A novel class of arbitrary high-order numerical schemes for fractional differential equations

TL;DR

This paper introduces a new high-order numerical scheme for solving time-fractional differential equations, combining stability, efficiency, and spectral accuracy, validated through various numerical examples.

Contribution

It establishes a novel equivalence between TFDEs and extended parametric differential equations, enabling high-order accurate schemes with low computational cost.

Findings

Achieves error order $O(\Delta t^{k} + M^{-m})$

Computational cost is $O(N)$ with minimal storage

Effective for both linear and nonlinear problems

Abstract

A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations (EPDE) by dimensional expanding, and establish the stability of EPDE. We apply BDF-$k$ formula for the temporal discretization, while we use the Jacobi spectral collocation method for the discretization of the extended direction. We analyze the stability of the proposed method and give rigorous error estimates with order $O(\Delta t^{k} + M^{-m})$, where $\Delta t$ and $M$ are time step size and number of collocation nodes in extended direction, respectively. Also, we point out that the computational cost and the storage requirement is essentially the same as the integer problems, namely, the computational cost and the storage of the present algorithm are $O(N)$ and $O(1)$, respectively, where $N$ is the total number of time step. We present several numerical examples, including both linear and nonlinear problems, to demonstrate the effectiveness of the proposed method and to validate the theoretical results