The development of higher-order numerical differential formulas of Caputo derivative and their applications (I)

Abstract

In this article, two different second-order numerical differential formulas have been derived for the Caputo derivatives CD0,tαft,(0<α<1) and CD0,tβft,(1<β<2) at point tk+1∕2, respectively. By applying these numerical differential formulas to the time fractional mixed sub-diffusion and diffusion–wave equations and combining with the fourth-order compact formula of the spatial derivative, then we get a difference scheme with convergence order Oτ2+h4, which improves the accuracy of the known algorithms in the literature for the same type of equation. Based on the discrete energy method, we prove that the developed difference scheme is unconditionally stable and convergent for α∈(0,1) and β∈1,−1+172. Finally, some numerical results are provided to testify the accuracy and efficiency of the difference scheme.