High-Order BDF Convolution Quadrature for Fractional Evolution Equations with Hyper-singular Source Term

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/IDm-BDFk method proposed by Shi and Chen (SIAM J Numer Anal 61:2559–2579, 2023) for the subdiffusion equation with a weakly singular source term. We prove that the kth-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.