Fast Evaluation of Linear Combinations of Caputo Fractional Derivatives and Its Applications to Multi-Term Time-Fractional Sub-Diffusion Equations

In the present work, linear combinations of Caputo fractional derivatives are fast evaluated based on the efficient sum-of-exponentials (SOE) approximation for kernels in Caputo fractional derivatives with an absolute error epsilon, which is a further work of the existing results in [13] (Commun. Comput. Phys., 21 (2017), pp. 650-678) and [16] (Commun. Comput. Phys., 22 (2017), pp. 1028-1048). Both the storage needs and computational amount are significantly reduced compared with the direct algorithm. Applications of the proposed fast algorithm are illustrated by solving a second-order multi-term time-fractional sub-diffusion problem. The unconditional stability and convergence of the fast difference scheme are proved. The CPU time is largely reduced while the accuracy is kept, especially for the cases of large temporal level, which is displayed by numerical experiments.