Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
For researchers in fractional calculus, this provides efficient algorithms for long-time integration of tempered fractional problems, though it is an incremental extension of existing methods.
This work extends fractional linear multistep methods to tempered fractional operators and develops two fast methods (Fast Method I and II) with linear complexity for discrete convolution approximation. Fast Method II outperforms Fast Method I in accuracy, efficiency, and simplicity, achieving O(Q) memory and O(Q n_T) computational cost.
In this work, we extend the fractional linear multistep methods in [C. Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional integral and derivative operators in the sense that the tempered fractional derivative operator is interpreted in terms of the Hadamard finite-part integral. We develop two fast methods, Fast Method I and Fast Method II, with linear complexity to calculate the discrete convolution for the approximation of the (tempered) fractional operator. Fast Method I is based on a local approximation for the contour integral that represents the convolution weight. Fast Method II is based on a globally uniform approximation of the trapezoidal rule for the integral on the real line. Both methods are efficient, but numerical experimentation reveals that Fast Method II outperforms Fast Method I in terms of accuracy, efficiency, and coding simplicity. The memory requirement and computational cost of Fast Method II are $O(Q)$ and $O(Qn_T)$, respectively, where $n_T$ is the number of the final time steps and $Q$ is the number of quadrature points used in the trapezoidal rule. The effectiveness of the fast methods is verified through a series of numerical examples for long-time integration, including a numerical study of a fractional reaction-diffusion model.