Abstract
AbstractThe computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require totalmemory andwork, whereNTandNSrepresent 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 derivativeof orderα∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernelt–1–αon the interval [Δt,T] with a uniform absolute errorε. We give the theoretical analysis to show that the number of exponentialsNexpneeded is of orderforT≫1 orforTH1 for fixed accuracyε. The resulting algorithm requires onlystorage andwork 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.
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