High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations

In this paper, a high order approximation with convergence order $$O( {{\tau ^{3 - \alpha }}})$$ to Caputo derivative $$_{C}D_{0,t}^{\alpha}f(t)$$ for $$\alpha\in(0,1)$$ is introduced. Furthermore, two high-order algorithms for Caputo type advection-diffusion equation are obtained. The stability and convergence are rigorously studied which depend upon the derivative order \alpha. The corresponding convergence orders are $$O(\tau^{3-\alpha}+h^2)$$ and $$O(\tau^{3-\alpha}+h^4)$$ where \tau is the time stepsize, h the space stepsize, respectively. Finally, numerical examples are given to support the theoretical analysis.