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The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total 
memory and 
work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative 
of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t–1–α on the interval [Δt, T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials Nexp needed is of order 
for T≫1 or 
for TH1 for fixed accuracy ε. The resulting algorithm requires only 
storage and 
work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.
