A Unified FastMemory-Saving Time-SteppingMethod for Fractional Operators and Its Applications

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time,especially for long-time integration,which taxes computational resources heavily for high-dimensional problems.Here,we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constant-order fractional operators,and identify the current pitfalls of such methods.In order to overcome the pitfalls,an improved sum-of-exponentials is developed and verified.We also present several sumof-exponentials for the approximation of the kernel function in variable-order fractional operators.Subsequently,based on the sum-of-exponentials,we propose a unified framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable orders.We test the fast method based on several benchmark problems,including fractional initial value problems,the time-fractional Allen-Cahn equation in two and three spatial dimensions,and the Schr¨odinger equation with nonreflecting boundary conditions,demonstrating the efficiency and robustness of the proposed method.The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.

Efficient Numerical Computation of Time-Fractional Nonlinear Schrodinger Equations in Unbounded Domain

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded domain.We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization procedure.Then,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is stated.To accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative.The resulting algorithm is highly efficient for long time simulations.Finally,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.

Approximating the Gaussian as a Sum of Exponentials and its Applications to the Fast Gauss Transform

We develop efficient and accurate sum-of-exponential(SOE)approximations for the Gaussian using rational approximation of the exponential function on the negative real axis.Six digit accuracy can be obtained with eight terms and ten digit accuracy can be obtained with twelve terms.This representation is of potential interest in approximation theory but we focus here on its use in accelerating the fast Gauss transform(FGT)in one and two dimensions.The one-dimensional scheme is particularly straightforward and easy to implement,requiring only twenty-four lines of MATLAB code.The two-dimensional version requires some care with data structures,but is significantly more efficient than existing FGTs.Following a detailed presentation of the theoretical foundations,we demonstrate the performance of the fast transforms with several numerical experiments.