolving the initial singularity and other nonlinear behaviors occurred away from the initial time. Compared with traditional local methods for the first-order derivative, the numerical analysis for nonlocal time-stepping schemes on non-uniform time meshes are challenging due to the convolution integral (nonlocal) form of fractional derivative. We develop a general framework for the stability and convergence analysis with three tools: a family of complementary discrete convolution kernels, a discrete fractional Gronwall inequality (DFGI) and a global (convolutional) consistency analysis, which is not limited to a specific time mesh by building a convolution structure of local truncation error. It seems that the present techniques are extendable to the variable-order, distributed-order diffusion equations and other nonlocal-in-time diffusion problems.
Prof Zhang is a Professor of School of Mathematics and Statistics, Wuhan University. He got his PhD degree in Hong Kong Baptist University in 2009. His research area include fast algorithms for fractional PEDs and numerical methods for PDEs.