An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative

TL;DR

This paper introduces an adaptive memory method for the Caputo fractional derivative that reduces computational memory requirements and improves accuracy by using non-uniform time distribution for convolution weights, demonstrated through sub-diffusion and viscoelastic simulations.

Contribution

A novel adaptive memory approach for the Caputo fractional derivative that enhances accuracy and efficiency by non-uniformly distributing past function evaluations.

Findings

Significantly reduces memory usage compared to previous methods.

Improves accuracy through convolution weights with non-uniform distribution.

Demonstrates superior performance in sub-diffusion and viscoelastic simulations.

Abstract

A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation and the creeping response of a fractional viscoelastic model are simulated to demonstrate the accuracy as well as the computational efficiency of the present method.