The computational operation and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total *O(MN) *memory and *O(MN ^{2})* work, where

*N*and

*M*represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, an efficient algorithm for the evaluation of the Caputo fractional derivative is presented. This algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the Abel kernel over the interval [dt, T] with a uniform absolute errorε. More importantly, the theoretical analysis is given to show that the number of exponentials

*P*needed is of order

*P = O(log N)*for

*T>>1*or

*P = O(log*for

^{2}N)*T≈1*for fixed accuracyεThe resulting algorithm only requires only

*O(MP)*storage and

*O(MNP)*work when numerically solving the time fractional PDEs. Furthermore, the stability and error analysis of the new scheme is addressed, and several numerical examples are provided to demonstrate the performance of our scheme. For example, fast algorithm is applied to the nonlinear subdiffusion equation, Fig 1. shows the fast algorithm reduces the computational cost significantly.

Fig. 1 The log-log (in base 10) plot of the CPU time (in seconds) versus the total number of time steps* N*.

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**References: **

[1] S. Jiang, J. Zhang, Q. Zhang and Z. Zhang, Commun. Comput. Phys., 21 (2017) 650-678.