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

TL;DR

The paper tackles numerical solution of time-fractional differential equations (TFDEs) with Caputo derivatives by transforming them into an equivalent integer-order extended parametric differential equation (EPDE) via dimensional expansion. It then discretizes the EPDE using a high-order BDF-$k$ time-stepping scheme and Jacobi spectral collocation in the extended θ-direction, yielding a stable method with error $O(\Delta t^{k} + M^{-m})$ and, crucially, $O(N)$ overall computational cost with $O(1)$ storage for fixed spectral size $M$. The authors prove stability of the EPDE and derive rigorous error estimates, supported by numerical tests on linear and nonlinear problems that confirm spectral accuracy in θ and high-order convergence in time, with robust long-time performance. This approach delivers a high-order, memory-efficient framework for TFDEs and suggests applicability to a broader class of nonlocal operators and fractional PDEs.

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 $O(Δt^{k} + M^{-m})$, where $Δt$ 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

Figures (4)

• Figure 1: Contours of $\rho(\sigma)$ for the BDF-$3$ scheme \ref{['fully scheme']} for different values of fractional order $\alpha$. The blue region is the "stable region".
• Figure 2: Example \ref{['example1']}. Convergence results for the spectral approximation of the extended $\theta$ direction for $\alpha = 0.2,\, 0.8$ and different times $T$. Here we set $\Delta t$ to be small enough. We obtain spectral accuracy for all test cases.
• Figure 3: Example \ref{['example1']}. Convergence results for the BDF-$k$ ($k = 3,4, 5$) schemes for different values of fractional order $\alpha = 0.2,\, 0.8$ at $T=1$ or $T=20$. Here we set $M = 30$. The results are consistent with Theorem \ref{['Theorem:fully error estimate']}.
• Figure 4: Example \ref{['example2']}. (a): Convergence results for the spectral approximation of the extended $\theta$ direction for $\alpha = 0.3,\, 0.7$ and $T = 1,100$. (b)-(d): Convergence results for the BDF-$k$ ($k = 3,4, 5$) schemes for $\alpha = 0.3,\, 0.7$ and $T =1, 100$, here we set $M = 30$.

Theorems & Definitions (21)

• Theorem 1
• proof
• Lemma 1
• Theorem 2
• proof
• Theorem 3
• Remark 1
• Remark 2
• Lemma 2
• proof