High-order BDF convolution quadrature for fractional evolution equations with hyper-singular source term

TL;DR

This paper develops a high-order BDF convolution quadrature method with a smoothing technique for hyper-singular source terms in fractional evolution equations, restoring convergence rates and confirming effectiveness through numerical experiments.

Contribution

The authors introduce a robust smoothing approach using Hadamard finite-part integrals for hyper-singular sources, proving high-order convergence for fractional evolution equations.

Findings

Restores k-th order convergence for hyper-singular sources in fractional equations.

Provides theoretical proof and numerical validation of the smoothing method.

Extends applicability to cases with non-compatible initial data.

Abstract

Anomalous diffusion in the presence or absence of an external force field is often modelled in terms of the fractional evolution equations, which can involve the hyper-singular source term. For this case, conventional time stepping methods may exhibit a severe order reduction. Although a second-order numerical algorithm is provided for the subdiffusion model with a simple hyper-singular source term $t^{\mu}$, $-2<\mu<-1$ in [arXiv:2207.08447], the convergence analysis remain to be proved. To fill in these gaps, we present a simple and robust smoothing method for the hyper-singular source term, where the Hadamard finite-part integral is introduced. This method is based on the smoothing/ID$m$-BDF$k$ method proposed by the authors [Shi and Chen, SIAM J. Numer. Anal., to appear] for subdiffusion equation with a weakly singular source term. We prove that the $k$th-order convergence rate can be restored for the diffusion-wave case $\gamma \in (1,2)$ and sketch the proof for the subdiffusion case $\gamma \in (0,1)$, even if the source term is hyper-singular and the initial data is not compatible. Numerical experiments are provided to confirm the theoretical results.