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

In this paper, a series of new high-order numerical approximations to α th ( 0 < α < 1 ) order Caputo derivative is constructed by using r th degree interpolation approximation for the integral function, where r ? 4 is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et?al., 2015; Li et?al., 2014), the convergence orders are O ( ? r + 1 - α ) , where ? is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection-diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O ( ? r + 1 - α + h 2 ) , where ? is the time stepsize, and h the space one. The stability and convergence of the proposed schemes for r = 4 are also considered. Without loss of generality, we only display the numerical examples for r = 4 , 5 , which support the numerical algorithms.